Linear Analysis on Manifolds

Home , Bundle metric, Elliptic complex, Fiber bundle, Lichnerowicz formula, Musical isomorphism

Pierre Albin Contents

Contents i

1 Differential operators on manifolds 3 1.1 Smooth manifolds ...... 3 1.2 Partitions of unity ...... 5 1.3 Tangent bundle ...... 5 1.4 Vector bundles ...... 8 1.5 Pull-back bundles ...... 9 1.6 The cotangent bundle ...... 11 1.7 Operations on vector bundles ...... 12 1.8 Sections of bundles ...... 14 1.9 Differential forms ...... 16 1.10 Orientations and integration ...... 18 1.11 Exterior derivative ...... 20 1.12 Differential operators ...... 21 1.13 Exercises ...... 23 1.14 Bibliography ...... 25

2 A brief introduction to Riemannian geometry 27 2.1 Riemannian metrics ...... 27 2.2 Parallel translation ...... 30 2.3 Geodesics ...... 34 2.4 Levi-Civita connection ...... 37

i ii CONTENTS

2.5 Associated bundles ...... 39 2.6 E-valued forms ...... 40 2.7 Curvature ...... 42 2.8 Complete manifolds of bounded geometry ...... 46 2.9 Riemannian volume and formal adjoint ...... 46 2.10 Hodge star and Hodge Laplacian ...... 48 2.11 Exercises ...... 51 2.12 Bibliography ...... 54

3 Laplace-type and Dirac-type operators 57 3.1 Principal symbol of a differential operator ...... 57 3.2 Laplace-type operators ...... 60 3.3 Dirac-type operators ...... 61 3.4 Clifford algebras ...... 64 3.5 Hermitian vector bundles ...... 69 3.6 Examples ...... 73 3.6.1 The Gauss-Bonnet operator ...... 73 3.6.2 The signature operator ...... 75 3.6.3 The ∂ operator ...... 76 3.6.4 The Dirac operator ...... 78 3.6.5 Twisted Clifford modules ...... 80 3.7 Supplement: Spin structures ...... 80 3.8 Exercises ...... 85 3.9 Bibliography ...... 88

4 Pseudodiﬀerential operators on Rm 89 4.1 Distributions ...... 89 4.2 Examples ...... 94 4.2.1 Fundamental solutions ...... 94 4.2.2 Principal values and ﬁnite parts ...... 95 CONTENTS iii

4.3 Tempered distributions ...... 96 4.3.1 Convolution ...... 96 4.3.2 Fourier transform ...... 98 4.4 Pseudodiﬀerential operators on Rm ...... 102 4.4.1 Symbols and oscillatory integrals ...... 102 4.4.2 Amplitudes and mapping properties ...... 108 4.5 Exercises ...... 116 4.6 Bibliography ...... 117

5 Pseudodiﬀerential operators on manifolds 119 5.1 Symbol calculus and parametrices ...... 119 5.2 Hodge decomposition ...... 123 5.3 The index of an elliptic operator ...... 127 5.4 Pseudodiﬀerential operators acting on L2 ...... 131 5.5 Sobolev spaces ...... 136 5.6 Exercises ...... 140 5.7 Bibliography ...... 144

6 Spectrum of the Laplacian 145 6.1 Spectrum and resolvent ...... 145 6.2 The heat kernel ...... 149 6.3 Weyl’s law ...... 156 6.4 Long-time behavior of the heat kernel ...... 158 6.5 Supplement: Determinant of the Laplacian ...... 159 6.6 Exercises ...... 161 6.7 Bibliography ...... 162

7 The Atiyah-Singer index theorem 163 7.1 Getzler rescaling ...... 163 7.2 Atiyah-Singer for Dirac-type operators ...... 171 7.3 Chern-Weil theory ...... 176 CONTENTS 1

7.4 Examples ...... 180 7.5 Atiyah-Singer for elliptic operators ...... 181 7.6 Exercises ...... 185 7.7 Bibliography ...... 187

Introduction

This text is a version of the lecture notes used during a course at the University of Illinois at Urbana-Champaign in Spring 2012 and a course at the University of Rome ‘La Sapienza’ in Summer 2014, as well as several reading courses. Many of the participants of the original course had attended a course on differentiable manifolds but did not have any experience with Riemannian geometry. The intention of the course was to introduce the techniques of geometric analysis on manifolds and establish the Atiyah-Singer index theorem. I learned these techniques from Rafe Mazzeo and Richard Melrose and indeed the construction of the heat kernel and proof of the local index theorem are taken from Melrose’s book on the Atiyah- Patodi-Singer index theorem. There are several textbooks that cover some of the same ground as these notes. Notably, John Roe, Elliptic operators, topology and asymptotic methods. Second edition. Pitman Research Notes in Mathematics Series, 395. Longman, Harlow, 1998., Nicole Berline, Ezra Getzler, Michele` Vergne, Heat kernels and Dirac operators. Corrected reprint of the 1992 original. Grundlehren Text Editions. Springer-Verlag, Berlin, 2004., H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, Spin geometry. Princeton Mathematical Series, 38. Princeton University Press, Princeton, NJ, 1989., David D. Bleecker and Bernhelm Booss-Bavnbek, Index Theory with Ap- plications to Mathematics and Physics. International Press, 2013. Beyond the feedback I received from the students taking the course, I received a helpful list of corrections from RaúlSánchez Galánand excellent suggestions from Chris Kottke whose own lecture notes on the subject can be found at his webpage http://ckottke.ncf.edu/neu/7376_sp16/. 2 CONTENTS Chapter 1

Diﬀerential operators on manifolds

1.1 Smooth manifolds

The m-dimensional Euclidean space Rm, the m-dimensional sphere Sm and the m- dimensional torus Tm = S1 × ... × S1 are all examples of smooth manifolds, the context where our studies will take place. A general manifold is a space constructed from open sets in Rm by patching them together smoothly. Formally, M is a manifold if it is a paracompact1 Hausdorﬀ topological space with an ‘atlas’, i.e., a covering by open sets {Vα} together with homeomorphisms

φα : Uα −→ Vα

m for some open subsets {Uα} of R . The integer m is known as the dimension of M and the maps φα : Uα −→ Vα (or sometimes just the sets {Vα}) are known as coordinate charts. An atlas is smooth if whenever Vαβ := Vα ∩ Vβ 6= ∅ the map

−1 −1 −1 (1.1) ψαβ := φβ ◦ φα : φα (Vαβ) −→ φβ (Vαβ)

−1 −1 m is a smooth diﬀeomorphism between the open sets φα (Vαβ), φβ (Vαβ) ⊆ R . (For ∞ k us smooth will always mean C .) If the maps ψαβ are all C -smooth, we say that the atlas is Ck.

If φα : Uα −→ Vα is a coordinate chart, it is typical to write

−1 (y1, . . . , ym) = φα : Vα −→ Uα and refer to the yi as local coordinates of M on Vα. 1 A topological space is paracompact if every open cover has an open reﬁnement in which each point is covered ﬁnitely many times.

3 4 CHAPTER 1. DIFFERENTIAL OPERATORS ON MANIFOLDS

Two smooth atlases are compatible (or equivalent) if their union is a smooth atlas. A smooth structure on M is an equivalence class of smooth atlases or, equivalently, a maximal smooth atlas on M. A manifold M together with a smooth structure is a smooth manifold. To indicate that a manifold M is m-dimensional we will sometimes write M m or, more often, we will refer to M as an m-manifold. A more general notion than manifold is that of a manifold with boundary. The former is deﬁned in the same way as a manifold, but the local model is

m m R− = {(x1, . . . , xn) ∈ R : x1 ≤ 0}. If M is a smooth manifold with boundary, its boundary ∂M consists of those points m in M that are in the image of ∂R− for some (and hence all) local coordinate charts. It is easy to check that ∂M is itself a smooth manifold (without boundary). The product of two manifolds with boundary is an example of a still more general notion, a manifold with corners. A map between two smooth manifolds (with or without boundary)

F : M −→ M 0 is smooth if it is smooth in local coordinates. That is, whenever

0 0 0 0 φα : Uα −→ Vα ⊆ M, φγ : Uγ −→ Vγ ⊆ M , are coordinate charts, the map

0 −1 0 (1.2) (φγ) ◦ F ◦ φα : Uα −→ Uγ is smooth (whenever it is defined). Notice that this definition is equivalent to making coordinate charts diffeomorphisms. The space of smooth maps between M and M 0 will be denoted C∞(M,M 0) and when M 0 = R, simply by C∞(M). Notice that a map F ∈ C∞(M,M 0) naturally induces a pull-back map

F ∗ : C∞(M 0) −→ C∞(M),F ∗(h)(p) = h(F (p)) for all h ∈ C∞(M 0), p ∈ M.

For us a manifold will mean a connected smooth manifold without boundary, unless otherwise speciﬁed. Similarly, unless we specify otherwise, a map between manifolds will be smooth.

A subset X ⊆ M is a submanifold of M of dimension k if among the m coordinates on M at each point of X there are k that restrict to give coordinates of 1.2. PARTITIONS OF UNITY 5

X. That is, at each point p ∈ X there is a coordinate chart of M, φ : U −→ V with q ∈ V, whose restriction to U ∩ Rk × {0} maps into X, φ : U ∩ k × {0} −→ V ∩ X. U∩Rk×{0} R These restricted charts then deﬁne a smooth k-manifold structure on X.

1.2 Partitions of unity

A manifold is deﬁned by patching together many copies of Rm. It is often useful to take a construction on Rm and transfer it to a manifold by carrying it out on each patch and then adding the pieces together.

For these constructions, it is very convenient that every open covering Vα of a smooth manifold carries a partition of unity. That is, a collection ωi of smooth functions ωi : M −→ R such that: i) At each p ∈ M only finitely many ωi(p) are non-zero ii) Each ωi is supported inside some Vα iii) The sum of the ωi is the constant function equal to one at each point p ∈ M. One remarkable fact that follows easily from the existence of partitions of unity is that any smooth manifold can be identified with a smooth submanifold of RN for some N. This is easy to show using a locally finite atlas with a corresponding partition of unity. Whitney’s embedding theorem shows that any smooth manifold M embeds into R2 dim M+1 and a compact smooth manifold into R2 dim M .

1.3 Tangent bundle

We know what a smooth function f : M −→ R is, what is its derivative? If γ is a smooth curve on M, that is, a smooth map γ : R −→ M, then f ◦ γ is a map from R to itself so its derivative is unambiguously deﬁned. We can take advantage of this to deﬁne the derivative of f at a point p ∈ M in terms of the curves through that point.

Indeed, let us say that two curves γ1 and γ2 are equivalent at p if γ1(0) = γ (0) = p and 2 ∂ −1 ∂ −1 (φ ◦ γ1) = (φ ◦ γ2) ∂t t=0 ∂t t=0 6 CHAPTER 1. DIFFERENTIAL OPERATORS ON MANIFOLDS for some (and hence any) coordinate chart φ : U −→ V ⊆ M containing p. It is easy to see that this is an equivalence relation and we refer to an equivalence class as a tangent vector at p ∈ M. The tangent space to M at p is the set of all tangent vectors at p and is denoted TpM. For the case of Euclidean space, the tangent space to Rm at any point is another copy of Rm, m m TpR = R . The tangent space of the m-sphere is also easy to describe. By viewing Sm as the unit vectors in Rm+1, curves on Sm are then also curves in Rm+1 so we have

m m+1 m+1 TpS ⊆ TpR = R , and a tangent vector at p will correspond to a curve on Sm precisely when it is orthogonal to p as a vector in Rm+1, thus

m m+1 TpS = {v ∈ R : v · p = 0}.

Clearly if γ1 is equivalent to γ2 at p then

∂ ∂ (f ◦ γ1) = (f ◦ γ2) ∂t t=0 ∂t t=0 for all f : M −→ R, so we can think of the derivative of f at p as a map from equivalence classes of curves through p to equivalence classes of curves through f(p), thus Dpf : TpM −→ Tf(p)R.

More generally, any smooth map F : M1 −→ M2 deﬁnes a map, known as its diﬀerential,

DpF : TpM1 −→ TF (p)M2 by sending the equivalence class of a curve γ on M1 to the equivalence class of the curve F ◦γ on M2. When the point p is understood, we will omit it from the notation. If the differential of a map is injective we say that the map is an immersion, if it is surjective we say that it is a submersion. If F : M1 −→ M2 is an injective immersion which is a homeomorphism between M1 and F (M1) then we say that F is an embedding. A connected subset X ⊆ M is a submanifold of M precisely when the inclusion map X,→ M is an embedding. If F is both an immersion and a submersion, i.e., if DF is always a linear isomorphism, we say that it is a local diffeomorphism. By the inverse function theorem, if F is a local diffeomorphism then every point p ∈ M1 has a neighborhood 1.3. TANGENT BUNDLE 7

V ⊆ M1 such that F V is a diffeomorphism onto its image. A diffeomorphism is a bijective local diffeomorphism. The chain rule is the fact that commutative diagrams

F1 M1 / M2

F2 F ◦F 2 1 ! M3 of smooth maps between manifolds induce, for each p ∈ M1, commutative diagrams between the corresponding tangent spaces,

DF1 TpM1 / TF1(p)M2

DF2 D(F ◦F ) 2 1 & TF2(F1(p))M3 i.e., that D(F2 ◦ F1) = DF2 ◦ DF1. Taking diﬀerentials of coordinate charts of M, we end up with coordinate charts on [ TM = TpM p∈M showing that TM, the tangent bundle of M, is a smooth manifold of twice the dimension of M. For Rm the tangent bundle is simply the product

m m m T R = R × R where the ﬁrst factor is thought of as a point in Rm and the second as a vector ‘based’ at that point; the tangent bundle of an open set in Rm has a very similar description, we merely restrict the ﬁrst factor to that open set. (We can think of M 7→ TM together with F 7→ DF as a covariant functor on smooth manifolds.)

If φ : U −→ V is a coordinate chart on M and p ∈ V ⊆ M, then Dpφ identifies m TpM with R . A different choice of coordinate chart gives a different identification m of TpM with R , but these identifications differ by an invertible linear map. This means that each TpM is naturally a vector space of dimension m, which however does not have a natural basis. Now that we know the smooth structure of TM, note that the differential of a smooth map between manifolds is a smooth map between their tangent bundles. There is also a natural smooth projection map

π : TM −→ M 8 CHAPTER 1. DIFFERENTIAL OPERATORS ON MANIFOLDS

m that sends elements of TpM to p and when M = R is simply the projection onto the ﬁrst factor. It is typical, for any subset Z ⊆ M, to use the notations

−1 π (Z) =: TM Z =: TZ M.

1.4 Vector bundles

A coordinate chart φ : U −→ V on a smooth m-manifold M induces a coordinate chart on TM and an identiﬁcation

−1 −1 Dφ m φ×id m (1.3) π (V) −−−−→ T U = U × R −−−−→V × R which restricts to each fiber π−1(p) to a linear isomorphism. We say that a coordinate chart of M locally trivializes TM as a ‘vector bundle’. A vector bundle over a manifold M is a locally trivial collection of vector spaces, one for each point in M. Formally a rank k real vector bundle consists of a smooth manifold E together with a smooth surjective map π : E −→ M such that the fibers π−1(p) form vector spaces over R and satisfying a local triviality condition: for any point p ∈ M and small enough neighborhood V of p there is a diffeomorphism k −1 Φ: V × R −→ π (V) whose restriction to any q ∈ V is a linear isomorphism

k −1 Φ q∈V : {q} × R −→ π (q). We refer to the pair (V, Φ) as a local trivialization of E. We will often refer to a rank k real vector bundle as a Rk-bundle. The discussion below of Rk-vector bundles applies equally well to complex vector bundles, or Ck-vector bundles, which are the deﬁned by systematically replacing Rk with Ck above. If X is a submanifold of M, and E −−→π M is a vector bundle, then

π|X E X −−−→ X is a vector bundle. A subset of a vector bundle is a subbundle if it is itself a bundle in a consistent fashion. That is, E1 is a subbundle of E2 if E1 and E2 are both bundles over M, E1 ⊆ E2, and the projection map E1 −→ M is the restriction of the projection map E2 −→ M. 1.5. PULL-BACK BUNDLES 9

A bundle morphism between two vector bundles

π1 π2 E1 −−→ M1,E2 −−→ M2 is a map F : E1 −→ E2 that sends fibers of π1 to fibers of π2 and acts linearly between fibers. Thus for any bundle morphism F there is a map F : M1 −→ M2 such that the diagram F (1.4) E1 / E2

π1 π2 F M1 / M2 commutes; we say that F covers the map F. We denote the restriction of F to the

ﬁber over p ∈ M1 by Fp :(E1)p −→ (E2)F (p). An invertible bundle morphism is called a bundle isomorphism. The trivial Rk-bundle over a manifold M is the manifold Rk × M with projection the map onto the second factor. We will use Rk to denote the trivial Rk-bundle. A bundle is trivial if it is isomorphic to Rk for some k. A manifold is said to be parallelizable if its tangent bundle is a trivial bundle, e.g., Rm is parallelizable. We will show later that Sm is not parallelizable when m is even. The circle S1 is parallelizable and hence (by exercise 1) so are all tori Tm. Some interesting facts which we will not show is that any three dimensional manifold is parallelizable and that the only parallelizable spheres are S1, S3 and S7. The fact that these sphere are parallelizable is easy to see by viewing them as the unit vectors in the complex numbers, the quaternions, and the octionians, respectively; the fact that these are the only parallelizable ones is not trivial.

1.5 Pull-back bundles

Any bundle morphism covers a map between manifolds (1.4). There is a sort of k converse to this: given a map between manifolds f : M1 −→ M2, and a R -bundle π E −−→ M2 the set ∗ f E = {(p, v) ⊆ M1 × E : f(p) = π(v)} ﬁts into the commutative diagram

fb f ∗E / E

π πb f M1 / M2 10 CHAPTER 1. DIFFERENTIAL OPERATORS ON MANIFOLDS where, for any (p, v) ∈ f ∗E,

π fb p o b (p, v) / v.

Moreover, if (U, Φ) is a local trivialization of E and V is a coordinate chart in M1 such that f(V) ⊆ U then

∗ −1 −1 k f E ∩ (V × π (U)) = {(p, (Φ f(p)) (x)) : p ∈ V, x ∈ R } and we can use this to introduce a C∞ manifold structure on f ∗E such that f ∗E ∩ −1 k ∗ πb k (V × π (U)) is diffeomorphic to V × R making f E −−→ M1 an R -bundle. The bundle f ∗E is called the pull-back of E along f. The simplest example of a pull-back bundle is the restriction of a bundle. If π π|X X is a submanifold of M, and E −−→ M is a vector bundle, then E X −−−→ X is (isomorphic to) the pull-back of E along the inclusion map X,→ M. We see directly from the definition that the pull-back bundle of a trivial bundle is again trivial, but note that the converse is not true. Given a smooth map f : X −→ Y, pull-back along f defines a functor between bundles on Y and bundles on X. Indeed, if E1 −→ Y and E2 −→ Y are bundles over Y and F : E1 −→ E2 is a bundle map between them then

∗ ∗ ∗ ∗ ∗ f F : f E1 −→ f E2, f F (p, v) = (p, F (p)v) for all (p, v) ∈ f E1

∗ ∗ is a bundle map between f E1 and f E2. Another useful relation between pull-backs and bundle maps is the following: π1 π2 If E1 −−→ M1 and E2 −−→ M2 are vector bundles and F : E1 −→ E2 is a bundle map so that F E1 / E2

π1 π2 F M1 / M2 ∗ then F is equivalent to a bundle map between E1 −→ M1 and F E2 −→ M1, namely ∗ E1 3 v 7→ (π1(v),F (v)v) ∈ F E2.

We will show later (Remark 3 on page 33) that if f : M1 −→ M2 and g : M1 −→ M2 are homotopic smooth maps, i.e., if there is a smooth map

H : [0, 1] × M1 −→ M2 s.t. H(0, ·) = f(·) and H(1, ·) = g(·) ∗ ∼ ∗ then f E = g E for any vector bundle E −→ M2. In particular, this implies that all vector bundles over a contractible space are trivial. 1.6. THE COTANGENT BUNDLE 11 1.6 The cotangent bundle

Every real vector space V has a dual vector space

∗ V = Hom(V, R) consisting of all the linear maps from V to R. Given a linear map between vector spaces T : V −→ W there is a dual map between the dual spaces

T ∗ : W ∗ −→ V ∗,T ∗(ω)(v) = ω(T (v)) for all ω ∈ W ∗, v ∈ V.

By taking the dual maps of the local trivializations of a vector bundle E −−→π M, the manifold E∗ = {(p, ω): p ∈ M, ω ∈ (π−1(p))∗} with the obvious projection E∗ −→ M is seen to be a vector bundle, the dual bundle to E. Every vector v ∈ Rk deﬁnes a dual vector v∗ ∈ (Rk)∗,

∗ k ∗ v : R −→ R, v (w) = v · w, and it is easy to see that the dual vectors of a basis of Rk form a basis of (Rk)∗ so that in fact v 7→ v∗ is a vector space isomorphism. In particular the dual bundle of an Rk-bundle is again a Rk-bundle.

Given vector bundles E1 −→ M1 and E2 −→ M2 and bundle map F : E1 −→ E2 covering a diﬀeomorphism F : M1 −→ M2, we get a dual bundle map

∗ ∗ ∗ ∗ −1 ∗ F : E2 −→ E1 ,F (q, θ) = (F (q), (Fp) θ).

The dual bundle to the tangent bundle is the cotangent bundle

T ∗M −→ M.

Remark 1. The assignation M 7→ T ∗M is natural, but is not a functor in the same way that M 7→ TM is. Rather, a map F : M1 −→ M2 between manifolds induces a map DF : TM1 −→ TM2 between their tangent bundles which, as explained above, ∗ we can interpret as a map DF : TM1 −→ F (TM2) between bundles over M1. Now ∗ ∗ ∗ ∗ we can take duals and we get a bundle map (DF ) : F (TM2 ) −→ TM1 . 12 CHAPTER 1. DIFFERENTIAL OPERATORS ON MANIFOLDS

Thus we can view M 7→ T ∗M as a functor but its target category2 should consist of vector bundles where a morphism between E1 −→ M1 and E2 −→ M2 consists ∗ of a map F : M1 −→ M2 and a bundle map Φ : F E2 −→ E1 and composition is

∗ (F1, Φ1) ◦ (F2, Φ2) = (F1 ◦ F2, Φ1 ◦ (F1 Φ2)).

1.7 Operations on vector bundles

We constructed the vector bundle E∗ −→ M from the vector bundle E −→ M by applying the vector space operation V 7→ V ∗ fiber-by-fiber. This works more generally whenever we have a smooth functor of vector spaces. Let Vect denote the category of finite dimensional vector spaces with linear isomorphisms, and let T be a (covariant) functor on Vect. We say that T is smooth if for all V and W in Vect the map

Hom(V,W ) −→ Hom(T (V ), T (W )) is smooth. Let E −−→π M be a vector bundle over M, deﬁne

−1 G Tp(E) = T (π (p)), T (E) = Tp(E) p∈M and let T (π): T (E) −→ M be the map that sends Tp(E) to p. We claim that T (π) T (E) −−−−→ M is a smooth vector bundle. Applying T ﬁbrewise to a local trivialization of E,

k −1 Φ: V × R −→ π (V), we obtain a map k −1 T (Φ) : V × T (R ) −→ T (π) (V), and we can use these maps as coordinate charts on T (E) to obtain the structure of a smooth manifold. Indeed, we only need to see that if two putative charts overlap then the map

−1 k T (Φ1) T (Φ2) k (V1 ∩ V2) × T (R ) −−−−−−−−−−→ (V1 ∩ V2) × T (R ) is smooth, and this follows from the smoothness of T .

2This is called the category of star bundles in Ivan Kola´rˇ, Peter Michor, Jan Slovak´ , Natural operations in diﬀerential geometry. Springer-Verlag, Berlin, 1993. 1.7. OPERATIONS ON VECTOR BUNDLES 13

Thus T (E) is a smooth manifold, T (π) is a smooth function, and

T (π) T (E) −−−−→ M satisﬁes the local triviality condition, so we have a new vector bundle over M. The same procedure works for contravariant functors (meaning that Hom(V,W ) is sent to Hom(T (W ), T (V ))) as well as functors with several variables. Therefore, for instance, we can deﬁne for vector bundles E −→ M and F −→ M the vector bundles:

i) Hom(E,F ) −→ M with ﬁber over p, Hom(Ep,Fp).

ii) E ⊗ F −→ M with ﬁber over p, Ep ⊗ Fp.

iii) S2E −→ M whose ﬁber over p is the space of all symmetric bilinear transformations from Ep × Ep to R.

iv) SkE −→ M whose ﬁber over p is the space of all symmetric k-linear transfor- k mations from Ep to R.

v) ΛkE −→ M whose ﬁber over p is the space of all antisymmetric k-linear k transformations from Ep to R.

We also obtain natural isomorphisms of vector bundles from isomorphisms of vector spaces. Thus, for instance, for vector bundles E,F, and G over M, we have3

i) E ⊕ F ∼= F ⊕ E ∼ ii) E ⊗ R = E

iii) E ⊗ F ∼= F ⊗ E

iv) E ⊗ (F ⊕ G) ∼= (E ⊗ F ) ⊕ (E ⊗ G)

∗ ∗ ∼ v) Hom(E,F ) = E ⊗ F (so E = Hom(E, R)) M vi) Λk(E ⊕ F ) ∼= (ΛiE ⊗ ΛjF ) i+j=k

vii) (ΛkE)∗ = Λk(E∗)

3Recall that R is the trivial R-bundle, R × M −→ M. 14 CHAPTER 1. DIFFERENTIAL OPERATORS ON MANIFOLDS 1.8 Sections of bundles

A section of a vector bundle E −−→π M is a smooth map s : M −→ E such that

E > s π

id M / M commutes. We will denote the space of smooth sections of E by C∞(M; E). This forms a vector space over R and, as such, it is inﬁnite-dimensional (unless we are in a ‘trivial’ situation where (dim M)(rank E) = 0). However it is also a module over C∞(M) (with respect to pointwise multiplication) and here it is ﬁnitely generated. For instance, the sections of a trivial bundle form a free C∞(M) module,

∞ n ∞ n C (M; R ) = (C (M)) .

We also introduce notation for functions and sections with compact support,

∞ ∞ Cc (M), Cc (M; E) and point out that the latter is a module over the former. One consequence of the module structure is that we can use a partition of unity on M to decompose a section into a sum of sections supported on trivializations of E. A section of the tangent bundle is known as a vector field. Thus a vector field on M is a smooth function that assigns to each point p ∈ M a tangent vector ∞ in TpM. A vector field V ∈ C (M; TM) naturally acts on a smooth function by differentiation. Indeed, let us write V (p) = [γp] with [γp] an equivalence class of paths representing V (p), then we have

∞ ∞ ∞ V : C (M) −→ C (M), (V f)(p) = ∂t t=0(f ◦ γp)(t) for all f ∈ C (M), p ∈ M.

If Vα is a coordinate chart on M with local coordinates y1, . . . , ym then we can write

m X ∂ V = ai(y1, . . . , ym) . Vα ∂y i=1 i Indeed, this is just the representation in the induced coordinates of TM with the identiﬁcation (1.3). This action of V on C∞(M) is linear and satisﬁes

V (fh) = V (f)h + fV (h), for all f, h ∈ C∞(M), 1.8. SECTIONS OF BUNDLES 15 i.e., it is a derivation of C∞(M). This derivation will vanish on all of C∞(M) only if the vector field is identically zero, so we can identify vector fields with a subset of the space of derivations of C∞(M). In fact, every derivation of C∞(M) is given by the action of a vector field. Indeed if we are working on Rn, we can write any smooth function f in terms of a fixed y ∈ R as X f(x) = f(y) + (xi − yi)gi(x), with gi smooth, gi(y) = ∂if(y) hence if D is a derivation on C∞(Rn) its value on f is X X Df(x) = D(xi)gi(x) + (xi − yi)Dgi(x) and taking the limit as x approaches y we get X Df(y) = D(xi)∂if(y).

Since y is arbiratry, this shows that the derivation D is given by the vector field P D(xi)∂i. For general M, we localize via a partition of unity and apply this argument in a coordinate chart. Thus we have an equivalence between vector fields on M and derivations on C∞(M). This is the ‘linearization’ of a similar fact: there is an equivalence between automorphisms of C∞(M) and diffeomorphisms of M.

An advantage of thinking of vector ﬁelds as operators on functions is that it makes sense to compose two vector ﬁelds

(VW )(f) = V (W (f)) : C∞(M) −→ C∞(M) the result is no longer a vector field, but is an example of a differential operator. However, if we form the commutator of two vector fields,

[V,W ]f = V (W (f)) − W (V (f)) then the result is again a vector field! In fact it is easy to check that this is a derivation on C∞(M). Moreover the commutator of vector fields satisfies the ‘Jacobi identity’

[[U, V ],W ] + [[V,W ],U] + [[W, U],V ] = 0 and hence the commutator, combined with the R-vector space structure, gives C∞(M,TM) the structure of a Lie algebra. 16 CHAPTER 1. DIFFERENTIAL OPERATORS ON MANIFOLDS

Remark 2. Whereas a smooth map F : M1 −→ M2 induces a map DF : TM1 −→ TM2, it is not always possible to ‘push-forward’ a vector ﬁeld. We say that two vector ∞ ∞ ﬁelds V ∈ C (M1; TM1), and W ∈ C (M2; TM2) are F -related if the diagram

DF TM1 / TM2 O O V W F M1 / M2 commutes, i.e., if DF ◦ V = W ◦ F. If F is a diffeomorphism, then we can push-forward a vector field: the vector ∞ −1 ∞ fields V ∈ C (M1; TM1) and F∗V = DF ◦ V ◦ F ∈ C (M2; TM2) are always ∞ F -related. Moreover, F∗[V1,V2] = [F∗V1,F∗V2] for all V1,V2 ∈ C (M1; TM1), so M 7→ C∞(M; TM) is a functor from smooth manifolds with diffeomorphisms to Lie algebras and isomorphisms.

1.9 Diﬀerential forms

Sections of the cotangent bundle of M,T ∗M, are known as diﬀerential 1-forms. More generally, sections of the k-th exterior power of the cotangent bundle, ΛkT ∗M, are known as diﬀerential k-forms. The typical notation for this space is

dim M M Ωk(M) = C∞(M;ΛkT ∗M), Ω∗(M) = Ωk(M) k=0 where by convention Ω0(M) = C∞(M). A differential form in Ωk(M) is said to have degree k. A differential k-form ω ∈ Ωk(M) can be evaluated on k-tuples of vector fields,

∞ k ω : C (M; TM) −→ R, it is linear in each variable and has the property that

ω(Vσ(1),...,Vσ(k)) = sign(σ)ω(V1,...,Vk) for any permutation σ of {1, . . . , k}. Diﬀerential forms of all degrees Ω∗(M) form a graded algebra with respect to the wedge product Ωk(M) × Ω`(M) 3 (ω, η) 7→ ω ∧ η ∈ Ωk+`(M) 1 X ω ∧ η(V ,...,V ) = sign(σ)ω(V ,...,V )η(V ,...,V ). 1 k+` k!`! σ(1) σ(k) σ(k+1) σ(k+`) σ 1.9. DIFFERENTIAL FORMS 17

This product is graded commutative (so sometimes commutative and sometimes anti-commutative)

ω ∈ Ωk(M), η ∈ Ω`(M) =⇒ ω ∧ η = (−1)k`η ∧ ω.

In particular if ω ∈ Ωk(M) with k odd, then ω ∧ ω = 0. Given a diﬀerential k-form, ω, we will write the operator ‘wedge product with ω,’ by ∗ ∗+k ∗ eω :Ω (M) −→ Ω (M), eω(η) = ω ∧ η, for all η ∈ Ω (M).

We say that a k-form is elementary if it is the wedge product of k 1-forms. Elementary k-forms are easy to evaluate

(ω1 ∧ · · · ∧ ωk)(V1,...,Vk) = det(ωi(Vj)).

Let p be a point in a local coordinate chart V on M with coordinates y1, . . . , ym. A ∗ basis of Tp M dual to the basis {∂y1 , . . . , ∂ym } of TpM is given by

1 m i i {dy , . . . , dy } where dy (∂yj ) = δj.

∗ A basis for the space of k-forms in Tp M is given by

i1 ik {dy ∧ · · · ∧ dy : i1 < ··· < ik}.

Any k-form supported in V can be written as a sum of elementary k-forms. Using a partition of unity we can write any k-form on M as a sum of elementary k-forms. Notice that anticommutativity of the wedge product implies that all elementary k- forms, and hence all k-forms, of degree greater than the dimension of the manifold vanish. There is also an interior product of a vector field and a differential form, which is simply the contraction of the differential form by the vector field. For every W ∈ C∞(M; TM), the interior product is a map

∗ ∗−1 iW :Ω (M) −→ Ω (M) k ∞ (iW ω)(V1,...,Vk−1) = ω(W, V1,...,Vk−1), for all ω ∈ Ω , {Vi} ⊆ C (M; TM). We will also use the notation iW ω = W y ω. Notice that antisymmetry of diﬀerential forms induces anticommutativity of the interior product in that iV iW = −iW iV ∞ 2 for any V,W ∈ C (M; TM). In particular, (iV ) = 0. 18 CHAPTER 1. DIFFERENTIAL OPERATORS ON MANIFOLDS

If F : M1 −→ M2 is a smooth map between manifolds, we can pull-back diﬀerential forms from M2 to M1,

∗ k k F :Ω (M2) −→ Ω (M1), ∗ k ∞ F ω(V1,...,Vk) = ω(DF (V1),...,DF (Vk)) for all ω ∈ Ω (M2),Vi ∈ C (M; TM1).

Pull-back respects the wedge product,

(1.5) F ∗(ω ∧ η) = F ∗ω ∧ F ∗η, and the interior product,

∗ ∗ F (DF (W ) y ω) = W y F (ω).

In the particular case of Rm, the pull-back of an m-form by a linear map T is (1.6) T ∗(dx1 ∧ · · · ∧ dxm) = (det T )(dx1 ∧ · · · ∧ dxm), directly from the deﬁnition of the determinant.

1.10 Orientations and integration

If W is an m-dimensional vector space, and (u1, . . . , um), (v1, . . . , vm) are ordered bases of W, we will say that they represent the same orientation if the linear map m that sends ui to vi has positive determinant. Equivalently, since Λ W is a one- dimensional vector space, we know that u1 ∧ · · · ∧ um and v1 ∧ · · · ∧ vm differ by multiplication by a non-zero real number and we say that they represent the same orientation if this number is positive. A representative ordered basis or non-zero m-form is said to be an orientation of W. The canonical orientation of Euclidean m space R corresponds to the canonical ordered basis, (e1, . . . , em). An Rk-vector bundle E −→ M is orientable if we can find a smooth orientation of its fibers, i.e., if there is a nowhere vanishing section of ΛkE −→ M. Notice that the latter is an R-bundle over M, so the existence of a nowhere vanishing section is equivalent to its being trivial. Thus E −→ M is orientable if and only if ΛkE −→ M is a trivial bundle. An orientation is a choice of trivializing section, and a bundle together with an orientation is said to be oriented. Note that, since Λk(E∗) = (ΛkE)∗,E is orientable if and only if E∗ is orientable. An m-manifold M is orientable if and only if its tangent bundle is orientable. Equivalently, a manifold is orientable when its cotangent bundle is orientable, so when there is a nowhere vanishing m-form. A choice of nowhere vanishing m-form 1.10. ORIENTATIONS AND INTEGRATION 19 is called an orientation of M or also a volume form on M. A coordinate chart φ : U −→ V is oriented if it pulls-back the orientation of M to the canonical orientation of Rm. A manifold is orientable precisely when it has an atlas of oriented coordinate charts. If M is oriented, there is a unique linear map, the integral, Z ∞ m : Cc (M;Λ M) −→ R M which is invariant under orientation-preserving diffeomorphisms and coincides in local coordinates with the Lebesgue integral. That is, if ω ∈ ΩmM is supported in a oriented coordinate chart φ : U −→ V then φ∗ω ∈ Ωm(Rm) and so we can take its Lebesgue integral. The assertion is that the result is independent of the choice of oriented coordinate chart, and the proof follows from (1.6) and the usual change of variable formula for the Lebesgue integral. By using a partition of unity, integration of any m-form reduces to integration of m-forms supported in oriented coordinate charts. It is convenient to formally extend the integral to a functional on forms of all degrees, by setting it equal to zero on all forms of less than maximal degree. What happens when a manifold is not orientable? At each point p ∈ M, the set of equivalence classes of orientations of TM, Op, has two elements, say o1(p) and o2(p). We define the orientable double cover of M to be the set

Or(M) = {(p, oi(p)) : p ∈ M, oi(p) ∈ Op}.

The manifold structure on M easily extends to show that Mf is a smooth manifold. Indeed, given a coordinate chart φ : U −→ V of M we set, for each q ∈ U,

−1 ∗ φe(q) = (φ(q), [(φ ) (e1 ∧ · · · ∧ em)]) ∈ Or(M) and we get a coordinate chart on M.f The mapping π : Mf −→ M that sends (p, oi(p)) to p is smooth and surjective. Moreover if V is a coordinate neighborhood in M −1 then π (V) = V1 ∪ V2 is the union of two disjoint open sets in Or(M) and π restricted to each Vi is a diﬀeomorphism onto V, hence the term ‘double cover’. It is easy to see that Or(M) is always orientable (at each point (p, oi(p)) choose the orientation corresponding to oi(p)). If M is connected and not orientable then Or(M) is connected and orientable, and we can often pass to Or(M) if we need an orientation. If M is orientable, then the same construction of Or(M) yields a disconnected manifold consisting of two copies of M. Orientations are one place where the theory of complex vector bundles dif- fers from that of general real vector bundles. Complex vector bundles are always orientable. 20 CHAPTER 1. DIFFERENTIAL OPERATORS ON MANIFOLDS

Indeed, ﬁrst note that a C-vector space W of complex dimension k has a canonical orientation as a R-vector space of real dimension 2k. We just take any basis of W over C, say b = {v1, . . . , vk}, and then take b = {v1, iv1, v2, iv2, . . . , vk, ivk} as 0 a basis for W over R. The claim is that if b = {w1, . . . , wk} is any other basis of W 0 over C, then b = {w1, iw1, . . . , wk, iwk} gives us the same orientation for W as a R- vector space, i.e., the change of basis matrix Bb from b to b0 has positive determinant. The proof is just to note that, if B is the complex change of basis matrix from b to b0 then det Bb = (det B)(det B) > 0. As the orientation is canonical, every Ck-vector bundle is orientable as a R2k-vector bundle.

1.11 Exterior derivative

If f is a smooth function on M, its exterior derivative is a 1-form df ∈ C∞(M; T ∗M) deﬁned by df(V ) = V (f), for all V ∈ C∞(M; TM). If ω ∈ C∞(M; T ∗M), its exterior derivative is a 2-form dω ∈ C∞(M;Λ2T ∗M) deﬁned by

dω(V,W ) = V (ω(W )) − W (ω(V )) − ω([V,W ]), for all V,W ∈ C∞(M; TM).

In the same way, if ω ∈ Ωk(M), we deﬁne its exterior derivative to be the (k+1)-form dω ∈ Ωk+1(M) deﬁned by

k X i dω(V0,...,Vk) = (−1) Vi(ω(V0,..., Vbi,...,Vp)) i=0 X i+j + (−1) ω([Vi,Vj],V0,..., Vbi,..., Vbj,...,Vp) i

The exterior derivative thus deﬁnes a graded map of degree +1,

d :Ω∗(M) −→ Ω∗(M).

It has the property that d2 = 0. It also distributes over the wedge product respecting the grading, in that

(1.7) ω ∈ Ωk(M), η ∈ Ω`(M) =⇒ d(ω ∧ η) = (dω) ∧ η + (−1)kω ∧ dη. 1.12. DIFFERENTIAL OPERATORS 21

The exterior derivative commutes with pull-back along a smooth map F : ∞ M1 −→ M2. Indeed, note that for a function on M2, f ∈ C (M2), this follows from the chain rule d(F ∗f)(V ) = d(f ◦F )(V ) = D(f ◦F )(V ) = Df(DF (V )) = df(DF (V )) = F ∗(df)(V ) and, using this, it follows for forms of arbitrary degree from (1.5) and (1.7). A k-form ω is called closed if dω = 0 and exact if there is a (k − 1)-form η such that dη = ω. The space of k-forms is an infinite dimensional vector space over R, as are the spaces of closed and exact forms. However the quotient vector space closed k-forms ker d :Ωk(M) −→ Ωk+1(M) Hk (M) = = , dR exact k-forms im (d :Ωk−1(M) −→ Ωk(M)) known as the k-th de Rham cohomology group of M, is finite dimensional for all k. The de Rham theorem identifies these vector spaces with the topological cohomology groups of M with R coefficients, which is one way we know that they are finite dimensional. These groups are one of the main objects of study in this course. We will come back to them after we build up more tools and in particular give an analytic proof of their finite dimensionality. If M is an oriented manifold with boundary, Stokes’ theorem is the fact that Z Z m−1 dω = ω, for all ω ∈ Ωc (M). M ∂M This compact statement generalizes the fundamental theorem of calculus, Green’s theorem, and Gauss’ theorem. If M does not have boundary, then Stokes’ theorem says that Z m−1 dω = 0, for all ω ∈ Ωc (M). M

1.12 Diﬀerential operators

It is clear what we mean by a diﬀerential operator of order n on Rm, namely a polynomial of degree n in the variables ∂y1 , ∂y2 , . . . , ∂ym ,

X j1 j2 jm (1.8) aj1,...,jm (y1, . . . , ym)∂y1 ∂y2 ··· ∂ym . j1,j2,...,jm j1+...+jm≤n

(The degree of this operator is the largest value of j1 + j2 + ... + jm for which aj1,...,jm is not zero, which we are assuming is n.) This operator is smooth when the 22 CHAPTER 1. DIFFERENTIAL OPERATORS ON MANIFOLDS

coefficient functions aj1,...,jm (y1, . . . , ym) are all smooth. The set of all differential n operators of order at most n is denoted Diff (Rm). This is a good moment to introduce multi-index notation. Let us agree, for all vectors α = (j1, . . . , jm), on the equivalence of the symbols

α j1 j2 jm ∂y = ∂y1 ∂y2 ··· ∂ym , and let us use |α| to denote j1 + ... + jm. Then we can write the expression above more succinctly as

n m X α P ∈ Diﬀ (R ) ⇐⇒ P = aα(y)∂y . |α|≤n

On a manifold, a diﬀerential operator of order at most n is a linear map

P : C∞(M) −→ C∞(M) that in any choice of local coordinates has the form (1.8). For instance, a vector ﬁeld V induces a linear map (a derivation) on smooth functions

V : C∞(M) −→ C∞(M) and this is exactly what we mean by a first order differential operator on M. In fact, a better description of differential operators of order at most n on M is to take the span of powers of C∞(M; TM),

Diffn(M) = span(C∞(M; TM))j 0≤j≤n where we agree that C∞(M; TM)0 = C∞(M). The exterior derivative is also a first order differential operator, but now between sections of vector bundles over M, e.g., for each k,

d : C∞(M;ΛkT ∗M) −→ C∞(M;Λk+1T ∗M).

In general, a diﬀerential operator of order at most n acting between sections of two vector bundles E and F is a linear map

C∞(M; E) −→ C∞(M; F ) that in any local coordinates that trivialize both E and F has the form (1.8) where we now require the functions aα(y) to be sections of Hom(E,F ). 1.13. EXERCISES 23

There is a very nice characterization (or deﬁnition if you prefer) of diﬀerential operators due to Peetre4. First, we say that a linear operator T : C∞(M; E) −→ C∞(M; F ) is a local operator if whenever s ∈ C∞(M; E) vanishes in an open set U ⊆ M, T s also vanishes in that open set,

s U = 0 =⇒ (T s) U = 0. It is easy to see that a diﬀerential operator is a local operator. Peetre showed that all linear local operators are diﬀerential operators.

1.13 Exercises

Exercise 1. Show that whenever M1 and M2 are smooth manifolds, M1 × M2 is too. Prove that T (M1 × M2) = TM1 × TM2. Exercise 2. Instead of deﬁning the cotangent bundle in terms of the tangent bundle, we can do things the other way around. At each point p of a manifold M, let

∞ Ip = {u ∈ C (M): u(p) = 0}

2 be the ideal of functions vanishing at p. Let Ip be the linear span of products of pairs of elements in Ip, and deﬁne

∗ 2 ∗ [ ∗ Tp M = Ip/Ip , T M = Tp M. p∈M Show that, together with the obvious projection map, this deﬁnes a vector bundle isomorphic to the cotangent bundle. Exercise 3. k π Show that a R -bundle E −−→ M is trivial if and only if there are k sections s1, . . . , sk such that, at each p ∈ M,

{s1(p), . . . , sk(p)} is linearly independent. Exercise 4. If E −−−→πE M is a subbundle of F −−−→πF M show that there is a bundle F/E −→ M −1 −1 whose fiber over p ∈ M is πF (p)/πE (p) 4 Jaak Peetre, Une caractérisationabstraite des opérateurs différentiels. Math. Scand. 7 1959, 211-218. 24 CHAPTER 1. DIFFERENTIAL OPERATORS ON MANIFOLDS

Exercise 5. Given vector bundles E −→ M and F −→ M, assume that f ∈ C∞(M; Hom(E,F )) is such that dim ker f(p) is independent of p ∈ M. Prove that the kernels of f form a vector bundle over M (a subbundle of E) and the cokernels of f also form a vector bundle over M (a quotient of F.) Exercise 6. (Poincaré’sLemma for 1-forms.) Let ω = a(x, y, z) dx + b(x, y, z) dy + c(x, y, z) dz be a smooth 1-form in R3 such that dω = 0. Define f : R3 −→ R by Z 1 f(x, y, z) = (a(tx, ty, tz)x + b(tx, ty, tz)y + c(tx, ty, tz)z) dt. 0 Show that df = ω. (Thus on R3 (also Rn) a 1-form ω satisfies dω = 0 ⇐⇒ ω = df for some f. It turns out that this is a reflection of the ‘contractibility’ of Rn.) Hint: First use dω = 0 to show that ∂b ∂a ∂c ∂a ∂b ∂c = , = , = ∂x ∂y ∂x ∂z ∂z ∂y and then use the identity

Z 1 ∂ a(x, y, z) = (a(tx, ty, tz)t) dt 0 ∂t Z 1 Z 1 ∂a ∂a ∂a = a(tx, ty, tz) dt + t x + y + z dt. 0 0 ∂x ∂y ∂z Exercise 7. Show that any smooth map F : M1 −→ M2 induces, for any k ∈ N0, a map in de Rham cohomology ∗ k k F :HdR(M2) −→ HdR(M1) Exercise 8. Show that the wedge product and the integral descend to cohomology. That is, show that if ω and η are closed diﬀerential forms then the value of R ω and the cohomology class of ω ∧ η only depends on the cohomology classes of ω and η. Exercise 9. Show that a linear map T : C∞(M; E) −→ C∞(M; F ) corresponds to a section of C∞(M; Hom(E,F )) if and only if T is linear over C∞(M). 1.14. BIBLIOGRAPHY 25 1.14 Bibliography

There are many wonderful sources for differential geometry. The first volume of Spivak, A comprehensive introduction to differential geometry is written in a con- versational style that makes it a pleasure to read. The other four volumes are also very good. I also recommend Lee, Introduction to smooth manifolds. For vector bundles, the books Milnor and Stasheff, Characteristic classes and Atiyah, K-theory are written by fantastic expositors, as is the book Bott and Tu, Differential forms in algebraic topology. 26 CHAPTER 1. DIFFERENTIAL OPERATORS ON MANIFOLDS Chapter 2

A brief introduction to Riemannian geometry

2.1 Riemannian metrics

A metric on a vector bundle E −→ M is a smooth family of inner products, one on each fiber of E. We make sense of smoothness by using the bundle S2(E) −→ M of symmetric bilinear forms on the fibers of E. Formally, a metric gE on E is a section of the vector bundle S2(E) −→ M that is non-degenerate and positive definite. In particular we have an C∞(M)-valued inner product on the sections of E, so a smooth map ∞ ∞ ∞ gE : C (M; E) × C (M; E) −→ C (M) that is linear in each entry, symmetric, and satisfies

gE(s, s) ≥ 0 and gE(s, s) = 0 ⇐⇒ s = 0.

A metric g on the tangent bundle of M is called a Riemannian metric, and the pair (M, g) is called a Riemannian manifold. Any vector bundle admits a metric. Indeed, on a local trivialization we can just use the Euclidean inner product. Covering the manifold with local trivializations, we can use a partition of unity to put these inner products together into a bundle metric.

Given a metric gE on E, we can use the inner product at a point p ∈ M to ∗ identify Ep and Ep . Indeed, we have

∗ (2.1) geE(p): Ep −→ Ep , (geE(p)(v))(w) = gE(p)(v, w) for all w ∈ Ep.

27 28 CHAPTER 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY

This extends to a bundle isomorphism

∗ geE : E −→ E . Conversely, any bundle isomorphism between E and E∗ defines by (2.1) an inner product on E (and on E∗), though not necessarily positive definite. In particular a Riemannian metric g on M induces a bundle isomorphism between TM and T ∗M, known as the ‘musical isomorphism’. A vector field V is associated a differential 1-form via g, denoted V [, and a 1-form ω is associated a vector field via g, denoted ω]. Incidentally, this is how the gradient is defined on a Riemannian manifold. Given a function f ∈ C∞(M), we get a one form df ∈ C∞(M; TM), and we use the metric to turn this into a vector field

∇f = (df)] ∈ C∞(M; TM).

Thus, the gradient satisﬁes (and is deﬁned by)

g(∇f, V ) = df(V ) = V (f), for all V ∈ C∞(M; TM).

Let us look at these objects in local coordinates. Let V be a local coordinate chart of M trivializing E, so that we have coordinates

(2.2) {y1, . . . , ym, s1, . . . , sk} where y1, . . . , ym are local coordinates on M, and s1, . . . , sk are linearly independent sections of E depending on y. A metric is a smooth family of k × k symmetric positive deﬁnite matrices

y 7→ gE(y) = ((gE)ab(y)) , where (gE)ab = gE(sa, sb).

Any section of E deﬁned over V, σ : V −→ E, can be written

k X a σ(y) = σ (y)sa a=1 for some smooth functions σa. It is common to write this more succinctly using the ‘Einstein summation convention’ as

a σ(y) = σ (y)sa 2.1. RIEMANNIAN METRICS 29 with the understanding that a repeated index that occurs as both a subindex and a a a superindex should be summed over. If σ = σ sa and τ = τ sa are two sections of E, their inner product with respect to gE is

a b gE(σ, τ) = σ τ (gE)ab.

It is very convenient, with regards to the Einstein convention, to denote the ab entries of the inverse matrix of (gE)ab by (gE) , thus for instance we have

ab a (gE) (gE)bc = δc .

∗ The metric gE determines a metric gE∗ on E by the relation

(2.3) gE∗ (geE(σ), geE(τ)) = gE(σ, τ), and in a local trivialization this corresponds to taking the inverse matrix of (gE)ab. Indeed, in these same coordinates (2.2) we have a natural local trivialization of E∗, namely

1 k j ∞ ∗ i i {y1, . . . , ym, s , . . . , s }, where s ∈ C (V; E ) satisfy s sj = δj.

Then the map geE satisﬁes b (geE)(sa)(sb) = gE(sa, sb) = gab, so (geE)(sa) = (gE)abs b ab and hence s = (gE) geE(sa). Thus we have a b ac bn (gE∗ )ab = gE∗ (s , s ) = gE∗ ((gE) geE(sc), (gE) geE(sn)) ac bn ac b ab = (gE) (gE) (gE)cn = (gE) δc = (gE) as required.

Given a smooth map F : M1 −→ M2 we can pull-back a metric g on M2 to 2 deﬁne a section of S (TM1) −→ M1,

∗ (F g)(V1,V2) = g(DF (V1),DF (V2)).

∗ If F is an immersion, then F g is a metric on M1. In particular, submanifolds of a Riemannian manifold inherit metrics. The simplest example of a Riemannian m-manifold is Rm with the usual Eu- clidean inner product on the ﬁbers of

m m m m T R = R × R −→ R . 30 CHAPTER 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY

From the previous paragraph it follows that all submanifolds of Rm inherit a metric from the one on Rm. A diﬃcult theorem of Nash, the Nash embedding theorem1, shows that all Riemannian manifolds can be embedded into RN for some N in such a way that the metric coincides with the one inherited from RN . So we can think of an arbitrary Riemannian manifold as a submanifold of a Euclidean space.

If (M1, g1) and (M2, g2) are Riemannian manifolds, a map F : M1 −→ M2 is a ∗ local isometry if it is a local diﬀeomorphism and F g2 = g1. An isometry between two Riemannian manifolds is a bijective local isometry. Riemannian geometry is the study of invariants of isometry classes of manifolds.

A metric on E −→ M allows us to deﬁne the length of a vector v ∈ Ep by p |v|gE = gE(p)(v, v) and the angle θ between two vectors v, w ∈ Ep satisﬁes

gE(p)(v, w) = |v|gE |w|gE cos θ.

Given a curve γ :[a, b] −→ M on M, and a Riemannian metric g, the length of γ is deﬁned to be Z b 0 `(γ) = |γ (t)|g dt. a This is independent of the chosen parametrization of γ. We can use the length of curves to deﬁne the distance between two points, p, q ∈ M, to be

(2.4) d(p, q) = inf{`(γ): γ ∈ C∞([0, 1],M), γ(0) = p, γ(1) = q} it is easy to see that

d(p, q) ≥ 0, d(p, q) = d(q, p), d(p, q) ≤ d(p, r) + d(r, q) and we will show below that d(p, q) = 0 =⇒ p = q, so that (M, d) is a metric space.

2.2 Parallel translation

On Rm, and more generally on a parallelizable manifold, it is clear what we mean by a constant vector ﬁeld. Given a tangent vector at a point in one of these spaces, we can ‘parallel translate’ it to the whole space by taking the constant vector ﬁeld.

1John Nash, The imbedding problem for Riemannian manifolds, Annals of Mathematics 63 (1), 1956, 20–63. 2.2. PARALLEL TRANSLATION 31

To try to make sense of this on a manifold, suppose we have a smooth curve γ : [0, 1] −→ M and a vector ﬁeld V ∈ C∞(M; TM), we would like to say that V is constant (or parallel) along γ if the map

V ◦ γ : [0, 1] −→ TM is constant. The problem with making sense of this is that TM is not a vector space but a collection of vector spaces, so V (γ(t)) ∈ Tγ(t)M belongs to a different vector space for each t. The same problem occurs with trying to make sense of a constant section of a vector bundle E −→ M. The solution is to use a connection, so called because it ‘connects’ different fibers of a vector bundle. A connection on a vector bundle E −→ M is a map

∇E : C∞(M; TM) × C∞(M; E) −→ C∞(M; E)

E E which is denoted ∇ (V, s) = ∇V s, is R-linear in both variables, i.e., for all ai ∈ R, ∞ ∞ Vi ∈ C (M; TM), si ∈ C (M; E),

E E E E E E ∇a1V1+a2V2 s = a1∇V1 s + a2∇V2 s, ∇V (a1s1 + a2s2) = a1∇V s1 + a2∇V s2 and satisﬁes, for f ∈ C∞(M),

E E E E (2.5) ∇fV s = f∇V s, ∇V (fs) = V (f)s + f∇V s.

E It follows that the value of ∇V s at a point p ∈ M depends only on the value of V E at p and the values of s in a neighborhood of p. We refer to ∇V s as the covariant derivative of s in the direction of V .

On M = Rm, vector ﬁelds are just maps V : Rm −→ Rm and so from calculus have a derivative DV (p): Rm −→ Rm at each point p ∈ M. The ‘ﬂat connection’ on T Rm has covariant derivative

∇U V (p) = DV (p)(U(p)).

n n n n If M ⊆ R is a submanifold of R , so that TpM ⊆ TpR = R , then a vector m field on M is a map V : M −→ R with the property that V (p) ∈ TpM for all p ∈ M. If we denote the orthogonal projection from the tangent space of Rn to the n n tangent space of M by πp : TpR −→ TpM, then the flat connection on R induces a connection on M by ∇U V (p) = πpDV (p)(U(p)). So in this case the covariant derivative is essentially the derivative from calculus ‘compressed’ onto the tangent space of M. This derivative is compatible with the 32 CHAPTER 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY metric induced on M by restricting the metric on Rn; it is an example of a ‘Levi- Civita connection’ as defined below.

In a local trivialization of E,

{y1, . . . , ym, s1, . . . , sk} a connection is determined by mk2 functions,

b Γia, where 1 ≤ i ≤ m, 1 ≤ a, b ≤ k known as the Christoﬀel symbols, deﬁned by

E b ∇ sa = Γ sb ∂yi ia where we are using the Einstein summation convention as above. Once we know the b functions Γia we can ﬁnd ∇V s at any point in this local trivialization by using (2.5). We can answer our original question by saying that, for a given curve γ : [a, b] −→ M a section of E, s : M −→ E, is parallel along γ (with respect to ∇E) if E ∇γ0(t)s = 0 at all points γ(t). Let us see write this equation in local coordinates as in (2.2). We can write a 0 i 0 s(t) := s(γ(t)) = σ (t)sa, γ (t) = (γ ) (t)∂yi and then E a i 0 a b ∇γ0(t)s = (∂tσ )sa + ((γ ) (t))σ Γiasb grouping together the coeﬃcients of sa, we see that this will vanish precisely when

a i 0 c a (2.6) ∂tσ + ((γ ) (t))σ Γic = 0 for all a.

This local expression shows that it is always possible to extend a vector in Ep to a covariantly constant section of E along any curve through p. Indeed, given any such curve γ :[a, b] −→ M, with p = γ(t0) and any vector s0 ∈ Eγ(t0), we can find s by solving (2.6) for its coefficients σ1, . . . , σk. As this is a linear system of ODE’s, there is a unique solution defined on all of [a, b]. We call this solution the parallel transport of s0 along γ. Parallel transport along γ defines a linear isomorphism between fibers of E −→ M,

Tγ : Ea −→ Eb. 2.2. PARALLEL TRANSLATION 33

This is particularly interesting when γ is a loop at a point p ∈ M. The possible linear automorphisms of Ep resulting from parallel transport along loops at p form a group, the holonomy group of ∇. Diﬀerent points in M yield isomorphic groups so, as long as M is connected, this group is independent of the base point p.

Remark 3. We can use parallel transport to show that pull-back bundles under homotopic maps are isomorphic. Indeed, assume that f : M1 −→ M2 and g : M1 −→ M2 are homotopic smooth maps, so that there is a smooth map

H : [0, 1] × M1 −→ M2 s.t. H(0, ·) = f(·) and H(1, ·) = g(·)

∗ ∼ ∗ and let E −→ M2 be a vector bundle. We want to show that f E = g E, and by ∗ considering H E we see that we can reduce to the case M2 = [0, 1] × M1, with

f = i0 : M1 −→ [0, 1] × M1, i0(p) = (0, p)

g = i1 : M1 −→ [0, 1] × M1, i1(p) = (1, p).

So let F −→ [0, 1] × M1 be a vector bundle. Choose a connection on F and for each p ∈ M1 let Tp be parallel transport along the curve t 7→ (t, p), then the map

∗ ∗ Φ: i0F −→ i1F, Φ(p, v) = (p, Tp(v)) is a vector bundle isomorphism.

E Next assume that we have both a metric gE and a connection ∇ on E. We will say that these objects are compatible if, for any sections σ and τ of E, and any vector ﬁeld V ∈ C∞(M; TM),

E E (2.7) V (gE(σ, τ)) = gE(∇V σ, τ) + gE(σ, ∇V τ). We also say in this case that ∇E is a metric connection. Parallel transport with respect to a metric connection is an isometry. That is, if γ :[a, b] −→ M is a curve on M and σ and τ are parallel along γ, then

∂tgE(σ(t), τ(t)) = 0. In fact this follows immediately from the metric condition since the derivative with respect to t is precisely the covariant derivative with respect to the vector field γ0(t). In local coordinates, the connection is metric if and only if the Christoffel symbols satisfy c c ∂yi (gE)ab = Γia(gE)cb + Γib(gE)ac. b Given the matrix-valued function gE, we can always find functions Γia that satisfy these relations. In other words given a metric on a trivial bundle, there is always 34 CHAPTER 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY a metric connection. Using a partition of unity it follows that for any metric on a bundle E there is a metric connection.

A connection on E is, for each vector field V ∈ C∞(M; TM), a first-order differential operator on sections of E,

E ∞ ∞ ∇V : C (M; E) −→ C (M; E).

If ∇E and ∇e E are two connections on E, then their diﬀerence is linear over C∞(M),

E E ∞ ∞ ∇V − ∇e V : C (M; E) −→ C (M; E) satisﬁes E E E E ∞ ∞ (∇V − ∇e V )(fs) = f(∇V − ∇e V )(s) for all f ∈ C (M), s ∈ C (M; E) and hence a zero-th order diﬀerential operator or equivalently, a section of the homomorphism bundle of E,

E E ∞ ∇V − ∇e V ∈ C (M; Hom(E)).

We can also incorporate the dependence on the vector ﬁeld V and write

∇E − ∇e E ∈ C∞(M; T ∗M ⊗ Hom(E)). Conversely, it is easy to see that whenever ∇E is a connection on E and Φ ∈ C∞(M; T ∗M ⊗ Hom(E)) is a section of T ∗M ⊗ Hom(E), the sum ∇E + Φ is again a connection on E. We say that the space of connections on E is an aﬃne space modeled on C∞(M; T ∗M ⊗ Hom(E)). In particular, any bundle has inﬁnitely many connections on it.

2.3 Geodesics

Every connection ∇ on the tangent bundle of a manifold M determines a class of curves known as its geodesics. A curve γ :(a, b) −→ M is a geodesic for ∇ if parallel transport along the curve preserves the tangent vector to the curve. That is, if

0 (2.8) ∇γ0(t)γ (t) = 0.

Heuristically, we are demanding that γ has no acceleration so γ should ‘curve’ only as much as the space requires. For example, if M = Rm with its ﬂat connection the geodesic equation says that Dγ0(t)(γ0(t)) = 0 2.3. GEODESICS 35 i.e., that the directional derivative of γ0(t) in the direction of γ0(t) vanishes. However this directional derivative is precisely ∂t, so the geodesic equation is

γ00(t) = 0 and the solutions are straight lines. If M ⊆ Rn has the connection induced by the ﬂat connection on Rn, the geodesic equation says that

0 0 00 πγ(t)Dγ (t)(γ (t)) = πγ(t)γ (t) = 0 thus a curve in M is a geodesic if and only if its acceleration as a curve in Rn is normal to M. In particular, if the intersection of M with a subspace of Rn that is everywhere orthogonal to M is a curve, then any constant speed parametrization of that curve must be a geodesic. The reason we must have constant speed is that the connection is compatible with the metric on M induced by the Euclidean metric on Rn, so any geodesic will have constant speed.

Geodesics exist in abundance. If p ∈ M and v ∈ TpM there exists a geodesic γ 0 for ∇ such that γ(0) = p and γ (0) = v. Indeed, choose local coordinates {y1, . . . , ym} centered at p and write γ(t) = (γ1(t), . . . , γm(t)) for a curve in this coordinate chart. The geodesic equation (2.8) in local coordinates is given by

2 k k i j (2.9) ∂t γ (t) + Γij∂tγ (t)∂tγ (t) = 0 and the claim is that there is a solution with γ(0) = 0 and γ0(0) = v. This is a semi-linear second order system of ODEs for the coeﬃcients γi(t), so we have the following existence result2

Lemma 1. For every point p0 on a manifold M endowed with a connection ∇, there is a neighborhood U of p0 and numbers ε1, ε2 > 0 such that: For all p ∈ U and tangent vector v ∈ TpM with |v|g < ε1 there is a unique geodesic

γ(p,v) :(−2ε2, 2ε2) −→ M, satisfying the initial conditions

0 (2.10) γ(p,v)(0) = p, γ(p,v)(0) = v.

Furthermore, the geodesics depend smoothly on the initial conditions.

2 For more detail, see Lemma 10.2 of John Milnor, Morse theory. Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963 36 CHAPTER 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY

The solution to the system (2.9) is unique up to its domain of definition. Since (2.9) is non-linear the geodesic may not be defined on all of R, but rather there is a maximal interval around zero, say (a, b), in which this solution is defined. With this domain, the solution is unique, so we say that for each (p, v) ∈ M there is a unique maximally extended geodesic satisfying (2.10). We will use γ(p,v) to refer to this maximally extended geodesic. By uniqueness we have the following homogeneity property

γ(p,λv)(t) = γ(p,v)(λt) for all λ ∈ R for which one of the two sides is deﬁned. Thus by making ε1 smaller in the lemma, we can take, e.g., ε2 = 1.

It is possible for two different connections to have the same geodesics. Indeed, note that we can rewrite (2.9) as 1 ∂2γk(t) + Γk + Γk ∂ γi(t)∂ γj(t) = 0 t 2 ij ji t t so only the ‘symmetric part’ of the Christoffel symbols are involved. The ‘antisymmetric part’ of the Christoffel symbols

k k k Tij = Γij − Γji deﬁne a tensor, known as the torsion tensor of the connection, which can be expressed invariantly by

∞ ∞ ∞ T : C (M; TM)×C (M; TM) −→ C (M; TM),T (U, V ) = ∇U V −∇V U −[U, V ]. Geometrically, this tensor provides a measure of how much a basis of the tangent space of M ‘twists’ when parallel translated along a geodesic. It is not hard to show3 that if two connections have the same geodesics and the same torsion then they coincide. Given any connection ∇ on TM with torsion tensor T, it follows that ∇ − T is the unique connection whose geodesics are those of ∇ and whose torsion tensor vanishes.

If M is a submanifold of X then a connection on TX induces a connection on TM. We say that M is totally geodesic (with respect to this connection) if all of the geodesics of M are also geodesics of X. For example a subset of Rn is totally geodesic (with respect to the ﬂat connection) if and only if it is a linear subspace. On any X, a geodesic is automatically a totally geodesic submanifold.

3See, e.g., Michael Spivak, A comprehensive introduction to diﬀerential geometry, vol II, Second edition. Publish or Perish, Inc., Wilmington, Del., 1979., Addendum 1 of Chapter 6. 2.4. LEVI-CIVITA CONNECTION 37 2.4 Levi-Civita connection

Among the inﬁnitely many connections on the tangent bundle of a Riemannian manifold, there is a unique connection that is compatible with the metric and satisﬁes an additional symmetry property. Lemma 2 (Fundamental lemma of Riemannian geometry). Let (M, g) be a Rie- mannian manifold, there is a unique metric connection ∇ on TM satisfying

∞ (2.11) ∇V W − ∇W V − [V,W ] = 0 for all V,W ∈ C (M; TM).

We say that a connection (necessarily on TM) that satisﬁes on (2.11) is symmetric or ‘torsion-free’. The unique metric torsion-free connection of a metric is called the Levi-Civita connection.

Proof. Given any three vector ﬁelds U, V, W ∈ C∞(M; TM), we can apply the metric property (2.7) to Ug(V,W ) + V g(W, U) − W g(U, V ) and then use the torsion-free property to cancel out all but one covariant derivative and arrive at the Koszul formula: 1 g(∇ V,W ) = Ug(V,W ) + V g(W, U) − W g(U, V ) U 2 − g(U, [V,W ]) + g(V, [W, U]) + g(W, [U, V ]) , which proves uniqueness. It is easy to check that this formula deﬁnes a symmetric metric connection, establishing existence.

Note that the coordinate vector fields ∂yi satisfy Schwarz’ theorem, [∂yi , ∂yj ] = 0, so from Koszul’s formula the Christoffel symbols of the Levi-Civita connection are given by 1 Γk = gk` (∂ g + ∂ g − ∂ g ) . ij 2 i `j j i` ` ij The geodesics of the Levi-Civita connection are also called the geodesics of the Riemannian metric. At each point p ∈ M we can define the exponential map by

expp : U ⊆ TpM −→ M, expp(v) = γ(p,v)(1) where U is the subset of TpM of vectors v for which γ(p,v)(1) is deﬁned. We know that U contains a neighborhood of the origin and that expp is smooth on U. 38 CHAPTER 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY

Let {y1, . . . , ym, ∂y1 , . . . , ∂ym } be local coordinates for TM over a coordinate chart V of p. Consider the smooth function F : U −→ M × M, deﬁned by

F (p, v) = (p, expp(v)).

Notice that, if {y1, . . . , ym, z1, . . . , zm} are the corresponding coordinates on V ×V ⊆ M × M, then

DF : T U −→ TM × TM satisﬁes DF (∂ ) = ∂ + ∂ ,DF (∂ ) = ∂ yi yi zi (∂yi ) zi

Id Id and hence has the form . In particular, we can apply the implicit function 0 Id theorem to F to conclude as follows:

Lemma 3. For each p ∈ M there is a neighborhood U of M and a number ε > 0 such that:

a) Any two points p, q ∈ U are joined by a unique geodesic in M of length less than ε.

b) This geodesic depends smoothly on the two points.

c) For each q ∈ U the map expq maps the open ε-ball in TqM diﬀeomorphically onto an open set Uq containing U.

Since expp is a diﬀeomorphism from Bε(0) ⊆ TpM into M, it deﬁnes a coordinate chart on M. These coordinates are known as normal coordinates.

At any point p ∈ M, one can show that if v ∈ TpM is small enough, the distance (2.4) between p and exp(v) is |v|g. Indeed minimizing the length of a curve with ﬁxed endpoints is equivalent to minimizing the ‘energy’ of the curve

Z b 0 2 E(γ) = |γ (t)|g dt a and as a problem in the calculus of variations this produces diﬀerential equations that characterize ‘critical curves’ for this functional. These equations, known as the Euler-Lagrange equations, are precisely the geodesic equations for the Levi-Civita connection. It follows that small enough segments of geodesics realize the distance between their endpoints. Note in particular that this establishes that d(p, q) = 0 =⇒ p = q and hence (M, d) is a metric space. 2.5. ASSOCIATED BUNDLES 39 2.5 Associated bundles

We saw above (2.3) that a metric on E determines a dual metric on E∗ by requiring that the bundle isomorphism ∗ geE : E −→ E be an isometry. A connection on E also naturally induces a connection on E∗. Given sections s of E and η of E∗, we can ‘contract’ or pair them to get a function η(s) on M, and we demand that the putative connection on E∗ satisfy the corresponding Leibnitz rule, E∗ E V (η(s)) = (∇V η)(s) + η(∇V s). E∗ Thus we deﬁne ∇V by

E∗ E (∇V η)(s) = V (η(s)) − η(∇V s).

If E −→ M and F −→ M are vector bundles over M endowed with metrics gE and gF then we have induced metrics

gE⊕F on E ⊕ F −→ M, gE⊗F on E ⊗ F −→ M determined by gE⊕F (σ ⊕ α, τ ⊕ β) = gE(σ, τ) + gF (α, β),

gE⊗F (σ ⊗ α, τ ⊗ β) = gE(σ, τ)gF (α, β), for all σ, τ ∈ C∞(M; E), α, β ∈ C∞(M; F ). Similarly, given connections ∇E and ∇F , we have induced connections

∇E⊕F on E ⊕ F −→ M, ∇E⊗F on E ⊗ F −→ M. determined by the corresponding Leibnitz rules,

E⊕F E F ∇V (σ ⊕ α) = ∇V σ ⊕ ∇V α E⊗F E F ∇V (σ ⊗ α) = ∇V σ ⊗ α + σ ⊗ ∇V α, for all V ∈ C∞(M; TM), σ ∈ C∞(M; E), and α ∈ C∞(M; F ). In particular, connections on E and F induce a connection on Hom(E,F ) = E∗ ⊗ F −→ M, satisfying

Hom(E,F ) F E (∇V Φ)(σ) = ∇V (Φ(σ)) − Φ(∇V σ) for all V ∈ C∞(M; TM), Φ ∈ C∞(M; Hom(E,F )), and σ ∈ C∞(M; E). 40 CHAPTER 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY

Using these induced connections, it makes sense to take the covariant derivative of a metric on E as a section of E∗ ⊗ E∗,

E∗⊗E∗ E E (∇V gE)(σ, τ) = V (gE(σ, τ)) − gE(∇V σ, τ) − gE(σ, ∇V τ).

Thus we ﬁnd that a connection is compatible with a metric if and only if the metric is covariantly constant with respect to the induced connection. We also have induced metrics and connections on pull-back bundles. Let f : M1 −→ M2 be a smooth map and E −→ M2 a vector bundle with metric gE and E connection ∇ . Any section of E, σ : M2 −→ E, can be pulled-back to a section of f ∗E, ∗ ∗ (f σ)(p) = (p, σ(f(p))) ∈ (f E)p, for all p ∈ M1.

∞ ∗ ∞ Moreover these pulled-back sections generate C (M1; f E) as a C (M1)-module in that ∞ ∗ ∞ ∗ ∞ C (M1; f E) = C (M1) · f C (M2; E). Thus, to specify a metric and connection on f ∗E, it suﬃces to describe their actions ∞ on products of functions on M1 and pulled-back sections. For any h1, h2 ∈ C (M1), ∞ and σ1, σ2 ∈ C (M2; E) we deﬁne

∗ ∗ gf ∗E(h1f σ1, h2f σ2)p = h1(p)h2(p)gE(σ1, σ2)f(p) f ∗E ∗ ∗ ∗ E ∇V h1f σ1 = V (h1)f σ1 + h1f (∇V σ1).

An example is the pull-back of a metric along the inclusion of a submanifold M,→ X. This produces the restriction of the metric on X to vectors tangent to M.

2.6 E-valued forms

We deﬁned a connection as a map

∇E : C∞(M; TM) × C∞(M; E) −→ C∞(M; E) but we can equivalently think of it as a map

∇E : C∞(M; E) −→ C∞(M; T ∗M) × C∞(M; E) = C∞(M; T ∗M ⊗ E).

The space C∞(M; T ∗M⊗E) is the 1-form part of the space of E-valued diﬀerential k-forms, Ωk(M; E) = C∞(M;ΛkT ∗M ⊗ E). 2.6. E-VALUED FORMS 41

This space is linearly spanned by ‘elementary tensors’, i.e., section of the form ω ⊗ s where ω ∈ Ωk(M) and s ∈ C∞(M; E). There is a wedge product between k-forms and E-valued `-forms given on elementary tensors by (ω ⊗ s) ∧ η = (ω ∧ η) ⊗ s and then extended by linearity. There is also a natural wedge product between a form in Ω∗(M; End E) and a form in Ω∗(M; E) resulting in another form in Ω∗(M; E). For elementary tensors ω ⊗ Φ ∈ Ω∗(M; End E), η ⊗ s ∈ Ω∗(M; E) the wedge product is given by (ω ⊗ Φ) ∧ (η ⊗ s) = (ω ∧ η) ⊗ Φ(s).

Suppose M is Riemannian manifold, so that we have the Levi-Civita connection on TM, and that we have a connection on E, then as explained above there is an induced connection on Ωk(M; E),

∇Ωk(M;E) :Ωk(M; E) −→ C∞(M; T ∗M ⊗ ΛkT ∗M ⊗ E). We can take the ﬁnal section, ‘antisymmetrize’ all of the cotangent variables, and end up with a section of Ωk+1(M; E). We think of the resulting map as an E-valued exterior derivative and denote it dE :Ωk(M; E) −→ Ωk+1(M; E). On elementary tensors ω ⊗ s, dE is given by dE(ω ⊗ s) = dω ⊗ s + (−1)deg(ω)ω ∧ ∇Es and in particular on Ω0(M; E) = C∞(M; E) coincides with ∇E. Unlike the case E = R, it is generally not true that (dE)2 = 0. Proposition 4. Let (M, g) be a Riemannian manifold, E −→ M a vector bundle, and let dE be the E-valued exterior derivative induced by a connection ∇E on E and the Levi-Civita connection on M. The map (dE)2 :Ωk(M; E) −→ Ωk+2(M; E) is given by wedge product with a diﬀerential form RE ∈ Ω2(M; End E). Moreover, if V,W ∈ C∞(M; TM), we have

E E E E E E R (V,W ) = ∇V ∇W − ∇W ∇V − ∇[V,W ]. 42 CHAPTER 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY

This 2-form with values in End E is is known as the curvature of the connection ∇E. We say that the connection is ﬂat if RE = 0.

Proof. First consider the eﬀect of applying ∇E and then ∇T ∗M⊗E to a section of E,

E T ∗M⊗E C∞(M; E) −−−→C∇ ∞(M; T ∗M ⊗ E) −−−−−−−→C∇ ∞(M; T ∗M ⊗ T ∗M ⊗ E).

By the Leibnitz rule deﬁning the connection on T ∗M ⊗ E, we have, for all V,W ∈ C∞(M; TM),

T ∗M⊗E E T ∗M⊗E E (∇ ∇ σ)(V,W ) = (∇V (∇ σ))(W ) = ∇E (∇E σ) − (∇Eσ)(∇ W ) = ∇E (∇E σ) − ∇E σ. V W V V W ∇V W We obtain dE by anti-symmetrizing, so we have

(dE)2(σ)(V,W ) = ∇E (∇E σ) − ∇E σ − ∇E (∇E σ) + ∇E σ V W ∇V W W V ∇W V E E E E E = ∇V ∇W − ∇W ∇V − ∇[V,W ] σ. This is what we wanted to prove for k = 0, and the same computation replacing E with ΛkT ∗M ⊗ E proves the result for general k.

The curvature of the Levi-Civita connection is referred to as the Riemannian curvature of (M, g).

2.7 Curvature

As explained in the previous section, the curvature of a Riemannian manifold (M, g) is R ∈ Ω2(M; End(TM)), deﬁned in terms of the Levi-Civita connection by

R(V,W ) = ∇V ∇W − ∇W ∇V − ∇[V,W ].

Together with the obvious (anti)symmetry R(V,W ) = −R(W, V ), we have the following: Proposition 5. Let (M, g) be a Riemannian manifold, and R the curvature of its Levi-Civita connection ∇, then for any V, W, X, Y ∈ C∞(M; TM) we have

a) (First Bianchi Identity) R(V,W )X + R(W, X)V + R(X,V )W = 0

b) g(R(V,W )X,Y ) = −g(R(V,W )Y,X)

c) g(R(V,W )X,Y ) = g(R(X,Y )V,W ) 2.7. CURVATURE 43

Proof. a) reduces by a straightforward computation to the Jacobi identity for the commutator of vector ﬁelds. b) is equivalent to proving that g(R(V,W )X,X) = 0, which we prove by noting that

g(∇V ∇W X − ∇W ∇V X,X)

= (V g(∇W X,X) − g(∇W X, ∇V X)) − (W g(∇V X,X) − g(∇V X, ∇W X)) 1 = [V,W ]g(X,X) = g(∇ X,X). 2 [V,W ] c) follows from the other symmetries. Starting with following consequences of the Bianchi identity:

g(R(V,W )X + R(W, X)V + R(X,V )W, Y ) = 0 g(R(W, X)Y + R(X,Y )W + R(Y,W )X,V ) = 0 g(R(X,Y )V + R(Y,V )X + R(V,X)Y,W ) = 0 g(R(Y,V )W + R(V,W )Y + R(W, Y )V,X) = 0 we add all of these equations up, and use the other symmetries to see that we get

2g(R(X,V )W, Y ) + 2g(R(Y,W )X,V ) = 0 and hence g(R(X,V )W, Y ) = g(R(W, Y )X,V ).

In local coordinates {y1, . . . , ym, ∂y1 , . . . , ∂ym } on the tangent bundle, the cur- ` vature is determined by the functions Rijk deﬁned by

` R(∂i, ∂j)∂k = Rijk∂`.

We can compute these functions explicitly in terms of the Christoﬀel symbols of the Levi-Civita connection,

` ` s ` s ` ` Rijk = ΓisΓjk − ΓjsΓik + ∂i(Γjk) − ∂j(Γik).

These formulæ simplify considerably if we use a normal coordinate system. Indeed, a very nice property of a these coordinates is that near the origin the co- eﬃcients of the metric agree to second order with the identity matrix up to second order. More precisely4:

4See, e.g., Proposition 1.28 of Nicole Berline, Ezra Getzler, Michele` Vergne Heat kernels and Dirac operators. Corrected reprint of the 1992 original. Grundlehren Text Edi- tions. Springer-Verlag, Berlin, 2004. x+363 pp. ISBN: 3-540-20062-2 44 CHAPTER 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY

Proposition 6. Let (M, g) be a Riemannian manifold, p ∈ M, and let {y1, . . . , ym} be normal coordinates centered at p. Then we have

k gij(0) = δij, Γij(0) = 0, 2 2 2 2 1 ∂ gj` ∂ gjk ∂ gik ∂ gi` Rijk`(0) = − + − 2 ∂yi∂yk ∂yi∂y` ∂yj∂y` ∂yj∂yk and the Taylor expansion of the functions gij at the origin has the form 1 X g (y) ∼ δ − R y y + O(|y|3). ij ij 3 ikj` k ` k,`

It follows that if (M, g) is locally isometric to Euclidean space then the curvature must vanish. The converse is also true and is essentially how Riemann intro- duced his curvature tensor5.

There is a lot of complexity in the curvature tensor, so it is standard to focus on one of its contractions. Let p ∈ M and v, w ∈ TpM, the Ricci curvature at v, w, Ric(v, w), is the trace of the endomorphism of TpM z 7→ R(v, z)w. This deﬁnes a symmetric tensor Ric ∈ C∞(M; T ∗M ⊗ T ∗M). The metric is also a symmetric section of T ∗M ⊗ T ∗M, so we can compare these two tensors. We say that a metric is Einstein if there is a constant λ such that Ric = λg.

The Ricci curvature corresponds to a linear self-adjoint map

Ricf : C∞(M; TM) −→ C∞(M; TM) determined by g(Ric(f V ),W ) = Ric(V,W ). The scalar curvature of a Riemannian manifold scal ∈ C∞(M) is the trace of Ricf . Finally, the sectional curvature of a Riemannian manifold (M, g) is a function that assigns to each 2-dimensional subspace of TpM a real number. Let Σ ⊆ TpM be a two dimensional space spanned by v, w ∈ TpM then g(R(v, w)v, w) K(Σ) = − g(v, v)g(w, w) − g(v, w)2

5See Michael Spivak, A comprehensive introduction to diﬀerential geometry, volume II, Sec- ond edition. Publish or Perish, Inc., Wilmington, Del., 1979. 2.7. CURVATURE 45 is independent of the choice of v, w. Notice that the denominator is the area of the parallelogram spanned by v and w. The sectional curvature determines the full curvature tensor. A manifold is said to have constant curvature if the sectional curvature is constant. If the sectional curvature at a point p ∈ M is constant and equal to K, then the Riemannian curvature at that point is given by

g(R(v, w)x, z)p = K(g(w, x)pg(v, z)p − g(v, x)pg(w, z)p), the Ricci curvature at p is (m − 1)Kgp and the scalar curvature is m(m − 1)K. It turns out that if this is true at each point of M, with a priori a diﬀerent K at each point, then in fact all of the K’s are equal and the manifold has constant curvature. The simply connected manifolds of constant curvature are, up to scale, just Euclidean space, the sphere, and hyperbolic space (see exercises 9 and 10). In two dimensions, the curvature is determined by the scalar curvature which, up to a factor of 2, coincides with the classical Gaussian curvature. In three dimensions, the curvature is determined by the Ricci curvature.

In local coordinates {y1, . . . , ym, ∂y1 , . . . , ∂ym } on the tangent bundle, the Ricci curvature is determined by the functions Ricij

s Ricij = Ric(∂i, ∂j) = Risj.

The map Ricf is given in local coordinates by

j j js Ric(f ∂i) = (Ric)f i ∂j, with (Ric)f i = g Ricis and hence the scalar curvature is given by

k jk jk s scal = (Ric)f k = g Ricjk = g Rjsk.

For the sake of intuition, recall what the sign of the Gaussian curvature tells us about a surface embedded in R3. Positive Gaussian curvature at a point means that the surface lies on a single side of its tangent plane at that point. Thus we can think of the sphere or a paraboloid. Negative Gaussian curvature at a point means that the surface locally looks like a saddle. Vanishing of the Gaussian curvature is true for a plane, but it is also true for a cylinder, since these are locally isometric. We should think of this as saying that there is at least one direction that looks Euclidean. For higher dimensional manifolds, these mental pictures are good descriptions of the behavior of sectional curvature. In fact the sectional curvature of the space Σ ⊆ TpM is the Gaussian curvature of the surface in M carved out by geodesics through p in the directions in Σ, i.e., the image of Σ under the exponential map. 46 CHAPTER 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY 2.8 Complete manifolds of bounded geometry

At each point p in a Riemannian manifold, the injectivity radius at p, injM (p), is the largest radius R such that

expp : BR(0) ⊆ TpM −→ M is a diﬀeomorphism. The injectivity radius of M is deﬁned by

injM = inf{injM (p): p ∈ M}. By Lemma 3, a compact manifold has a positive injectivity radius. We say that a manifold is complete if there is a point p ∈ M for which expp is defined on all of TpM (though it need not be injective on all of TpM). The nomenclature comes from the Hopf-Rinow theorem, which states that the following assertions are equivalent: a) The metric space (M, d), with d defined by (2.4), is complete. b) Any closed and bounded subset of M is compact. c) There is a point p ∈ M for which expp is defined on all of TpM. d) At any point p ∈ M, expp is defined on all of TpM. Moreover in this case one can show that any two points p, q ∈ M can be joined by a geodesic γ with d(p, q) = `(γ). A Riemannian manifold (M, g) is called a manifold with bounded geometry if: 1) M is complete and connected

2) injM > 0 3) For every k ∈ N0, there is a constant Ck such that k |∇ R|g ≤ Ck.

Every compact Riemannian manifold has bounded geometry. Many non- compact manifolds, such as Rm and Hm, have bounded geometry. It is often the case that constructions that one can do on Rm can be carried out also on manifolds of bounded geometry but that one runs into trouble trying to carry them out on manifolds that do not have bounded geometry.

2.9 Riemannian volume and formal adjoint

On Rm with the Euclidean metric and the canonical orientation, there is a natural choice of volume form

dvolRm = dy1 ∧ · · · ∧ dym. 2.9. RIEMANNIAN VOLUME AND FORMAL ADJOINT 47

This assigns unit volume to the unit square. The volume of a parallelotope generated by m-vectors z1, . . . , zm is equal to the determinant of the change of basis matrix from y to z, which we can express as the ‘Gram determinant’ q det(zi · xj) .

It turns out that a Riemannian metric g on an orientable manifold M also determines a natural volume form, dvolg, determined by the property that, at any p ∈ M and any local coordinates in which gij(p) = δij we have dvolg(p) = dvolRm . In arbitrary oriented local coordinates, x1, . . . , xm, dvol(p) should be the volume of the parallelotope generated by ∂xi and hence q dvolg(p) = det(gij(p)) dx1 ∧ · · · ∧ dxm.

The volume form is naturally associated to the metric in that, e.g.,

∗ dvolf ∗g = f dvolg and one can check that the volume form is covariantly constant with respect to the Levi-Civita connection.

Given a metric gE on a bundle E −→ M and a Riemannian metric on M, we obtain an inner product on sections of E by setting:

∞ ∞ h·, ·iE : C (M; E) × C (M; E) −→ R, Z ∞ hσ, τiE = gE(σ, τ) dvolg, for all σ, τ ∈ C (M; E). M ∞ This is a non-degenerate, positive definite inner product on C (M; E) because gE is a non-degenerate positive definite inner product on each Ep. Using this inner product, we can define the Lp-norms in the usual way Z 1/p ∞ p k·k p : C (M; E) −→ , ksk p = |s| dvol L (M;E) R L (M;E) gE g M p p and then define the L space of sections of E,L (M; E), by taking the k·kLp(M;E)- closure of the smooth functions with finite Lp(M; E)-norm. These are Banach spaces, and we will mostly work with L2(M; E), which is also a Hilbert space. If we have a Riemannian manifold (M, g) and two vector bundles E −→ M, F −→ M with bundle metrics, we say that two linear operators

T : C∞(M; E) −→ C∞(M; F ),T ∗ : C∞(M; F ) −→ C∞(M; E) 48 CHAPTER 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY are formal adjoints if they satisfy

∗ ∞ ∞ hT σ, τiF = hσ, T τiE, for all σ ∈ Cc (M; E), τ ∈ Cc (M; F ).

(We use the word formal because we are not working with a complete vector space like L2 or Lp but just with smooth functions, this avoids some technicalities.) As a simple example, the formal adjoint of a scalar diﬀerential operator on R is easy to compute

∗ X k X k k ∞ ak(x)∂x u(x) = (−1) ∂x(ak(x)u(x)), for all u ∈ C (R).

The formal adjoint of any diﬀerential operator is similarly computed using integration by parts in local coordinates (the compact supports allow us not to worry about picking up ‘boundary terms’). We say that a linear operator is formally self-adjoint if it coincides with its formal adjoint. For instance, if T and T ∗ are formal adjoints then TT ∗ and T ∗T are formally self-adjoint. Moreover, these operators are positive semideﬁnite since on compactly supported sections

hT ∗T s, si = hT s, T si = |T s|2.

In the next section we will compute the formal adjoints of d and ∇E, which will lead to natural Laplace-type operators.

2.10 Hodge star and Hodge Laplacian

We have already looked at the space of diﬀerential forms on M and seen that the smooth structure on M induces a lot of structure on Ω∗(M), such as the exterior product, interior product and exterior derivative. Now we suppose that (M, g) is a Riemannian manifold and we will see that there is still more structure on Ω∗(M). First, we have already noted above that a metric on M induces metrics on associated vector bundles, such as Λ∗T ∗M. It is easy to describe this metric in terms of the metric on T ∗M when dealing with elementary k-forms, namely

1 k 1 k i j i j ∗ gΛkT ∗M (ω ∧ · · · ∧ ω , θ ∧ · · · θ )p = det gT ∗M (ω , θ )p , for all {ω , θ } ⊆ Tp M.

On any oriented Riemannian m-manifold, we deﬁne the Hodge star operator

∗ :Ωk(M) −→ Ωm−k(M) 2.10. HODGE STAR AND HODGE LAPLACIAN 49 by demanding, for all ω, η ∈ Ωk(M),

ω ∧ ∗η = gΛkT ∗M (ω, η) dvolg .

Thus, for example, in R4 with Euclidean metric and canonical orientation, with y1, . . . , y4 an orthonormal basis, ∗1 = dy1 ∧ dy2 ∧ dy3 ∧ dy4 ∗dy1 = dy2 ∧ dy3 ∧ dy4, ∗dy2 = −dy1 ∧ dy3 ∧ dy4, ∗dy1 ∧ dy2 = dy3 ∧ dy4, ∗dy2 ∧ dy4 = −dy1 ∧ dy3 ∗dy1 ∧ dy2 ∧ dy3 = dy4, ∗dy1 ∧ dy2 ∧ dy3 ∧ dy4 = 1 and so on. Note that ∗ is symmetric in that

ω ∧ ∗η = gΛkT ∗M (ω, η) dvolg = gΛkT ∗M (η, ω) dvolg = η ∧ ∗ω.

The Hodge star is almost an involution,

ω ∈ Ωk(M) =⇒ ∗(∗ω) = (−1)k(m−k)ω.

The Hodge star acts on diﬀerential forms and we have Z hω, ηiΛ∗T ∗M = ω ∧ ∗η. M This lets us get a formula for the formal adjoint of the exterior derivative, ∗ k−1 k d = δ. Namely, whenever ω ∈ Ωc (M) and η ∈ Ωc (M), Z Z k−1 hdω, ηiΛkT ∗M = (dω) ∧ ∗η = d(ω ∧ ∗η) − (−1) ω ∧ d ∗ η M M Z Z = (−1)k ω ∧ d ∗ η = (−1)k ω ∧ (−1)(m−k+1)(k−1) ∗2 d ∗ η M M m(k+1)+1 = hω, (−1) ∗ d ∗ ηiΛk−1T ∗M and so, when acting on Ωk(M),

δ = d∗ = (−1)m(k+1)+1 ∗ d∗ = (−1)k ∗−1 d ∗ .

A form is called coclosed if it is in the null space of δ. (Functions are thus automatically coclosed.) In particular on one-forms we have

ω ∈ Ω1(M) =⇒ δω = − ∗ d ∗ ω, 50 CHAPTER 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY

1 and, if ω ∈ Ωc (M), then Stokes’ theorem (on manifolds with boundary) says that Z Z Z Z δ(ω) dvolg = − ∗d ∗ ω dvolg = − d(∗ω) = − ∗ω. M M M ∂M Hence on manifolds without boundary, Z δ(ω) dvolg = 0. M

The Hodge Laplacian on k-forms is the diﬀerential operator

k k ∆k :Ω (M) −→ Ω (M), ∆k = dδ + δd.

Note that this operator is formally self-adjoint. Elements of the null space of the Hodge Laplacian are called harmonic diﬀerential forms, and we will use the notation k k H (M) = {ω ∈ Ω (M) : ∆kω = 0} for the vector space of harmonic k-forms. A form that is closed and coclosed is automatically harmonic. Conversely, if k ω ∈ Ωc (M) is compactly supported and harmonic, then

0 = h∆ω, ωiΛkT ∗M = hdω, dωiΛk+1T ∗M + hδω, δωiΛk−1T ∗M and so ω is both closed and coclosed. In particular, if M is compact, we have a natural map k k H (M) −→ HdR(M) since each harmonic form is closed. We will show later that this map is an isomorphism by showing that every de Rham cohomology class on a compact manifold contains a unique harmonic form. Since d2 = 0 and δ2 = 0, we can also write the Hodge Laplacian as

2 ∆k = (d + δ) .

Also, notice that

∗∆k = ∆m−k∗,

d∆k = dδd = ∆k+1d, δ∆k = δdδ = ∆k−1δ and hence ω harmonic implies dω and δω harmonic, and is equivalent to ∗ω harmonic. In particular, Hk(M) = Hm−k(M), 2.11. EXERCISES 51 which we call Poincar´eduality for harmonic forms.

Let us describe the Laplacian on functions ∆ = ∆0 in a little more detail. Since δ vanishes on functions, we have

∆ = δd = − ∗ d ∗ d.

A closed function is necessarily a constant, so on compact manifolds all harmonic functions are constant. In local coordinates, {y1, . . . , ym}, the Laplacian is given by 1 ∂ p ij ∂f ij ij k ∆f = −p det gij g = − g ∂i∂j − g Γij∂k f. det gij ∂yj ∂yi

On Euclidean space our Laplacian is

m X ∂2f ∆ = − ∂y2 i=1 i which is, up to a sign, the usual Laplacian on Rm. With our sign convention, the Laplacian is sometimes called the ‘geometer’s Laplacian’ while the opposite sign convention yields the ‘analyst’s Laplacian’. The Laplacian on functions on a Rie- mannian manifold is sometimes called the Laplace-Beltrami operator to emphasize that there is a Riemannian metric involved.

2.11 Exercises

Exercise 1. Show that if (M1, g1) and (M2, g2) are Riemannian manifolds, then so is (M1 × M2, g1 + g2). Exercise 2. Let Rk −→ M be the trivial Rk bundle on M. Identify a section of Rk with a map M −→ Rk. Thus every section has a natural derivative given by a bundle map TM −→ T Rk. Use this derivative to define a flat connection on Rk. Exercise 3. It is tempting to think that the interior product and the exterior derivative will define a covariant derivative on T ∗M without having to use a metric, say by

∞ ∗ ∞ ∇e V ω = V y dω, for all ω ∈ C (M; T M),V ∈ C (M; TM).

Show that this is not a covariant derivative on T ∗M. 52 CHAPTER 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY

Exercise 4. Show that parallel transport determines the connection. Exercise 5. Consider the ﬂat torus Tm = Rm/Zm with the metric it inherits from Rm. Compute the curvature and geodesics of Tm. Exercise 6. If γ(t) is a geodesic on a Riemannian manifold show that γ0(t) has constant length. Exercise 7. a) Determine the geodesics on the sphere. b) All of the points of the sphere have the same injectivity radius, what is it? c) Consider the geodesic triangle on the sphere S2 with a vertex at the ‘north pole’ and other vertices on the ‘equator’, suppose that at the north pole the edge form an angle θ. Describe the parallel translation map along this triangle. Hint: Because the sides are geodesics, any vector parallel along the sides forms a constant angle with the tangent vector of the edge. Exercise 8. Give an example of two curves on a Riemannian manifold with the same endpoints and with diﬀerent parallel transport maps. Exercise 9. Consider the 2-dimensional sphere of radius r centered at the origin of R3, 2 3 Sr = {(y1, y2, y3) ∈ R : |y| = r}. 3 2 The Euclidean metric on R induces the ‘round’ metric on Sr. Compute the sectional curvatures of this metric. Hint: Use the spherical coordinates (θ, ψ) that parametrize the sphere via φ(θ, φ) = (r cos θ cos ψ, r cos θ sin φ, r sin θ).

Notice that any curve on S2 can be parametrized γ(t) = (θ(t), ψ(t)) and the length of γ0(t) is rp(θ0)2 + (ψ0)2 cos2 θ . 2 2 2 2 2 2 Hence the round metric in these coordinates is gSr = r dθ + r cos θdψ . Exercise 10. m One model of hyperbolic space of dimension m, (H , gHm ) is the space m m R+ = {(x, y1, . . . , ym−1) ∈ R : x > 0} with the metric dx2 + |dy|2 g m = . H x2 Compute the Christoﬀel symbols and the sectional curvatures of Hm. 2.11. EXERCISES 53

Exercise 11. (Riemann’s formula) Let K be a constant. Show that a neighborhood of the origin m in R with coordinates {y1, . . . , ym} endowed with the metric |dy|2 K 22 1 + 4 |y| has constant curvature K. Exercise 12. Use stereographic projection to write the metric on the sphere as a particular case of Riemann’s formula. Exercise 13. a) Show that any diﬀerential k-form α on M × [0, 1] has a ‘standard decomposition’

α = αt(s) + ds ∧ αn(s) where ds is the unique positively oriented unit 1-form on [0, 1] and, for each s ∈ [0, 1], αt(s) is a diﬀerential k-form on M and αn(s) is a diﬀerential (k − 1)-form on M. What is the standard decomposition of dα in terms of αt and αn? b) For each s ∈ [0, 1] let

is : M −→ M × [0, 1], is(p) = (p, s).

We can think of the pull-back of a diﬀerential form by is as the ‘restriction of a form on M × [0, 1] to M × {s}’. Consider the map π∗ that takes k-forms on M × [0, 1] to (k − 1)-forms on M given by

Z 1 π∗(α) = αn(s) ds. 0 Show that for any diﬀerential k-form α on M × [0, 1]

∗ ∗ i1α − i0α = dπ∗α + π∗(dα). c) Show that whenever [0, 1] 3 t 7→ ωt is a smooth family of closed k-forms on a manifold M, the de Rham cohomology class of ωt is independent of t. Exercise 14. (Poincar´e’sLemma) A manifold M is (smoothly) contractible if there is a point p0 ∈ M, and a smooth map H : M × [0, 1] −→ M such that

H(p, 1) = p, H(p, 0) = p0 for all p ∈ M. 54 CHAPTER 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY

H is known as a ‘contraction’ of M. a) Show that Rn is contractible and any manifold is locally contractible (i.e., every point has a contractible neighborhood). b) Let M be a contractible smooth n-dimensional manifold, and let ω be a closed diﬀerential k-form on M (i.e., dω = 0). Prove that there is a (k − 1)-form η on M such that dη = ω. ∗ ∗ ∗ (Hint: Let H be a contraction of M, and let α = H ω. Then ω = (H ◦ i1) ω = i1α ∗ and i0α = 0. Check that η = π∗α works.) Exercise 15. Let M be an n-dimensional manifold with boundary ∂M. A (smooth) retraction of M onto its boundary is a smooth map φ : M −→ ∂M such that

φ(ζ) = ζ, whenever ζ ∈ ∂M.

Prove that if M is compact and oriented, then there do not exist any retractions of M onto its boundary. R (Hint: Let ω be an (n − 1)-form on ∂M with ∂M ω > 0 (how do we know there is such an ω?), notice that dω = 0 (why?). Assume that φ is a retraction of M onto its boundary, and apply Stokes’ theorem to φ∗ω to get a contradiction.) If φ : M −→ ∂M is a continuous retraction, one can approximate it by a smooth retraction, so there are in fact no continuous retractions from M to ∂M. Exercise 16. (Brouwer fixed point theorem) Let B be the closed unit ball in Rn, show that any continuous map F : B −→ B has a fixed point (i.e., there is a point ζ ∈ B such that F (ζ) = ζ). (Hint: If F : B −→ B is a smooth map without any fixed points, let φ(ζ) be the point on ∂B where the ray starting at F (ζ) and passing through ζ hits the boundary. Show that φ is continuous and then apply the continuous version of the previous exercise.) Exercise 17. i Let Vi be a local orthonormal frame for TM with dual coframe θ . Show that X X i d = eθ ∇Vi , δ = − iVi ∇Vi .

2.12 Bibliography

My favorite book for Riemannian geometry is the eponymous book by Gallot, Hulin, and Lafontaine. Some other very good books are Riemannian geometry and geometric analysis by Jost, and Riemannian geometry by Petersen. 2.12. BIBLIOGRAPHY 55

The proof of the Nash embedding theorem by Nash features a powerful extension of the implicit function theorem which has found many other applications. For an account of the embedding theorem, including both this extension and a method due to Matthias Gunther to circumvent it, see Andrews, Ben, Notes on the isometric embedding problem and the Nash-Moser implicit function theorem, Surveys in analysis and operator theory (Canberra, 2001), 157–208, Proc. Centre Math. Appl. Austral. Nat. Univ., 40, Austral. Nat. Univ., Canberra, 2002. Perhaps the best place to read about connections is the second volume of Spi- vak, A comprehensive introduction to differential geometry which includes transla- tions of articles by Gauss and Riemann. Spivak does a great job of presenting the various approaches to connections and curvature. In each approach, he explains how to view the curvature as the obstruction to flatness. An excellent concise introduction to Riemannian geometry, which we follow when discussing geodesics, can be found in Milnor, Morse theory. Another excellent treatment is in volume one of Taylor, Partial differential equations. 56 CHAPTER 2. A BRIEF INTRODUCTION TO RIEMANNIAN GEOMETRY Chapter 3

Laplace-type and Dirac-type operators

3.1 Principal symbol of a diﬀerential operator

Let M be a smooth manifold, E −→ M,F −→ M real vector bundles over M, and D a differential operator of order ` acting between sections of E and sections of F, D ∈ Diff`(M; E,F ). In any coordinate chart V of M that trivializes both E and F, D is given by X α D = aα(y)∂y |α|≤` ∞ where {y1, . . . , ym} are the coordinates on V and aα ∈ C (V; Hom(E,F ) V ). This expression for D changes quite a bit if we choose a different set of coordinates. However it turns out we can make invariant sense of the ‘top order part’ of D, X α aα(y)∂y , |α|=` as a constant coefficient differential operator on each fiber of the tangent space of M. It is more common to take the dual approach and interpret this as a function on 1 m the cotangent bundle of M as follows: Let {y1, . . . , ym, dy , . . . , dy } be the induced coordinates on the cotangent bundle of M, π : T ∗M −→ M, over V and define the j principal symbol of D at the point ξ = ξjdy of the cotangent bundle, to be

` X α ∞ ∗ ∗ σ`(D)(y, ξ) = i aα(y)ξ ∈ C (T M; π Hom(E,F )). |α|=` That is, we just ignore all the terms in the expression for the diﬀerential operator

57 58 CHAPTER 3. LAPLACE-TYPE AND DIRAC-TYPE OPERATORS of order less than ` and replace differentiation with a polynomial in the cotangent variables. Why the factor of i? A key fact that we will exploit later is that the Fourier n transform on R interchanges differentiation by yj with multiplication by iξj, Z Z −iy·ξ −iy·ξ e ∂yj f(y) dy = e (iξj)f(y) dy. Rn Rn Of course here one should be working with complex valued functions, and soon we will be. For the moment, just think of the power of i as a formal factor. Another way of defining the principal symbol, which has the advantage of ∗ being obviously independent of coordinates, is: At each ξ ∈ Tp M, choose a function f such that df(p) = ξ and then set

−` −itf itf (3.1) σ`(D)(p, ξ) = lim t e ◦ D ◦ e (p) ∈ Hom(Ep,Fp). t→∞

By the Leibnitz rule, e−itf ◦ D ◦ eitf is a polynomial in t of order ` and the top order term at p is of the form

` X α1 αm ` X α1 αm (it) aα(p)(∂y1 f)(p) ··· (∂ym f)(p) = (it) aα(p)ξ1 ··· ξm |α|=` |α|=` which shows that (3.1) is independent of the choice of f and hence well-defined. Notice that the principal symbol of a differential operator is not just a smooth function on T ∗M, but in fact a homogeneous polynomial of order ` on each fiber of T ∗M, we will denote these functions by

∗ ∗ ∞ ∗ ∗ P`(T M; π Hom(E,F )) ⊆ C (T M; π Hom(E,F )).

Replacing the exponentials in (3.1) with their Taylor expansions it is easy to ﬁnd that, e.g.,

t2 e−itf ◦ D ◦ eitf = D + it[D, f] − [[D, f], f] + O(t3). 2 Hence for an operator of order one or two, it is easy to carry out the diﬀerentiations in (3.1) and see that 1 σ (D)(p, ξ) = i [D, f] , σ (D)(p, ξ) = − [[D, f] , f] , 1 2 2 respectively, for any f such that df(p) = ξ. A similar formula holds for higher order operators. 3.1. PRINCIPAL SYMBOL OF A DIFFERENTIAL OPERATOR 59

The principal symbol of order ` vanishes if and only if the operator is actually of order ` − 1, so we have a short exact sequence

`−1 ` σ` ∗ ∗ 0 −→ Diﬀ (M; E,F ) −→ Diﬀ (M; E,F ) −−→ P`(T M; π Hom(E,F )) −→ 0.

One very important property of the principal symbol map is that it is a homomorphism. That is, if

∞ ∞ ∞ ∞ D1 : C (M; E1) −→ C (M; E2),D2 : C (M; E2) −→ C (M; E3) are diﬀerential operators of orders `1 and `2 respectively, then

∞ ∞ D2 ◦ D1 : C (M; E1) −→ C (M; E3) is a diﬀerential operator with symbol

σ`1+`2 (D2 ◦ D1)(p, ξ) = σ`2 (D2)(p, ξ) ◦ σ`1 (D1)(p, ξ), as follows directly from (3.1).

This is especially useful for scalar operators on Ei = R since then multiplication ∗ in P`1+`2 (T M) is commutative, so we always have

σ`1 (D1)(p, ξ) ◦ σ`2 (D2)(p, ξ) = σ`2 (D2)(p, ξ) ◦ σ`1 (D1)(p, ξ), even though D1 ◦ D2 is generally diﬀerent from D2 ◦ D1. It is also true that the symbol map respects adjoints in that, if D and D∗ are formal adjoints, then ∗ ∗ σ`(D ) = σ`(D) where on the right we take the adjoint in Hom, so in local coordinates just the matrix transpose. We can use (3.1) to ﬁnd the symbols of d and δ by noting that

[d, f] ω = d(fω) − fdω = edf ω,

[δ, f] ω = δ(fω) − fδω = −idf ] ω hence k ∗ k+1 ∗ σ1(d)(p, ξ) = ieξ :Λ Tp M −→ Λ Tp M k ∗ k−1 ∗ σ1(δ)(p, ξ) = −iiξ] :Λ Tp M −→ Λ Tp M. Then the symbol of the Hodge Laplacian is

2 (3.2) σ2(∆k)(p, ξ) = (σ1(d)σ1(δ) + σ1(δ)σ1(d)) (p, ξ) = eξiξ] + iξ] eξ = |ξ|g 60 CHAPTER 3. LAPLACE-TYPE AND DIRAC-TYPE OPERATORS

k ∗ 2 i.e., it is given by multiplying vectors in Λ Tp M by the scalar |ξ|g.

A diﬀerential operator is called elliptic if σ`(D)(p, ξ) is invertible at all ξ 6= 0. Hence ∆k is an elliptic diﬀerential operator, d and δ are not elliptic, and d ± δ are elliptic. One of the main aims of this course is to understand elliptic operators. One particularly interesting property of an elliptic operator D on a compact manifold is that its ‘index’

ind D = dim ker D − dim ker D∗ is well-defined (in that both terms in the difference are finite) and remarkably stable, as we will show below. Before doing that, however, we will start by constructing examples of elliptic operators.

3.2 Laplace-type operators

A second-order diﬀerential operator on a Rk-bundle E −→ M over a Riemannian ∗ manifold (M, g) is of Laplace-type if its principal symbol at ξ ∈ Tp M is scalar 2 multiplication by |ξ|g. Thus by (3.2) the Hodge Laplacian ∆k is, for every k, a Laplace-type operator. A general Laplace-type operator H can be written in any local coordinates as

ij k k ∞ H = −g ∂yi ∂yj + a ∂yk + b, with a , b ∈ C (V; Hom(E,E)).

If E −→ M is a vector bundle with connection ∇E, then we can form the composition

E T ∗M⊗E C∞(M; E) −−−→C∇ ∞(M; T ∗M ⊗ E) −−−−−−−→C∇ ∞(M; T ∗M ⊗ T ∗M ⊗ E) and then take the trace over the two factors of T ∗M to deﬁne the connection Laplacian,

∆∇ = − Tr(∇T ∗M⊗E∇E): C∞(M; E) −→ C∞(M; E).

(This is also known as the Bochner Laplacian.) Explicitly, we have

(∇T ∗M⊗E∇Es)(V,W ) = ∇E ∇E − ∇E s V W ∇V W and so in local coordinates we have ∆∇s = − gij∇E ∇E − gijΓk ∇E s. ∂yi ∂yj ij ∂yk 3.3. DIRAC-TYPE OPERATORS 61

Thus this is a Laplace-type operator. Directly from the formula in local coordinates we see that the Laplacian on functions is the connection Laplacian for the Levi-Civita connection ∆∇ = ∆. It turns out that any Laplace-type operator on a bundle E −→ M is equal to a connection Laplacian plus a zero-th order diﬀerential operator1. Proposition 7. Let (M, g) be a Riemannian manifold, E −→ M a vector bundle, and H a Laplace-type operator on sections of E. There exists a unique connection ∇E on E and a unique section Q ∈ C∞(M; Hom(E)) such that

H = ∆∇ + Q.

3.3 Dirac-type operators

For the purpose of ‘quantizing’ electromagnetism, the physicist P. A. M. Dirac sought out a diﬀerential operator D with the property that D2 is the Laplacian (albeit in ‘Lorentzian signature’). He was working over R4 and realized that if he allowed his operator to act on vectors instead of scalars then he could carry this out. Indeed, he found that there are constant four-by-four matrices γ0, γ1, γ2, γ3 such that

i 2 X 2 (γ ∂i) = Id4×4 ∂i .

Expanding the left hand side, now known as the Dirac operator, shows that it suffices to find matrices such that ( 2 Id if i = j γiγj + γjγi = 4×4 0 if i 6= j and it is simple enough to find these matrices. The fact that he needed to pass to vector valued functions led Dirac to the introduction of ‘spinors’. Much later Atiyah and Singer, in studying the integrality of an a priori ratio- nal number, rediscovered the Dirac operator in a general form on manifolds. This notion has been of great utility in physics and is central to most analytic approaches to the Atiyah-Singer index theorem.

Let E −→ M and F −→ M be Rk-vector bundles over a Riemannian manifold (M, g). A first order differential operator D ∈ Diff1(M; E,F ) is a Dirac-type

1For the proof of this proposition see, e.g., Proposition 2.5 of Nicole Berline, Ezra Get- zler, Michele` Vergne Heat kernels and Dirac operators. Corrected reprint of the 1992 original. Grundlehren Text Editions. Springer-Verlag, Berlin, 2004. x+363 pp. ISBN: 3-540-20062-2 62 CHAPTER 3. LAPLACE-TYPE AND DIRAC-TYPE OPERATORS operator if D∗D and DD∗ are Laplace-type operators. If E = F and D = D∗, we say that D is a symmetric Dirac-type operator. If D ∈ Diﬀ1(M; E,F ) is a Dirac-type operator, then

0 D∗ D = ∈ Diﬀ1(M; E ⊕ F ) e D 0 is a symmetric Dirac-type operator. We have seen one example of a symmetric Dirac-type operator so far, the de Rham operator:

m M d + δ ∈ Diff1(M;Ω∗(M)), where Ω∗(M) = Ωk(M) = C∞(M;Λ∗T ∗M) k=0 Indeed, we know that (d + δ)∗ = d + δ and (d + δ)2 = ∆, the Hodge Laplacian on forms of all degrees. Analogously to how Dirac found that he needed to pass to vector-valued functions to find a differential operator that squared to the Laplacian, we will show that the existence of a Dirac-type operator on sections of a bundle E places strong conditions on E. Indeed, we can use the symbol of an arbitrary symmetric Dirac operator D to uncover some algebraic structure on E. For every p ∈ M, and ξ ∈ TpM, let

θ(p, ξ) = σ1(D)(p, ξ): Ep −→ Ep, and, when the point p is ﬁxed, let θ(ξ) = θ(p, ξ). Since D is symmetric, we have

θ(ξ) = θ(ξ)∗ and since it is a Dirac-type operator, we have

θ(ξ)2 = g(ξ, ξ) Id .

∗ We can ‘polarize’ this identity by applying it to ξ + η ∈ TpM ,

θ(ξ + η)2 = θ(ξ)2 + θ(ξ)θ(η) + θ(η)θ(ξ) + θ(η)2 = g(ξ + η, ξ + η) Id = (g(ξ, ξ) + 2g(ξ, η) + g(η, η)) Id and hence θ(ξ)θ(η) + θ(η)θ(ξ) = 2g(ξ, η) Id .

More generally, suppose we have two vector spaces V and W, a quadratic form q on V, and a linear map cl : V −→ End(W ) 3.3. DIRAC-TYPE OPERATORS 63 such that2

(3.3) cl (v) cl (v) = −q(v) Id . We call cl Cliﬀord multiplication by elements of V. (Note that we put no restrictions on q, it need not be positive or non degenerate.) 1 As above, we can polarize (3.3). Deﬁne q(v, w) = 2 (q(v + w) − q(v) − q(w)) and apply (3.3) to v + w to obtain

(3.4) cl (v) cl (w) + cl (w) cl (v) = −2q(v, w) Id .

The map cl canonically extends to the Cliﬀord algebra of (V, q), which is deﬁned as follows. First let O V = R ⊕ V ⊕ (V ⊗ V ) ⊕ (V ⊗ V ⊗ V ) ⊕ ... and note that the map cl extends to a map clb : N V −→ End(W ). This extension vanishes on elements of the form v ⊗ w + w ⊗ v + 2q(v, w) Id and hence on the ideal generated by these elements n O o Icl (V ) = span κ ⊗ (v ⊗ w + w ⊗ v + 2q(v, w) Id) ⊗ χ : κ, χ ∈ V .

The quotient is the Cliﬀord algebra of (V, q), O Cl(V, q) = V/Icl (V ), it has an algebra structure induced by the tensor product in N V which satisﬁes v · w + w · v = −2q(v, w) whenever v, w ∈ V. By construction, Cl(V, q) has the universal property that any linear map cl : V −→ End(W ) satisfying (3.3) extends to a unique algebra homomorphism cl : Cl(V, q) −→ End(W ), with cl (1) = Id .

This universal property makes the Cliﬀord algebra construction functorial. Indeed, if (V, q) and (V 0, q0) are vector spaces with quadratic forms and T : V −→ V 0 is a linear map such that T ∗q0 = q then there is an induced algebra homomorphism

Te : Cl(V, q) −→ Cl(V 0, q0).

2Some authors use a diﬀerent convention, namely cl (v) cl (v) = q(v) Id . These are related by multiplication by i, or by replacing q with −q. 64 CHAPTER 3. LAPLACE-TYPE AND DIRAC-TYPE OPERATORS

Uniqueness of Te shows that T]◦ S = Te ◦ Se for any quadratic-form preserving linear map S. One advantage of functoriality is that we can extend this construction to vector bundles: Given a vector bundle F −→ M and a bundle metric gF there is a vector bundle Cl(F, gF ) −→ M whose ﬁber over p ∈ M is Cl(Fp, gF (p)). In particular, coming back to our ﬁrst order Dirac-type operator D ∈ C∞(M; E), we can conclude that its principal symbol σ1(D) induces an algebra bundle morphism (3.5) cl : Cl(T ∗M, g) −→ End(E) with cl (1) = Id, cl (ξ) = iθ(ξ) for all ξ ∈ T ∗M.

A bundle E −→ M together with a map (3.5) is called a Cliﬀord module. (It should really be called a bundle of Cliﬀord modules, but the abbreviation is common.) In the case of the de Rham operator we have

∗ ∗ ∗ ∗ σ1(d + δ)(p, ξ) = i(eξ − iξ] ):Λ Tp M −→ Λ Tp M, and an induced algebra homomorphism

∗ ∗ ∗ (3.6) cl : Cl(T M, g) −→ End(Λ Tp M).

3.4 Cliﬀord algebras

Let us consider the structure of the algebra Cl(V, q) associated to a quadratic form q on V. If e1, . . . , ek is a basis for V then any element of Cl(V, q) can be written as a polynomial in the ei. In fact, using (3.4) we can choose the polynomial so that no exponent larger than one is used, and so that the ei occur in increasing order; thus any ω ∈ Cl(V, q) can be written in the form

X α1 α2 αm (3.7) ω = aαe1 · e2 ··· em

αj ∈{0,1} with real coeﬃcients aα. This induces a ﬁltration on Cl(V, q), Cl(k)(V, q) = {ω as in (3.7), with |α| ≤ k}, which is consistent with the algebraic structure,

Cl(k)(V, q) · Cl(`)(V, q) ⊆ Cl(k+`)(V, q). 3.4. CLIFFORD ALGEBRAS 65

The map (3.6), which was induced by the symbol of the de Rham operator, makes sense algebraically for any Cliﬀord algebra,

M : Cl(V, q) −→ End(Λ∗V ).

There is a distinguished element 1 ∈ Λ∗V, and evaluation at one induces

(3.8) Mf : Cl(V, q) −→ Λ∗V, Mf(ω) = M(ω)(1).

Clearly if we think of v ∈ V as an element of Cl(V, q), we have Mf(v) = v. Moreover, it is not hard to see that Mf is an isomorphism of vector spaces. In fact, Λ∗V is the graded algebra associated to the ﬁltered algebra Cl(V, q), i.e.,

Cl(k)(V, q)/ Cl(k−1)(V, q) ∼= ΛkV.

If we set q = 0, then we recognize Cl(V, 0) = Λ∗V as algebras. In general the algebras Cl(V, q) and Λ∗V only coincide as vector spaces. N The Cliﬀord algebra Cl(V, q) is naturally Z2-graded. Indeed, note that V N N has a Z2-grading as an algebra with even part (V ⊗V ) and odd part V ⊗ (V ⊗V ). N Since the ideal Icl (V ) is contained in the even part of V, the Z2-grading descends to Cl(V, q). Thus we have a decomposition

Cl(V, q) = Cl(V, q)0 ⊕ Cl(V, q)1 compatible with Cliﬀord multiplication in that

Cl(V, q)i · Cl(V, q)j = Cl(V, q)i+j mod 2.

Moreover as vector spaces we have

∼ even ∼ odd Cl(V, q)0 = Λ V, Cl(V, q)1 = Λ V.

Let us identify a few of the low-dimensional Cliﬀord algebras. From the vector space isomorphism Cl(V, q) ∼= Λ∗V we know that dim Cl(V, q) is a power of two. Let us agree that R itself, with its usual algebra structure, is the unique one-dimensional Cliﬀord algebra.

In dimension two, let us identify Cl(R, gR) and Cl(R, −gR). Let e1 be a unit vector in R, so that both of these Clifford algebras are generated by 1, e1. In the 2 first case, where the quadratic form is the usual Euclidean norm, we have e1 = −1, and it is easy to see that ∼ Cl(R, gR) = C 66 CHAPTER 3. LAPLACE-TYPE AND DIRAC-TYPE OPERATORS as R-algebras. In the second case, where the quadratic form is minus the usual 1 1 Euclidean norm, the elements v = 2 (1 + e1) and w = 2 (1 − e1) satisfy v · w = 0, v · v = v, w · w = w and we can identify ∼ Cl(R, −gR) = R ⊕ R as R-algebras, where the summands consist of multiples of v and w, respectively. 2 2 Next consider Cl(R , ±gR2 ). Let e1, e2 be an orthonormal basis of (R , gR2 ). 2 Using the universal property, we can define an isomorphism f between Cl(R , ±gR2 ) and a four dimensional algebra A by specifying the images of e1 and e2 and verifying that f(ei) · f(ei) = ∓1, i ∈ {1, 2}. 2 For Cl(R , −gR2 ) the map 0 1 1 0 e 7→ , e 7→ 1 1 0 2 0 −1 extends linearly to an algebra isomorphism 2 ∼ 2 Cl(R , −gR2 ) = End(R ). 2 Similarly, we can identify Cl(R , gR2 ) with the quaternions by extending the map

e1 7→ i, e2 7→ j

(so, e.g., e1 · e2 maps to k). Thus (real) Clifford algebras generalize both the complex numbers and the quaternions. It is possible to compute all of the Clifford algebras of finite dimensional vector spaces3 and an incredible fact is that they recur (up to ‘suspension’) with periodicity eight in the sense that, for example, n+8 n 16 Cl(R , gRn+8 ) = Cl(R , gRn ) ⊗ End(R ).

It is useful to consider the complexiﬁed Cliﬀord algebra

Cl(V, q) = C ⊗ Cl(V, q) as these complex algebras have a simpler structure than Cl(V, q). In particular, we next show that they repeat with periodicity two. m Let us restrict attention to V = R and q = gRm , and abbreviate m m Cl(R , gRm ) = Cl(m), Cl(R , gRm ) = Cl(m). 3See for instance, H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, Spin geometry. Princeton Mathematical Series, 38. Princeton University Press, Princeton, NJ, 1989. 3.4. CLIFFORD ALGEBRAS 67

Proposition 8. There are isomorphisms of complex algebras

∼ 2 ∼ 2 Cl(1) = C , Cl(2) = End(C ) ∼ and Cl(m + 2) = Cl(m) ⊗ Cl(2) and so ∼ 2k ∼ 2k 2k Cl(2k) = End(C ), Cl(2k + 1) = End(C ) ⊕ End(C ).

2 Proof. The ﬁrst two identities are just the complexiﬁcations of Cl(R, gR) and Cl(R , gR2 ) computed above. To prove periodicity, embed Rm+2 into Cl(m) ⊗ Cl(2) by picking an orthonormal basis {e1, . . . , em+2} and taking

ej 7→ icl (ej) ⊗ cl (em+1)cl (em+2) for 1 ≤ j ≤ m,

ej 7→ 1 ⊗ cl (ej) for j > m.

Then the universal property of Cl(m + 2) leads to the isomorphism above.

Given a vector space with quadratic form (V, q), let e1, . . . , em be an oriented orthonormal basis and deﬁne the chirality operator

m d e Γ = i 2 e1 ··· em ∈ Cl(V, q). This is an involution (i.e., Γ2 = Id), and is independent of the choice of oriented orthonormal basis used in its deﬁnition. Moreover,

Γ · v = −v · Γ if dim V even, Γ · v = v · Γ if dim V odd.

From the explicit description of Cl(m), we can draw some immediate consequences about the representations of this algebra4. Indeed if m is even, say m = 2k, k then Cl(2k) = End(C2 ) and up to isomorphism there is a unique irreducible Cl(2k)- module, ∆/ , and it has dimension 2k,

Cl(2k) = End(∆)/ .

k k If m is odd, say m = 2k + 1, then Cl(2k + 1) = End(C2 ) ⊕ End(C2 ) and up to isomorphism there are two irreducible Cl(2k + 1) modules ∆/ ±,

Cl(2k + 1) = End(∆/ +) ⊗ End(∆/ −) 4For an alternate approach, ﬁrst determining the representations and then deducing the explicit description of Cl(m), see Chapter 4 of John Roe Elliptic operators, topology and asymptotic methods. Second edition. Pitman Research Notes in Mathematics Series, 395. Longman, Harlow, 1998. 68 CHAPTER 3. LAPLACE-TYPE AND DIRAC-TYPE OPERATORS each of dimension 2k distinguished by

Γ = ±1. ∆/ ±

Any ﬁnite dimensional complex representation of Cl(2k) must be a direct sum of copies of ∆/ , or equivalently of the form

W = ∆/ ⊗ W 0 for some auxiliary ‘twisting’ vector space W 0. We can recover W 0 from W via

0 W = HomCl(2k)(∆/ ,W ), that is, W 0 can be identiﬁed with the homomorphisms from ∆/ to W that commute with Cliﬀord multiplication. In the same fashion we have

0 End(W ) = Cl(2k) ⊗ End(W ) = Cl(2k) ⊗ EndCl(2k)(W ).

One can similarly decompose representations of Cl(2k + 1) in terms ∆/ ±. Although Cl(2k) = End(∆)/ follows directly from the proposition, it can be useful to have a more concrete representative of ∆/ . So let us describe how to con- m struct ∆/ from (R , gRm ) or indeed any m-dimensional vector space V with a deﬁnite quadratic form, q. Let e1, . . . , em be an oriented orthonormal basis of V and consider the subspaces of V ⊗ C, 1 K = span{ηj = √ (e2j−1 − ie2j): j ∈ {1, . . . , m}} 2 1 K = span{η = √ (e2j−1 + ie2j): j ∈ {1, . . . , m}}. j 2

We can use the fact that V ⊗ C = K ⊕ K to deﬁne a natural homomorphism

∗ cl : V ⊗ C −→ End(Λ K) (3.9) √ √ cl (k) = 2 (ek), cl k = 2 (ik) for k ∈ K.

This satisfies the Clifford relations and so extends to a homomorphism of Cl(V, q) into End(Λ∗K). Comparing dimensions, we see this is in fact an isomorphism so we can take ∆/ = Λ∗K. We refer to this representation as the spin representation. Below, we will use the consequence that the metric q on V induces a Hermitian metric on ∆/ compatible with Clifford multiplication. 3.5. HERMITIAN VECTOR BUNDLES 69 3.5 Hermitian vector bundles

Recall that on a complex vector space V a Hermitian inner product is a map

h·, ·i : V × V −→ C that is linear in the ﬁrst entry and satisﬁes

hv, wi = hw, vi, for all v, w ∈ V.

It is positive deﬁnite if hv, vi ≥ 0, and moreover hv, vi = 0 implies v = 0.

A Hermitian metric hE on a complex vector bundle E −→ M is a smooth Hermitian inner product on the ﬁbers of E. Compatible connections are deﬁned in the usual way, namely, a connection ∇E on E is compatible with the Hermitian metric hE if

E E ∞ dhE(σ, τ) = hE(∇ σ, τ) + hE(σ, ∇ τ), for all σ, τ ∈ C (M; E).

One natural source of complex bundles is the complexiﬁcation of real bundles. If F −→ M is a real bundle of rank k then F ⊗ C is a complex bundle of rank k. We can think of F ⊗ C as F ⊕ F together with a bundle map

J : F ⊕ F / F ⊕ F (u, v) / (−v, u) satisfying J 2 = − Id . (Such a map is called an almost complex structure.) A bundle metric gF on F −→ M induces a Hermitian metric hF on F ⊗ C by

hF ((u1, v1), (u2, v2)) = gF (u1, u2) + gF (v1, v2) + i (gF (u2, v1) − gF (u1, v2))

F thus the ‘real part’ of hF is gF . Similarly a connection on F, ∇ , induces a connection F on F ⊗ C, namely the induced connection on F ⊕ F. If ∇ is compatible with gF , then the induced connection on F ⊗ C is compatible with hF . A Riemannian metric on a manifold induces a natural Hermitian inner product on C∞(M; C) or more generally on sections of a complex vector bundle E endowed with a Hermitian bundle metric, hE. In the former case, Z ∞ hf, hi = f(p)h(p) dvolg for all f, h ∈ C (M; C) M and in the latter, Z ∞ hσ, τiE = hE(σ, τ) dvolg for all σ, τ ∈ C (M; E) M 70 CHAPTER 3. LAPLACE-TYPE AND DIRAC-TYPE OPERATORS

(though the integrals need not converge if M is not compact). A complex vector bundle E over a Riemannian manifold (M, g) is a complex Cliﬀord module if there is a homomorphism, known as Cliﬀord multiplication,

∗ cl : Cl(T M, g) −→ End(E).

Cliﬀord multiplication is compatible with the Hermitian metric hE if

∞ ∗ ∞ hE(cl (θ) σ, τ) = −hE(σ, cl (θ) τ), for all θ ∈ C (M; T M), σ, τ ∈ C (M; E). Cliﬀord multiplication is compatible with a connection ∇E if

[∇E, cl (θ)] = cl (∇θ) , where on the right hand side we are using the Levi-Civita connection. A connection compatible with Clifford multiplication is known as a Clifford connection. Let us define a Dirac bundle to be a complex Clifford module, E −→ M, with E a Hermitian metric hE and a metric connection ∇ , both of which are compatible with Clifford multiplication. Any Dirac bundle has a generalized Dirac operator

1 ðE ∈ Diﬀ (M; E) given by E ∞ ∇ ∞ ∗ icl ∞ ðE : C (M; E) −−−→C (M; T M ⊗ E) −−−→C (M; E). Note that this is indeed a Dirac-type operator, since

E [ðE, f] = [icl ◦ ∇ , f] = icl (df) shows that 2 2 σ1(ðE)(p, ξ) = cl (ξ) , σ2(ðE)(p, ξ) = |ξ|g. It is also a self-adjoint operator thanks to the compatibility of the metric with the connection and Cliﬀord multiplication. 2 We know that ðE is a Dirac-type operator and hence can be written as a connection Laplacian up to a zero-th order term. There is an explicit formula for this term which is often very useful. First given two vector ﬁelds V,W ∈ C∞(M; TM), the curvature R(V,W ) can be thought of as a two-form

Re(V,W ) ∈ Ω2(M), Re(V,W )(X,Y ) = g(R(V,W )Y,X) for all X,Y ∈ C∞(M; TM) and so we have an action of R(V,W ) on E by cl Re(V,W ) . 3.5. HERMITIAN VECTOR BUNDLES 71

Next note that for any complex Cliﬀord module, E, (for simplicity over an even dimensional manifold) and a contractible open set U ⊆ M we have

∼ / ∗ / E U = ∆ ⊗ HomCl(T M)(∆,E U ) where ∆/ denotes the trivial bundle ∆/ × U −→ U. Correspondingly we have

∗ End(E) = Cl(T M) ⊗ EndCl(T ∗M)(E) locally. However both sides of this equality deﬁne bundles over M and the natural map from the right hand side to the left hand side is locally an isomorphism so these are in fact globally isomorphic bundles. Proposition 9. The curvature RE ∈ Ω2(M; End(E)) of a Cliﬀord connection ∇E on ∗ E −→ M decomposes under the isomorphism End(E) = Cl(T M) ⊗ EndCl(T ∗M)(E) into a sum

E 1 E R (V,W ) = 4 cl Re(V,W ) + Rb (V,W ),

∗ E cl Re(V,W ) ∈ Cl(T M), Rb (V,W ) ∈ EndCl(T ∗M)(E) for any V,W ∈ C∞(M; TM).

E 2 We call Rb ∈ Ω (M; EndCl(T ∗M)(E)) the twisting curvature of the Cliﬀord connection ∇E.

Proof. From the compatibility of the connection with Cliﬀord multiplication, and the deﬁnition of the curvature in terms of the connection, it is easy to see that for any V,W ∈ C∞(M; TM), ω ∈ C∞(M; T ∗M), and s ∈ C∞(M; E),

RE(V,W )cl (ω) s = cl (ω) RE(V,W )s + cl (R(V,W )ω) s where R(V,W )ω denotes the action of R(V,W ) as a map from one-forms to one- forms. We can write the last term as a commutator

1 1 cl (R(V,W )ω) s = 4 cl Re(V,W ) cl (ω) s − 4 cl (ω) cl Re(V,W ) s

i ∗ since we have, in any local coordinates θ for T M with dual coordinates ∂i, cl Re(V,W ) cl (θp) − cl (θp) cl Re(V,W ) i j p p i j = g(R(V,W )∂i, ∂j)cl θ · θ · θ − g(R(V,W )∂i, ∂j)cl θ · θ · θ j i = −2g(R(V,W )∂p, ∂j)cl θ + 2g(R(V,W )∂i, ∂p)cl θ i = 4g(R(V,W )∂i, ∂p)cl θ . 72 CHAPTER 3. LAPLACE-TYPE AND DIRAC-TYPE OPERATORS

So we conclude that E 1 R − 4 cl (R) commutes with Cliﬀord multiplication by one-forms, and hence it commutes with all Cliﬀord multiplication.

2 ∇ With this preliminary out of the way we can now compare ðE and ∆ . The proof is through a direct computation in local coordinates5.

Theorem 10 (Lichnerowicz formula). Let E −→ M be a Dirac bundle, ∆∇ the E E0 2 Bochner Laplacian of the Cliﬀord connection ∇ ,R ∈ Ω (M; EndCl(T ∗M)(E)) the E twisting curvature of ∇ and scalM the scalar curvature of M, then

2 ∇ E0 1 ðE = ∆ + cl Rb + 4 scalM

i where we deﬁne (in local coordinates ∂i for TM and dual coordinates θ )

E0 X E i j cl Rb = Rb (∂i, ∂j)cl θ · θ . i

A Z2-graded Dirac bundle is a Dirac bundle E with a decomposition

E = E+ ⊕ E− that is orthogonal with respect to the metric, parallel with respect to the connection, and odd with respect to the Cliﬀord multiplication – by which we mean that

cl (v) interchanges E+ and E−, for all v ∈ T ∗M.

We can equivalently express this in terms of an involution on E :A Z2 grading on a Dirac bundle is γ ∈ C∞(M; End(E)) satisfying γ2 = Id, γcl (v) = −cl (v) γ, ∇Eγ = 0, γ∗ = γ.

We make E into a Z2-graded Dirac-bundle by deﬁning

E± = {v ∈ E : γ(v) = ±v}.

5See Theorem 3.52 of Nicole Berline, Ezra Getzler, Michele` Vergne Heat kernels and Dirac operators. Corrected reprint of the 1992 original. Grundlehren Text Edi- tions. Springer-Verlag, Berlin, 2004. x+363 pp. ISBN: 3-540-20062-2 3.6. EXAMPLES 73

If M is even dimensional, then any complex Cliﬀord module over M has a Z2-grading given by Cliﬀord multiplication by the chirality operator

γ = cl (Γ) but there are often other Z2-gradings as well. (Note that Γ is a well-deﬁned section of Cl(T ∗M) by naturality.)

In the presence of a Z2 grading, the generalized Dirac operator is a graded (or odd) operator,

− 0 ðE ± 1 ± ∓ + ∗ − ðE = + , with ðE ∈ Diﬀ (M; E ,E ), (ðE) = ðE. ðE 0

+ The index of the Dirac-type operator ðE is the diﬀerence

+ + − ind ðE = dim ker ðE − dim ker ðE. and often this is an interesting geometric, or even topological, quantity. Generalized Dirac operators show up regularly in geometry, we will describe several natural examples in the next section. A simple consequence of the Lichnerowicz formula is that

E0 1 + cl Rb + 4 scalM > 0 =⇒ ker ðE = {0} =⇒ ind ðE = 0.

+ E0 1 So the vanishing of the index of ðE is an obstruction to having cl Rb + 4 scalM > 0. This is most interesting when the former is a topological invariant.

3.6 Examples

3.6.1 The Gauss-Bonnet operator

Our usual example is the de Rham operator

∗ ∗ ðdR = d + δ :Ω (M) −→ Ω (M), although this is a real generalized Dirac operator, so our conventions will have to be slightly diﬀerent from those above. As we mentioned above, in this case the Cliﬀord module is the bundle of forms of all degrees,

Λ∗T ∗M −→ M. 74 CHAPTER 3. LAPLACE-TYPE AND DIRAC-TYPE OPERATORS

Cliﬀord multiplication by a 1-form ω ∈ C∞(M; T ∗M) is given by

cl (ω) = (eω − iω] ), this is compatible with both the Riemannian metric on Λ∗T ∗M and the Levi-Civita connection, and we have cl ◦ ∇ = d + δ. Thus Λ∗T ∗M is a (real) Dirac bundle with generalized Dirac operator the de Rham operator.

This Dirac bundle is Z2-graded. Let us define M M ΛevenT ∗M = ΛkT ∗M, ΛoddT ∗M = ΛkT ∗M k even kodd and correspondingly Ωeven(M) and Ωodd(M). Notice that Clifford multiplication ex- changes these subbundles and that the covariant derivative with respect to a vector field preserves these subbundles, so

even even odd odd odd even ðdR :Ω (M) −→ Ω (M), ðdR :Ω (M) −→ Ω (M).

even The operator ðdR is the Gauss-Bonnet operator. A form in the null space of d + δ is necessarily in the null space of the Hodge Laplacian ∆ = (d + δ)2 and so is a harmonic form. Conversely any compactly supported harmonic form is in the null space of d + δ. So on compact manifolds,

∗ even even odd ker(d + δ) = H (M), ind ðdR = dim H (M) − dim H (M). The latter is the Euler characteristic of the Hodge cohomology of M,

χHodge(M), an important topological invariant of M. Notice that on odd-dimensional compact manifolds, the Hodge star is an isomorphism between Heven(M) and Hodd(M), so the Euler characteristic of the Hodge cohomology vanishes. The Lichnerowicz formula in this case is known as the Weitzenb¨ock formula, and simpliﬁes to

∗ X k i j ∆k = ∇ ∇ − g(R(ei, ej)ek, e`)eθ ie` eθ iθ ijk`

k where ek is a local orthonormal frame with dual coframe θ . In particular on functions we get ∗ ∆0 = ∇ ∇ 3.6. EXAMPLES 75 and on one-forms we get Bochner’s formula

∗ ∆1 = ∇ ∇ + Ricf where Ricf is the endomorphism of T ∗M associated to the Ricci curvature via the metric. In particular, if the Ricci curvature is bounded below by a positive constant then ∆1 does not have null space, i.e.,

Ric > 0 =⇒ H1(M) = 0.

3.6.2 The signature operator

Now consider the complexiﬁcation of the diﬀerential forms on M,

∗ ∗ ∗ ∗ ΛCT M = Λ T M ⊗ C and the corresponding space of sections, Ω∗ (M). Cliﬀord multiplication is as before, C but now we can allow complex coeﬃcients,

1 cl (ω) = i (eω − iω] ), this is compatible with both the Riemannian metric on Λ∗ T ∗M (extended to a C Hermitian metric) and the Levi-Civita connection, and we have

icl ◦ ∇ = d + δ.

Since Λ∗ T ∗M is a complex Cliﬀord module, we have an involution given by C Cliﬀord multiplication by the chirality operator Γ. One can check that, up to a factor of i, Γ coincides with the Hodge star,

1 k(k−1)+ dim M (Γ) : Ωk (M) −→ Ωm−k(M), (Γ) = i 2 ∗ . cl C C cl If M is even dimensional, this induces a grading of Λ∗ T ∗M as a Dirac bundle, Z2 C

∗ ∗ ∗ ∗ ∗ ∗ ΛCT M = Λ+T M ⊕ Λ−T M. The de Rham operator splits as

− 0 ðsign d + δ = + ðsign 0

+ and we refer to ðsign as the Hirzebruch signature operator. 76 CHAPTER 3. LAPLACE-TYPE AND DIRAC-TYPE OPERATORS

The index of this operator

+ + − ind ðsign = dim ker ðsign − dim ker ðsign is again an important topological invariant of M. It is always zero if dim M is not a multiple of four, and otherwise it is equal to the signature of the quadratic form

1 dim M H 2 (M) / C R u / M u ∧ u

It is called the signature of M, sign(M).

3.6.3 The ∂ operator

A real manifold M of dimension m = 2k is a complex manifold if it has a collection of local charts parametrized by open sets in Ck for which the transition charts are holomorphic. This yields a decomposition of the complexiﬁed tangent bundle

1,0 0,1 (1,0) 0,1 TM ⊗ C = T M ⊕ T M, v ∈ Tp M ⇐⇒ v ∈ Tp M.

By duality there is a similar decomposition of the cotangent bundle

∗ 1,0 01, T M ⊗ C = Λ M ⊕ Λ M.

In local complex coordinates, z1, . . . , zm, which we can decompose into real and imaginary parts zj = xj + iyj. We can deﬁne

dzj = dxj + idyj, dzj = dxj − idyj

1,0 0,1 and then Λ M is spanned by {dzj} and Λ M is spanned by {dzj}. More generally we set

Λp,qM = Λp(Λ1,0M) ∧ Λq(Λ0,1M) and we have a natural isomorphism

k ∗ M p,q Λ T M ⊗ C = Λ M. p+q=k

We denote the space of sections of Λp,qM by Ωp,qM, and speak of forms of bidegree (p, q). 3.6. EXAMPLES 77

The exterior derivative naturally maps

d :Ωp,qM −→ Ωp+1,qM ⊕ Ωp,q+1M and we can decompose it into the Dolbeault operators

d = ∂ + ∂ ∂ :Ωp,qM −→ Ωp+1,qM, ∂ :Ωp,qM −→ Ωp,q+1M.

It follows from d2 = 0 that

2 ∂2 = ∂ = ∂∂ + ∂∂ = 0 and from the Leibnitz rule for d that ∂(ω ∧ η) = ∂ω ∧ η + (−1)|ω|ω ∧ ∂η ∂(ω ∧ η) = ∂ω ∧ η + (−1)|ω|ω ∧ ∂η.

In local coordinates, for ω ∈ Ωp,qM

X J K ω = αJK dz ∧ dz =⇒ |J|=p,|K|=q X αJK X αJK ∂ω = dz` ∧ dzJ ∧ dzK , and ∂ω = dz` ∧ dzJ ∧ dzK . ∂z` ∂z` `,|J|=p,|K|=q `,|J|=p,|K|=q

The Dolbeault complex is given by

(3.10) 0 −→ Ω0,0M −−→∂ Ω0,1M −−→∂ ... −−→∂ Ω0,mM −→ 0 and its cohomology is called the Dolbeault cohomology of M. We deﬁne the Hodge- Dolbeault cohomology to be

∗ ∗ H0,q(M) = {ω ∈ Ω0,pM :(∂∂ + ∂ ∂)ω = 0}.

As in the real case, we will show below that the Hodge-Dolbeault cohomology coincides with the Dolbeault cohomology. A complex Hermitian manifold M is a K¨ahler manifold if the splitting

1,0 0,1 TM ⊗ C = T M ⊕ T M is preserved by the Levi-Civita connection. On a K¨ahlermanifold, the bundle M Λ0,∗M = Λ0,kM k 78 CHAPTER 3. LAPLACE-TYPE AND DIRAC-TYPE OPERATORS is a Dirac bundle with respect to the Hermitian metric on M, the Levi-Civita connection, and the Cliﬀord multiplication √ cl (ω) = 2 (eω0,1 − i(ω1,0)] ).

The corresponding generalized Dirac operator is

√ ∗ 2 (∂ + ∂ ).

0,∗ There is a natural Z2 grading on Λ M given by Λ0,∗M = Λ0,evenM ⊕ Λ0,oddM.

The index of the corresponding non-self-adjoint operator is the Euler characteristic of the Hodge-Dolbeault cohomology.

3.6.4 The Dirac operator

If E −→ M is a complex Cliﬀord module over manifold M of even dimension, say m = 2k, then Cliﬀord multiplication

∗ cl : Cl(T M, g) −→ End(E) is, for each p ∈ M, a ﬁnite dimensional complex representation of Cl(2k) on Ep. We ∗ can decompose Ep in terms of the unique irreducible representation of Cl(Tp M), ∆/ ,

Ep = ∆/ ⊗ HomCl(2k)(∆/ ,Ep).

If it happens that Ep = ∆/ as Cliﬀord modules for all p ∈ M we say that E is a SpinC bundle over M. If a SpinC bundle satisﬁes the symmetry condition

HomCl(T ∗M)(E,E) is trivial we say that E is a spin bundle over M. We will generally denote a spin bundle over M by

(3.11) S/ −→ M.

Not every manifold admits a spin bundle, and some manifolds admit many. For example, a genus g Riemann surface admits 22g distinct spin bundles. All spheres in dimension greater than two admit spin bundles. All compact orientable manifolds of dimension three or less admit spin bundles. The complex projective space CPk admits a spin bundle if and only if k is odd. Additionally, any (almost) complex 3.6. EXAMPLES 79 manifold has a spinC bundle. The existence of spin bundles is discussed in supple- mentary section 3.7.

The Riemannian metric and Levi-Civita connection induce the structure of a Dirac bundle on any spin bundle. Indeed, we saw above that the metric on a 2k-dimensional vector space V with positive quadratic form induces a metric on the corresponding ∆/ so it suﬃces to describe the induced compatible connection. In a local coordinate chart U that trivializes T ∗M, if we choose an orthonormal frame 1 m θ , . . . , θ for TM U , the Levi-Civita connection is determined by an m × m matrix of one-forms ω deﬁned through

i j ∇θ = ωijθ .

(The fact that the connection respects the metric is reﬂected in the symmetry ωij = / −ωji.) We can use ω to deﬁne a one-form valued in End(S) U , namely

1 X i j ω = ωijcl θ · θ e 2 i

∞ ∞ ð : C (M; S/) −→ C (M; S/). As before Cliﬀord multiplication by the chirality operator Γ deﬁnes an involution exhibiting a Z2-grading on S,/ S/ = S/+ ⊕ S/−, and a corresponding splitting of the Dirac operator

− 0 ð ð = + . ð 0

The Lichnerowicz formula for the Dirac operator is particularly interesting since there is no twisting curvature, so we get 1 2 = ∆∇ + scal . ð 4 M Thus if the metric has positive scalar curvature then ð2 is strictly positive and in particular does not have null space. We will show that the index of ð+ is a topological 80 CHAPTER 3. LAPLACE-TYPE AND DIRAC-TYPE OPERATORS invariant. Thus the vanishing of ind ð+ is a topological necessary condition for the existence of a metric of positive scalar curvature6 on M.

3.6.5 Twisted Cliﬀord modules

If E −→ M is a Dirac bundle and F −→ M is any Hermitian bundle with a unitary connection, then E ⊗F −→ M is again a Dirac bundle if we take the tensor product metric and connection and extend Cliﬀord multiplication to E ⊗ F by acting on the ﬁrst factor. We say that this is the Dirac bundle E twisted by F. If M admits a spin bundle S/ −→ M, then any Dirac bundle E −→ M is a twist of S./ Indeed, we have

E = S/ ⊗ HomCl(T ∗M)(S,E/ ).

In the case of a twisted spin bundle S/ ⊗ F −→ M, the ‘twisting curvature’ in the Lichnerowicz formula is precisely the curvature of the connection on F. The index of a twist of the signature operator is known as a twisted signature. Twisted Dirac and twisted signature operators play an important rôlein the heat equation proof of the index theorem. For example if M admits a spin bundle S/ −→ M then the Clifford module Λ∗T ∗M ⊗ C must be a twist of S,/ and indeed ∗ ∗ ∼ ∗ ∼ / /∗ / ΛCT M = Cl(T M) = End(S) = S ⊗ S. The generalized Dirac operator d + δ on Λ∗ T ∗M is the twisted Dirac operator ∗ C obtained by twisting S/ with S/ . The two different Z2 gradings correspond to whether we treat S/∗ as a -graded bundle (which yields the splitting into ΛevenT ∗M and Z2 C ΛoddT ∗M) or as an ungraded bundle (which yields the splitting into Λ∗ T ∗M and C + ∗ ∗ Λ−T M).

3.7 Supplement: Spin structures

We can think of a vector bundle over a manifold M as a smooth family of vector spaces parametrized by M and in the same spirit ask about smooth families of

6In contrast, any manifold of dimension at least three admits a metric of constant negative scalar curvature, see Thierry Aubin, Métriquesriemanniennes et courbure. J. Differential Geometry 4, 1970, 383-424. The same is true for manifolds with boundary and for non-compact manifolds. For compact surfaces, the sign of the integral of the scalar curvature is topologically determined by the Gauss-Bonnet theorem. 3.7. SUPPLEMENT: SPIN STRUCTURES 81 manifolds parametrized by M. The analogue is the concept of a (locally trivial) fiber bundle7, or fibration. A fiber bundle is a surjective smooth map φ : M −→ B between manifolds with the property that each of the fibers φ−1(p) is diffeomorphic to a fixed manifold F and each point p ∈ B has a neighborhood U for which there is a diffeomorphism HU participating in a commutative diagram

HU φ−1(U) / F × U

φ πU # U | We call M the total space, B the base of the fiber bundle and F the typical fiber. We denote the fiber bundle by

F − M −→ B where the lack of arrow-head between F and M recalls that there is no canonical map from F to M. (There are many inclusions F,→ M, one for each b ∈ B and ∼ −1 diffeomorphism F = φ (b), but no natural choice.) We refer to each pair (U,HU ) as a local trivialization of φ. Every fiber bundle map is a surjective submersion of smooth manifolds. An important theorem of Ehresmann8 establishes the converse, every surjective submersion between closed manifolds is a fibration. Even if M is not compact, this characterization holds among proper maps. The simplest example of a fiber bundle with base B and fiber F is the Cartesian product and the canonical projection,

B × F −→ B.

This is the ‘trivial fiber bundle’ over B with fiber F. Locally all fiber bundles are of this form so we often think of them as being ‘twisted products’ of B and F.

A vector bundle E of rank k over a manifold X is a ﬁber bundle

k R − E −→ X with the extra property that in each local trivialization (U,FU ),FU restricts to a linear map between φ−1(p) and Rk × {p}, for each p ∈ U. 7The British spelling ‘fibre bundle’ is also common. 8Charles Ehresmann, Les connexions infinitésimalesdans un espace fibrédifférentiable, Col- loque de Topologie, Bruxelles (1950), 29-55. 82 CHAPTER 3. LAPLACE-TYPE AND DIRAC-TYPE OPERATORS

Another way of thinking about this extra requirement is to look at transitions from one trivialization to another. If F − M −→ B is a ﬁber bundle and (U,HU ), (V,HV ) are local trivializations with U ∩ V= 6 ∅, then

−1 HV ◦ HU : F × U ∩ V −→ F × U ∩ V is a diﬀeomorphism that projects to the identity map on U ∩ V. Hence it must have the form F × U ∩ V / F × U ∩ V

(f, x) / (hUV (x)f, x) ∞ for some hUV ∈ C (U ∩ V, Diff(F )). We call hUV the transition function corresponding to the trivializations (U,HU ), (V,HV ). A vector bundle is a fiber bundle with typical fiber Rk in which the transition functions, which are a priori diffeomorphisms of Rk, are linear maps.

Another important class of fiber bundles are principal bundles9. If G is a Lie group10 then a principal G-bundle is a fiber bundle G − P −→ B with typical fiber G with the additional requirement that the transition functions, which are a priori diffeomorphisms of G, are given by left multiplication by an element of G.

It turns out that the concepts of vector bundles and principal bundles are φ closely related. If Rn − E −−→ X is a vector bundle, a ‘frame’ for E at x ∈ X is an ordered basis of the vector space φ−1(x) and we denote the set of frames for E at x by Fr(Ex). The space

Fr(E) = {(x, v): x ∈ X and v ∈ Fr(Ex)} is called the frame bundle of E. It is easy to see that it is a smooth manifold (e.g., by identifying it with a subbundle of ⊕nE −→ X) and that the natural map Fr(E) −→ X is a ﬁber bundle. The transition maps are linear transformations from one basis to another, i.e., elements of GL(n) = GLn(R). Identifying Fr(Ex) with GL(n) we ﬁnd that GL(n) − Fr(E) −→ X

9Serre, in his lecture ‘How to write mathematics badly’ (available online), mentions that one of his favorite misspellings is ‘principle bundle’. This suggests that you have principles, but they depend on parameters. 10A Lie group is a group that is also a smooth manifold in such a way that multiplication and inversion are smooth functions. 3.7. SUPPLEMENT: SPIN STRUCTURES 83 is a principal bundle. This bundle is called the associated principal GL(n) bundle of E.

If E is orientable then we obtain an oriented vector bundle by demanding that the transition maps be orientation preserving. The same construction of an associated principal bundle yields a principal bundle with group

GL+(n) = {T ∈ GL(n) : det T > 0}.

We refer to the passage from the GL(n) principal bundle to the GL+(n) principal bundle as a reduction of the structure group. If we choose a bundle metric then we can restrict attention to orthogonal frames, and thus reduce the structure group to

t O(n) = On(R) = {A ∈ GLn(R): AA = Id}. If the bundle is also oriented, then we can further reduce the structure group to

SO(n) = SOn(R) = {A ∈ O(n) : det A = 1} by considering only oriented orthogonal frames. In terms of the groups themselves, notice that O(n) is a deformation retract of GL(n) (e.g., through the polar decomposition of an invertible matrix). Topologi- cally the change from GL(n) to O(n) is insigniﬁcant, reﬂecting the fact that we can always endow a bundle with a bundle metric. The group SO(n) is the connected component of the identity of O(n). Topologically, there is an obstruction to reducing the structure group to SO(n), namely the bundle must be orientable. There is a natural ‘next step’ in terms of reducing the structure group, we should pass to the universal cover of SO(n). This is called the spin group, Spin(n).

The Spin group Spin(n) is closely related to the Cliﬀord algebra Cl(n) = n Cl(R , gRn ). Indeed we can identify the Lie algebra of SO(n), so(n), consisting of skew-symmetric n × n matrices, with the vector space

−1 2 C2(n) = Mf (Λ V ) ⊆ Cl(n) with Mf the vector space isomorphism (3.8) by using the map

n τ : C2(n) −→ End(R ), τ(a)(v) = a · v − v · a and noting that its image consists of skew-symmetric endomorphisms. Endowing C2(n) with a Lie bracket coming from the commutator of Cliﬀord multiplication 84 CHAPTER 3. LAPLACE-TYPE AND DIRAC-TYPE OPERATORS

yields a Lie algebra isomorphic to so(n). Exponentiating C2(n) within Cl(n) yields a group called the spin group and denoted Spin(n). It can be identiﬁed with the multiplicative subgroup of Cl(n) generated by multiplication by pairs of unit vectors. One can show11 that Spin(n) is simply connected and hence is the universal cover of SO(n). The latter has fundamental group Z2, so Spin(n) −→ SO(n) is a double cover and the kernel of this map can be determined to be {±1} ⊆ Spin(n). Every representation ρ : SO(n) −→ End(V ) yields a representation of Spin(n) by pull-back. However there are representations of Spin(n) that do not descend to representations of SO(n). Assume that n is even and consider the spin representation (3.9) of Cl(n) on End(Λ∗K). The decomposition Λ∗K = ΛevenK ⊕ ΛoddK is preserved by elements of C2(n) and hence by Spin(n). It follows that the restrictions

cl even : Spin(n) −→ End(ΛevenK), and cl odd : Spin(n) −→ End(ΛoddK) are irreducible representations of Spin(n). They are called the half-spin representations. Since cl ∗(−1) 6= cl ∗(1), these representations do not descend to representations of SO(n).

Now let M be an oriented Riemannian manifold and SO(n) − F rSO(TM) −→ M be the principal SO(n)-bundle of oriented orthonormal frames. Every local trivialization of this principal bundle is isomorphic to SO(n) × U with U ⊆ M and so has two lifts to Spin(n) × U, differing by multiplication by −1. It is occasionally possible to make these choices consistently and lift the entire frame bundle. A spin structure on M is a principal Spin(n) bundle Spin(n) − P −→ M with a bundle map P −→ F rSO(TM) which on each fiber is the covering map Spin(n) −→ SO(n). If M admits a spin structure, then M together with a fixed spin structure is called a spin manifold. If M is a spin manifold, then the vector bundle associated to the principal Spin(n)-bundle and the spin representation is called the spin bundle and denoted S/ −→ M. The Clifford bundle of M can be obtained from the principal bundle P −→ M using the representation of Spin(n) on Cl(n) and hence the action of Cl(n) on the spin representation shows that S/ is a Clifford module. The existence and number of spin structures is determined topologically. Namely12,

11See, e.g., Proposition 3.10 of Nicole Berline, Ezra Getzler, Michele` Vergne, Heat kernels and Dirac operators. Corrected reprint of the 1992 original. Grundlehren Text Editions. Springer-Verlag, Berlin, 2004. 12For a proof, see Theorem 1.7 in H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, Spin geometry. Princeton Mathematical Series, 38. Princeton University Press, Princeton, NJ, 1989. 3.8. EXERCISES 85 an orientable manifold M has a spin structure if and only if its ‘second Stieﬀel- Whitney class’ 2 w2(M) ∈ H (M; Z2) vanishes. In that case, the diﬀerent spin structures are parametrized by elements of Hom(π1(M), Z2).

C The group spin (n) is the quotient of Spin(n) × U(1) by Z2 = {±1}. The notion of a spinC structure is similar but weaker. Let us say that a rank k oriented bundle E −→ M has a spin structure if its associated principal SO(k) bundle has a lift to a principal Spin(k) bundle. Thus a manifold has a spin structure when its tangent bundle does. A manifold has a spinC structure if there exists a complex line bundle L −→ M such that TM ⊕ L −→ M has a spin structure. This is again topologically obstructed, but the obstruction is much weaker: w2(M) must be the mod 2 reduction of an integral cohomology class13.

3.8 Exercises

Exercise 1. a) Show that the scalar linear differential operators on Rm form a (non-commutative) algebra. Verify that the commutator PQ − QP of a differential operator of order ` and a differential operator of order `0 has at most order ` + `0 − 1. b) Show that the linear differential operators with constant coefficients form a commutative subalgebra which is isomorphic to the polynomial algebra R[ξ1, . . . , ξm]. Exercise 2. Verify that for P ∈ Diffk(M; E,F ) and Q ∈ Diff`(M; F,G), the operator PQ is in k+` Diff (M; E,G) and its principal symbol satisfies σk+`(P ◦ Q) = σk(P ) ◦ σ`(Q). Exercise 3. Let E −→ M and F −→ M be bundles endowed with metrics, lying over the Riemannian manifold M. Verify that the formal adjoint of P ∈ Diffk(M; E,F ),P ∗, k ∗ ∗ is an element of Diff (M; F,E) and its principal symbol satisfies σk(P ) = σk(P ) . Exercise 4. Let M be a Riemannian manifold with boundary, and E −→ M, and F −→ M bundles endowed with bundle metrics. The formal adjoint of an operator P : C∞(M; E) −→ C∞(M; F ) is defined by requiring ∗ ∞ ◦ ∞ ◦ hP σ, τiF = hσ, P τiE, for all σ ∈ Cc (M ; E), τ ∈ Cc (M ; F ). 13Theorem D.2 in H. Blaine Lawson, Jr. and Marie-Louise Michelsohn, Spin geometry. Princeton Mathematical Series, 38. Princeton University Press, Princeton, NJ, 1989. 86 CHAPTER 3. LAPLACE-TYPE AND DIRAC-TYPE OPERATORS

Let ν be the inward-pointing unit normal vector field at the boundary. Define 1 dvol∂M by iν dvolM ∂M . Show that if P ∈ Diff (M; E,F ) is a first-order differential operator then Z ∗ [ hP σ, τiF − hσ, P τiE = gF (iσ1(P )(p, ν )σ, τ) dvol∂M , ∂M ∞ ∞ for all σ ∈ Cc (M; E), τ ∈ Cc (M; F ). Hint: Use a partition of unity to reduce to a coordinate chart, then compute in local coordinates with ∂x1 , . . . , ∂xm−1 tangent to ∂M and ∂xm = ν, the inward pointing normal vector. Exercise 5. Show that End(Cm) has no proper two-sided ideals. Hint: Suppose M0 belongs to such an ideal I and v0 6= 0 belongs to the range of m M0. Show that every v ∈ C belongs to the range of some M ∈ I, and hence that every one-dimensional projection belongs to I. Exercise 6. a) Suppose a compact oriented manifold M has non-negative Ricci curvature – i.e., that g(Ric(f ω), ω) ≥ 0 for all ω ∈ Ω1(M). Show that every harmonic one-form η satisfies ∇η = 0. As this is a first order differential equation for η, a harmonic one-form is determined by its ‘initial condition’ at a point in M. Conclude that, if M is also connected, dim H1(M) ≤ dim M. b) Show that on any connected compact oriented Riemannian manifold dim H0(M) = 1 and so by Poincaréduality dim Hm(M) = 1. c) From (b), we see that on a compact oriented Riemannian surface (i.e., two dimen- 1 sional manifold) χHodge(M) = 2−dim H (M). Show that if M is a compact oriented surface admitting a metric of non-negative Gaussian curvature then χHodge(M) ≥ 0. (It turns out that only the sphere and the torus have non-negative Euler characteristic.) Exercise 7. Let M and M 0 be compact oriented Riemannian manifolds. i)Show that

k X dim Hk(M × M 0) = (dim Hj(M))(dim Hk−j(M 0)) j=0 when M×M 0 is endowed with the product metric (we will soon show that dim Hk(M) is independent of the choice of metric). ii) Conclude that

0 0 χHodge(M × M ) = χHodge(M)χHodge(M ). 3.8. EXERCISES 87

k ∗ 0 Lk j ∗ k−j ∗ 0 Hint: Use the fact that Λ T (M × M ) = j=0(Λ T M) ∧ (Λ T M ), show that with respect to this decomposition the Hodge Laplacian for the product metric satisﬁes k X ∆k,M×M 0 = (∆j,M ⊗ Id + Id ⊗∆k−j,M 0 ), j=0 and conclude that if

j j j j j j 0 {α , . . . , α } is a basis of H (M) and {β , . . . , β 0 } is a basis of H (M ) 1 bj (M) 1 bj (M )

j k−j k 0 then {αs ⊗ βt : 0 ≤ j ≤ k} is a basis of H (M × M ). Exercise 8. Let M and M 0 be compact oriented Riemannian manifolds. Show that signature is multiplicative, sign(M × M 0) = sign(M)sign(M 0), where the signature on M × M 0 is defined using the product metric (we will soon show that the signature is independent of the choice of metric). Hint: Express dM×M 0 and δM×M 0 in terms of dM , δM , dM 0 , and δM 0 . Choose bases of ∗ ∗ 0 ∗ 0 Λ±T M and Λ±T M and use them to construct bases of Λ±T (M × M ). Exercise 9. We can interpret any vector field V on M as a first order differential operator. What is its principal symbol? Exercise 10. Let E −→ M and F −→ M be two vector bundles over M. Show that, for any k 14 k ∈ N, the assignment U 7→ Diff (M; E U ,F U ) is a sheaf on M. Exercise 11. If G − P −→ M is a principal G bundle and f : X −→ M is a smooth map between manifolds, show that there is a principal G-bundle G − f ∗P −→ X and a map f ∗P −→ P such that f ∗P / P

f X / M

14A sheaf F of vector spaces is: for every open set U ⊆ M a vector space F (with F (∅) = 0), for every inclusion of open sets, U ⊆ V, a restriction map ρUV : F (V) −→ F (U) satisfying

U ⊆ V ⊆ W =⇒ ρWU = ρVU ◦ ρWV and, whenever {Vi} is an open cover of U and we are given si ∈ F (Vi) such that

ρVi,Vi∩Vj (si) = ρVj ,Vi∩Vj (sj) for all i, j, there exists a unique s ∈ F (U) such that ρU,Vi (s) = si for all i. 88 CHAPTER 3. LAPLACE-TYPE AND DIRAC-TYPE OPERATORS commutes. The principal bundle f ∗P −→ X is known as the pull-back of P −→ M. Exercise 12. An smooth action of a Lie group G on a manifold M is a smooth homomorphism A : G −→ Diﬀ(M). The action is free if whenever g ∈ G and x ∈ M satisfy A(g)(x) = x we have g equal to the identity of the group. The orbit of a point x ∈ M is the set G.x = {A(g)x : g ∈ G} and the orbit space of the action is the space M/G = {G.x : x ∈ M} with the smooth structure induced by the projection map p : M −→ M/G i.e, a function on M/G is smooth if and only if its pull-back to M is smooth. Show that if G − P −→ M is a principal G-bundle there is an action of G on P whose orbit space is M. Exercise 13. Let G − P −→ M be a principal G bundle, V a vector space, and ρ : G −→ Iso(V ) a linear representation of G on V. Show that there is a natural ‘diagonal’ action of G on P × V and denote its orbit space by P ×G V. Show that V − P ×G V −→ M is a vector bundle over M. Exercise 14.

1 1 2 a) Show that the map S −→ S given by z 7→ z is a principal Z2-bundle. b) Show that if G is a Lie group and H is a subgroup then G −→ G/H is a principal H-bundle.

3.9 Bibliography

The books Spin geometry by Lawson and Michelson, and Heat kernels and Dirac operators by Berline, Getzler, and Vergne treat Dirac operators in detail. Other good sources are Elliptic operators, topology and asymptotic methods by Roe which gives a very nice short treatment of many of the topics of this course, and volume two of Partial diﬀerential equations by Taylor. There is also a very nice treatment of these topics in Melrose, The Atiyah- Patodi-Singer index theorem, where they are systematically extended to non-compact manifolds with asymptotically cylindrical ends. Chapter 4

m Pseudodiﬀerential operators on R

4.1 Distributions

It is an unfortunate fact that most functions are not differentiable. Happily, there is a way around this through duality. For instance, consider integration by parts for smooth1 functions, Z Z f 0(t)φ(t) dt = − f(t)φ0(t) dt R R ∞ ∞ with, say, f ∈ C (R) and φ ∈ Cc (R). We notice that for the right hand side to make sense we do not need for f to be differentiable and then decide to use this identity as a definition of the derivative of f. So for a possibly non-differentiable f we will say that a function h is a weak derivative of f if Z Z 0 ∞ f(t)φ (t) dt = − h(t)φ(t) dt, for all φ ∈ Cc (R). R R Obviously if f is differentiable then f 0 is its weak derivative. Generally the taking of weak derivatives will force us to expand our notion of ‘function’. ∞ For example, for f(t) = |t| we have, for any φ ∈ Cc (R),

Z Z 0 Z ∞ f(t)φ0(t) dt = − tφ0(t) dt + tφ0(t) dt R −∞ 0 Z 0 Z ∞ Z 0 ∞ = −[tφ(t)]−∞ + φ(t) + [tφ(t)]0 − φ(t) dt = − sign(t)φ(t) dt −∞ 0 R

1As in the previous chapter, we will work with complex valued functions and complex vector bundles. Thus for instance the symbol C∞(M) will now mean C∞(M; C).

89 90 CHAPTER 4. PSEUDODIFFERENTIAL OPERATORS ON RM where sign(t) is the function equal to −1 for negative t and equal to 1 for positive t. Notice that f is continuous but its weak derivative is not. If we want to take a second weak derivative we compute Z Z ∞ Z ∞ 0 0 0 0 ∞ sign(t)φ (t) dt = − φ (t) + φ (t) dt = −[φ(t)]∞ + [φ(t)]0 = −2φ(0). R 0 0 This shows that the weak derivative of of sign(t) is a multiple of the Dirac delta ‘function’, δ0, deﬁned by Z ∞ φ(t)δ0(t) dt = φ(0), for all φ ∈ Cc (R). R Of course there is no function satisfying the deﬁnition of the delta function. Instead we should think of δ0 as a linear functional ∞ δ0 : Cc (R) −→ C, δ0(φ) = φ(0). This is the motivating example of a distribution. A distribution on Rm is a continuous linear functional on smooth functions of compact support, ∞ m T : Cc (R ) −→ C. Continuity here means that for every compact set K ⊆ Rm there are constants C and ` satisfying

X α ∞ |T (φ)| ≤ C sup |∂ φ| for all φ ∈ Cc (K). K |α|≤`

Similarly, let (M, g) be an oriented Riemannian2 manifold. A distribution on M is a continuous linear functional on smooth functions of compact support,

∞ T : Cc (M) −→ C, such that for every compact set K ⊆ M there are constants C and ` satisfying

X α ∞ |T (φ)| ≤ C sup |∇ φ|g for all φ ∈ Cc (K). K |α|≤` We will denote3 the set of distributions by C−∞(M). If the same integer ` can be used for all compact sets K, we say that T is of order `. (A distribution of order `

2We do not need to have a Riemannian metric or an orientation to define the space of distributions. It does make things simpler, though, since we can use the Riemannian volume form to carry out integrations (really to trivialize the ‘density bundle’) and the gradient to measure regularity. 3In the literature one often finds the set of distributions denoted D0(M). This is because Laurent Schwartz, who first defined distributions, denoted the smooth functions of compact support by D(M). 4.1. DISTRIBUTIONS 91

` has a unique extension to a continuous linear functional on Cc(M), but we shall not use this.) We can restate the continuity condition in terms of sequences: A linear func- ∞ ∞ tional T on C (M) is a distribution if and only if whenever φj ∈ Cc (M) is a sequence supported in a ﬁxed compact subset of M that is converging uniformly to zero together with all of its derivatives, we have T (φj) −→ 0. The Hermitian pairing that the metric induces on C∞(M; C) yields an inclusion

C∞(M) / C−∞(M)

h / Th Z ∞ Th(φ) = hφ, hi = φ(p)h(p) dvol for all φ ∈ Cc (M). M Indeed, any smooth function h ∈ C∞(M) naturally induces a distribution (of order zero) T h Z ∞ Th(φ) = hφ, hi = φ(p)h(p) dvol for all φ ∈ Cc (M). M

More generally, we get a distribution (of order zero) Th, from any function h for which the pairing hφ, hi can be guaranteed to be ﬁnite,

1 ∞ Lloc(M) = {h measurable : hφ, hi < ∞ for all φ ∈ Cc (M)}.

Similarly, any measure on M, µ, deﬁnes a distribution Tµ of order zero Z ∞ Tµ(φ) = φ dµ for all φ ∈ Cc (M). M The converse is true as well4, any distribution of order zero is a measure on M. An example of a distribution that is not a measure is

0 ∞ T (φ) = φ (0) for all φ ∈ Cc (M).

As a very convenient abuse of notation, the pairing between a smooth function of compact support f and a distribution T is denoted

∞ −∞ h·, ·i : Cc (M) × C (M) −→ C hφ, T i = T (φ) and then in another abuse we identify Th and h.

4See Theorem 2.1.6 of Lars Hormander,¨ The analysis of linear partial diﬀerential operators. I. Distribution theory and Fourier analysis. Classics in Mathematics. Springer-Verlag, Berlin, 2003. 92 CHAPTER 4. PSEUDODIFFERENTIAL OPERATORS ON RM

Now we can proceed to extend many concepts and operations of C∞(M) to C−∞(M) by duality using the pairing h·, ·i. For instance, we can multiply a distribution T by a smooth function f ∈ C∞(M) by deﬁning

∞ hφ, fT i = hφf, T i, for all φ ∈ Cc (M).

We say that a distribution T vanishes on an open set U if

∞ hφ, T i = 0, for all φ ∈ Cc (U). The support of a distribution supp(T ) is the set of points in M having no neighborhood on which T vanishes. Equivalently, a point ζ is not in the support of T if there is a neighborhood of ζ on which T vanishes. The distributions of 5 −∞ compact support will be denoted Cc (M). A distribution of compact support can be evaluated not just on functions of compact support but on all of C∞(M). In fact, distributions of compact support form the continuous dual of the space C∞(M). Although it makes no sense to talk about the value of a distribution at a point, we can restrict a distribution to an open set U by restricting its domain to functions with compact support in U. We say that two distributions S and T coincide on an open set U if ∞ hφ, Si = hφ, T i for all φ ∈ Cc (U). If on each set of an open cover of M we have a distribution, and these distributions coincide on overlaps, then we can use a partition of unity to construct a distribution that restricts to the given ones. It is also true that two distributions coincide if and only if they coincide on open sets covering M. A closely related concept to the support of a distribution is that of the singular support of a distribution sing supp(T ), deﬁned by specifying its complement: a point ζ ∈ M is not in the singular support of T if there is a smooth function f that coincides with T in a neighborhood of ζ. We can apply a linear operator F to T by using its formal adjoint F ∗,

∗ ∞ hφ, F (T )i = hF (φ),T i, for all φ ∈ Cc (M). In particular, we can apply a diﬀerential operator to a distribution, e.g.,

∞ hφ, ∆T i = h∆φ, T i, for all φ ∈ Cc (M). The weak derivatives we discussed above are computed in exactly this way. Thus a general function in, say, L1(M) is not differentiable as a function, but is infinitely differentiable as a distribution! 5Schwartz’ notation was E0(M), and one often finds this in the literature. 4.1. DISTRIBUTIONS 93

If E −→ M is a vector bundle over M we deﬁne distributional sections of E ∞ ∗ to be continuous linear functionals on Cc (M; E ). As above, there is a natural way of thinking of a smooth section τ of E as a distributional section Tτ of E, namely Z ∞ ∗ Tτ (ω) = ω(τ) dvolM , for all ω ∈ Cc (M; E ). M ∗ If we assume that E is endowed with a metric, gE, we can identify E with E and ∞ identify distributional sections of E with continuous linear functionals on Cc (M; E). We can then extend our abuse of notation and denote the evaluation of a distribution T on a compactly supported section σ by

hσ, T iE = T (σ) and also denote Tτ merely by τ. All of the operations on distributions mentioned above extend to distributional sections. In particular we can talk about the support of a distributional section and −∞ then the distributional sections of compact support, Cc (M; E). These distributions have a simple description

−∞ −∞ ∞ Cc (M; E) = Cc (M) ⊗ C (M; E). C∞(M) Indeed, every simple tensor

−∞ ∞ T ⊗ σ ∈ Cc (M) ⊗ C (M; E) C∞(M) deﬁnes an element in the dual to C∞(M; E∗) by

(T ⊗ σ)(ω) = T (ω(σ)), for all ω ∈ C∞(M; E∗),

−∞ ∞ −∞ and so we have an inclusion Cc (M) ⊗C∞(M) C (M; E) −→ Cc (M; E). To see −∞ that this map is surjective, start with S ∈ Cc (M; E) and note that S deﬁnes, given any section ω ∈ C∞(M; E∗), a compactly supported distribution on M via

∞ C (M) 3 φ 7→ S(φω) ∈ C.

Since the support of S is compact, we can cover it with ﬁnitely many sets {Ui} that ∗ ∗ trivialize E , choose {si1, . . . , sik} trivializing sections of E supported in Ui, and then write X X S(ω) = S( ωijsij) = S(sij)(ω(sij)) i,j i,j ! X ∞ ∗ = S(sij) ⊗ sij (ω), for all ω ∈ C (M; E ). i,j 94 CHAPTER 4. PSEUDODIFFERENTIAL OPERATORS ON RM 4.2 Examples

4.2.1 Fundamental solutions

k Let P be a constant coefficient differential operator on Rm,P ∈ Diff (Rm),

X α P = aα(x)D . |α|≤k

We say that a distribution Q ∈ C−∞(Rm) is a fundamental solution of the diﬀerential operator P if PQ = δ0. Two fundamental solutions will diﬀer by an element of the null space of P. 2 We have already seen that, in R, ∂x|x| = −2δ0, hence 1 2 − 2 |x| is a fundamental solution of ∆ = −∂x. On Rm, the fundamental solution of the Laplacian is given by ( (2π)−1 log |x| if m = 2 Q = m − 1 |x|2−m if m > 2 (m−2) Vol(Sm−1) ∞ m Indeed, note that on |x| > 0, ∆Qm = 0, so we can write, for any φ ∈ Cc (R ), Z hφ, ∆Qmi = h∆φ, Qmi = lim Qm∆φ dx ε→0 |x|>ε Z Z = lim Qm∆φ − φ∆Qm dx = − lim Qm∂rφ − φ∂rQm dS ε→0 |x|>ε ε→0 |x|=ε  Z −1 log ε∂rφ − φε − lim dS if m = 2  ε→0 2π = |x|=ε Z ε2−m∂ φ − φ(2 − m)ε1−m − lim r dS if m > 2  m−1 ε→0 |x|=ε (2 − m) Vol(S ) = φ(0) where in the fourth equality we have used Green’s formula. If we identify R2 with C by identifying (x, y) with z = x+iy then the operator 1 ∂ = 2 (∂x + i∂y) on complex valued functions has as null space the space of holomorphic functions. 1 A similar application of Green’s theorem allows us to identify πz as a fundamental solution for ∂. 4.2. EXAMPLES 95

4.2.2 Principal values and ﬁnite parts

Let us see a few more examples of distributions. The function x 7→ xα is locally integrable on R if α > −1. For α = −1 we deﬁne the (Cauchy) principal value of 1 1 −∞ x , PV( x ) ∈ C (R) by Z Z ∞ 1 φ(x) φ(x) − φ(−x) hφ, PV( x )i = lim dx = dx ε→0 |x|>ε x 0 x Z ∞ Z 1 0 ∞ = φ (tx) dt dx, for all φ ∈ Cc (R). 0 −1

This is indeed an element of C−∞(R), since whenever supp φ ⊆ [−C,C] we have Z ∞ Z 1 0 0 φ (tx) dt dx ≤ 2C sup |φ |, 0 −1

1 so PV( x ) is a distribution of order at most one (in fact of order one). Notice that 1 ∞ x PV( x ) = 1 since, for any φ ∈ Cc (R), Z Z 1 1 φ(x)x hφ, x PV( x )i = hxφ, PV( x )i = lim dx = φ(x) dx = hφ, 1i. ε→0 |x|>ε x It is then easy to see that

−∞ 1 T ∈ C (R), xT = 1 =⇒ T = PV( x ) + Cδ0,C ∈ C.

1 1 so in this sense PV( x ) is a good replacement for x as a distribution. What about higher powers of x−1? Simply taking the principal value will not work for x−2 since we have Z φ(x) Z φ(x) + φ(−x) 2φ(0) 2 dx = 2 dx = + ( bounded ) |x|>ε x x>ε x ε

−2 1 so instead we deﬁne the (Hadamard) ﬁnite part of x , FP( x2 ) by Z Z ∞ 1 φ(x) − φ(0) φ(x) − 2φ(0) + φ(−x) hφ, FP( x2 )i = lim 2 dx = 2 dx ε→0 |x|>ε x 0 x Z ∞ Z 1 Z 1 00 ∞ = s φ (stx) dt ds dx, for all φ ∈ Cc (R). 0 0 −1 As before, this is clearly a distribution of order at most two and we have

−∞ 2 1 T ∈ C (R), x T = 1 =⇒ T = FP( x2 ) + C0δ0 + C1∂xδ0,C0,C1 ∈ C. 96 CHAPTER 4. PSEUDODIFFERENTIAL OPERATORS ON RM

1 We can continue this and define FP( xn ) for any n ∈ N, n ≥ 2, by Z Pn−2 1 (k) k 1 φ − k=0 k! φ (0)x hφ, FP( xn )i = lim n dx. ε→0 |x|>ε x This defines a distribution of order at most n and moreover n−1 −∞ n 1 X (k) T ∈ C (R), x T = 1 =⇒ T = FP( xn ) + Ci∂x δ0, {Ci} ⊆ C. k=0 One can similarly define finite parts of non-integer powers of x and also define finite parts in higher dimensions.

4.3 Tempered distributions

On Rm, the class of Schwartz functions S(Rm) sits in-between the smooth functions and the smooth functions with compact support ∞ m m ∞ m Cc (R ) ⊆ S(R ) ⊆ C (R ) m n ∞ m α β o S(R ) = f ∈ C (R ): pα,β(f) = sup |x ∂ f| < ∞, for all α, β Rm Correspondingly the dual spaces satisfy −∞ m 0 m −∞ m Cc (R ) ⊆ S (R ) ⊆ C (R ) 0 m −∞ m S (R ) = T ∈ C (R ): pα,β(f) → 0 for all α, β =⇒ T (fj) → 0 . The dual space of S, S0(Rm), is known as the space of tempered distributions. Notice that these inclusions are strict. Any Lp function will be in S0(Rm) but −∞ m will only be in Cc (R ) if it has compact support. On the other hand, the function 2 e|x| deﬁnes a distribution in C−∞(Rm) but not a tempered distribution.

4.3.1 Convolution

The convolution of two functions in S(Rm) is Z Z (f ∗ h)(x) = f(x − y)h(y) dy = h(x − y)f(y) dy = (h ∗ f) Rm Rm and deﬁnes a bilinear map S(Rm) × S(Rm) −→ S(Rm). The commutativity is particularly clear from the expression ZZ hφ, f ∗ hi = f(x)h(y)φ(x + y) dx dy, 4.3. TEMPERED DISTRIBUTIONS 97 which we can also write as Z Z Z Z hφ, f ∗ hi = h(y) f(x)φ(x + y) dxdy = h(y) f(z − y)φ(z) dzdy

= hfˇ∗ φ, hi, where fˇ(x) = f(−x).

This allows us to use duality to deﬁne convolution of a tempered distribution with a Schwartz function

m 0 m 0 m ∗ : S(R ) × S (R ) −→ S (R ) m hφ, f ∗ Si = hfˇ∗ φ, Si for all f, φ ∈ S(R ) and similarly ∞ m −∞ m −∞ m ∗ : Cc (R ) × C (R ) −→ C (R ) ˇ ∞ m hφ, f ∗ Si = hf ∗ φ, Si for all f, φ ∈ Cc (R ). For instance, notice that we have Z ˇ hφ, f ∗ δ0i = hf ∗ φ, δ0i = f(x)φ(x) dx = hφ, fi Rm m and so f ∗ δ0 = f for all f ∈ S(R ). If P is any constant coeﬃcient diﬀerential operator, and f, h ∈ S(Rm) then P (f ∗ h) = P (f) ∗ h = f ∗ P (h) and so by duality we also have

m 0 m P (f ∗ S) = P (f) ∗ S = f ∗ P (S), for all f ∈ S(R ),S ∈ S (R ).

Putting these last two observations together, suppose that S ∈ S0(Rm) is a fundamental solution of the constant coeﬃcient diﬀerential operator P, so that P (S) = δ0, then we have

m P (f ∗ S) = f ∗ P (S) = f ∗ δ0 = f, for all f ∈ S(R ). Thus a fundamental solution for P is tantamount to a right inverse for P, even though P is generally not invertible.

In particular, with Qm as above and m > 2 for simplicity,

0 m m ∞ m u ∈ S (R ), f ∈ S(R ), and ∆u = f =⇒ u = f ∗ Qm ∈ C (R ) Z 1 Z f(y) (4.1) u(x) = Qm(x − y)f(y) dy = m−1 m−2 dy Rm (2 − m) Vol(S ) Rm |x − y| 98 CHAPTER 4. PSEUDODIFFERENTIAL OPERATORS ON RM

Remark 4. We can also use convolution to prove that test functions are dense −∞ 0 −∞ in distributions, be they Cc , S , or C . Indeed, ﬁrst say that a sequence φn ∈ ∞ m Cc (R ) is an approximate identity if Z 1 φn ≥ 0, φn(x) dx = 1, φn(x) = 0 if |x| > n , Rd ∞ m and note that, as distributions, φn → δ0. Let χn ∈ Cc (R ) approximate the constant function one, and notice that, for any distribution S, the sequence

∞ m Sn ∈ Cc (R ),Sn(x) = χn(x)(φn ∗ S)(x) converges as distributions to S.

4.3.2 Fourier transform

Recall that Fourier transform of a function f ∈ L1(Rm) is deﬁned by 1 m ∞ m F : L (R ) −→ L (R ) Z 1 −ix·ξ F(f)(ξ) = fb(ξ) = n/2 f(x)e dx (2π) Rm and satisﬁes for f ∈ S(Rm),

F(∂xj f)(ξ) = iξjF(f)(ξ), ∂ξj F(f)(ξ) = −iF(xjf)(ξ).

1 To simplify powers of i, let us deﬁne Ds = i ∂s so that these identities become

F(Dxj f)(ξ) = ξjF(f)(ξ),Dξj F(f)(ξ) = −F(xjf)(ξ), and indeed for any multi-indices α, β,

α β |β| α β (4.2) ξ Dξ F(f)(ξ) = (−1) F(Dx x f)(ξ). In particular we have the very nice result that

m m F : S(Rx ) −→ S(Rξ ).

The formal adjoint of F, F ∗, is deﬁned by

∗ ∞ m hF(f), hi m = hf, F (h)i m , for all f, h ∈ C ( ) Rξ Rx c R and since we have Z Z 1 −ix·ξ hF(f), hi m = F(f)(ξ)h(ξ) dξ = e f(x)h(ξ) dξdx Rξ m/2 Rm (2π) Rm×Rm 4.3. TEMPERED DISTRIBUTIONS 99 we conclude that Z ∗ 1 ix·ξ F (h)(x) = m/2 h(ξ)e dξ. (2π) Rm ∗ Using the fact Dj = Dj we ﬁnd

α β ∗ |α| ∗ α β (4.3) x Dx F (φ)(x) = (−1) F (Dξ ξ φ)(x) and hence ∗ m m F : S(Rξ ) −→ S(Rx ).

Proposition 11 (Fourier inversion formula). As a map on S(Rm), the Fourier transform F is invertible with inverse F ∗.

Proof. We want to show that, for any f ∈ S(Rm) we have F ∗(F(f)) = f, i.e., Z Z eix·ξ e−iy·ξf(y) dy dξ = (2π)mf(x) but note that the double integral does not converge absolutely and hence we can not change the order of integration. We get around this by introducing a function ψ ∈ S with ψ(0) = 1, letting ψε(x) = ψ(εx) and then using Lebesgue’s dominated convergence theorem to see that

∗ ∗ F (h) = lim F (ψεh). ε→0 Hence the integral we need to compute is given by Z Z ZZ lim eix·ξψ(εξ) e−iy·ξf(y) dy dξ = lim ei(x−y)·ξψ(εξ)f(y) dξ dy ε→0 ε→0 Z Z m/2 y−x dy m/2 = (2π) lim f(y)ψb( ) m = (2π) lim f(x + εη)ψb(η) dη ε→0 ε ε ε→0 Z = (2π)m/2f(x) ψb(η) dη.

−|x|2/2 R m/2 If we take ψ(x) = e , then ψb(η) = ψ(η) and ψb(η) dη = (2π) so the result follows. One can prove FF ∗ = Id similarly.

We can extend the Fourier transform and its inverse to tempered distributions in the usual way

∗ 0 m 0 m F, F : S (R ) −→ S (R ) ∗ ∗ m hφ, F(T )i = hF (φ),T i, hφ, F (T )i = hF(φ),T i for all φ ∈ S(R ) 100 CHAPTER 4. PSEUDODIFFERENTIAL OPERATORS ON RM and these are again mutually inverse and satisfy (4.2), (4.3). In particular, we can compute that Z ∗ ∗ 1 −m/2 hφ, F(δ0)i = hF (φ), δ0i = F (φ)(0) = m/2 φ(x) dx = hφ, (2π) i (2π) Rm −m/2 so that F(δ0) = (2π) . Then, using (4.2) and (4.3), we can deduce that α |α| −m/2 α ∗ m/2 ∗ α |α| m/2 α F(D δ0) = (−1) (2π) x , F (1) = (2π) δ0, F (x ) = (−1) (2π) D δ0.

Another consequence of Proposition 11 is that, for any φ ∈ S(Rm), 2 ∗ 2 kφbkL2 = hF(φ), F(φ)i = hF F(φ), φi = hφ, φi = kφkL2 thus F has a unique extension to an invertible map

2 m 2 m F : L (R ) −→ L (R ) and this map is an isometry – a fact known as Plancherel’s theorem. The Fourier transform is a very powerful tool for understanding differential k operators. Note that if P ∈ Diff (Rm), X α P = aα(x)D , |α|≤k and the coefficients aα(x) are bounded with bounded derivatives, then for any φ ∈ S(Rm) (or even S0(Rm)) we have P φ = F ∗(p(x, ξ)Fφ) X α with p(x, ξ) = aα(x)ξ . |α|≤k

The polynomial p(x, ξ) is called the total symbol of P and will be denoted σtot(P ). One consequence which can seem odd at ﬁrst is that a diﬀerential operator acts by integration! Indeed, we have 1 ZZ (4.4) (P φ)(x) = F ∗(σ (P )Fφ)(x) = ei(x−y)·ξp(x, ξ)φ(y) dy dξ tot (2π)m although note that this integral is not absolutely convergent. Notice the commonality between (4.4) and (4.1). In each case we have a map of the form Z (4.5) A(f)(x) = KA(x, y)f(y) dy Rm where f and KA, the integral kernel of A, may have low regularity. In fact, this describes a very general class of maps. 4.3. TEMPERED DISTRIBUTIONS 101

Theorem 12 (Schwartz kernel theorem). Let M and M 0 be manifolds, E −→ M and E0 −→ M 0 vector bundles, and let HOM(E,E0) −→ M × M 0

0 0 0 be the bundle whose ﬁber at (ζ, ζ ) ∈ M × M is Hom(Eζ ,Eζ0 ). If ∞ −∞ 0 0 A : Cc (M; E) −→ C (M ; E ) is a bounded linear mapping, there exists

−∞ 0 0 KA ∈ C (M × M ; Hom(E,E )) known as the Schwartz kernel of A or the integral kernel of A such that

hψ, A(φ)i = hψ φ, KAi where ∞ 0 0 0 ψ φ ∈ Cc (M × M ), (ψ φ)(ζ, ζ ) = ψ(ζ)φ(ζ ). We will not prove this theorem6 and we will not actually use it directly. Rather we view as an inspiration to study operators by describing their Schwartz kernels. In fact, it is common to purposely confuse an operator and its Schwartz kernel. Note 1 0 that if the integral kernel is given by a function, say KA ∈ Lloc(M × M ), then A is given by (4.5). A simple example is the kernel of the identity map

−∞ KId ∈ C (M × M), KId = δdiagM and we abuse notation by writing Z 0 0 f(ζ) = δdiagM f(ζ ) dvol(ζ ). M Note that on Rm we can also write its integral kernel in terms of the Fourier transform Z ∗ 1 i(x−y)·ξ (4.6) Id = F F =⇒ KId = m e dξ. (2π) Rm Similarly we can write the kernel of a diﬀerential operator P ∈ Diﬀ∗(M) as

−∞ ∗ KP ∈ C (M × M), KP = P δdiagM or, on Rm, 1 Z (4.7) K = ei(x−y)·ξp(x, ξ) dξ. P (2π)m 6For a proof, see section 5.2 of Lars Hormander,¨ The analysis of linear partial diﬀerential operators. I. Distribution theory and Fourier analysis. Classics in Mathematics. Springer-Verlag, Berlin, 2003. 102 CHAPTER 4. PSEUDODIFFERENTIAL OPERATORS ON RM 4.4 Pseudodiﬀerential operators on Rm

4.4.1 Symbols and oscillatory integrals

In both (4.6) and (4.7) we have integral kernels of the form 1 Z ei(x−y)·ξp(x, ξ) dξ (2π)m where p(x, ξ) is a polynomial in ξ of degree k and the integral makes sense as a distribution. The way we deﬁne pseudodiﬀerential operators is to take a similar integral, but replace p(x, ξ) with a more general function. Let U ⊆ Rm be a (regular7) open set and r ∈ R a real number. A function a ∈ C∞(U × Rm) is a symbol of order r on U, r m a ∈ S (U; R ) if for all multi-indices α, β there is a constant Cα,β such that

α β p 2 r−|β| sup |Dx Dξ a(x, ξ)| ≤ Cα,β( 1 + |ξ| ) . x∈U

Notice that the symbol condition is really only about the growth of a and its derivatives as |ξ| → ∞. So we always have

r m r0 m 0 S (U; R ) ⊆ S (U; R ) if r < r and, if a is smooth in (x, ξ) and compactly supported (or Schwartz) in ξ, then

\ r m −∞ m a ∈ S (U; R ) = S (U; R ). r∈R

For example, if r ∈ N then a polynomial of degree r is an example of a symbol of order r. Also, for arbitrary r ∈ R, we can take a function a(x, θ) deﬁned for |θ| = 1, extend it to (x, ξ) with |ξ| > 1 by

r ξ a(r, ξ) = |ξ| a(x, |ξ| ) for |ξ| > 1 and choose any smooth extension of a for |ξ| < 1 and we obtain a symbol of order r. It is easy to see that pointwise multiplication deﬁnes a map

r m r m r +r m S 1 (U; R ) × S 2 (U; R ) −→ S 1 2 (U; R ) 7Recall that an open set is regular if it is equal to the interior of its closure. 4.4. PSEUDODIFFERENTIAL OPERATORS ON RM 103 and that diﬀerentiation deﬁnes a map

α β r m r−|β| m Dx Dξ : S (U; R ) −→ S (U; R ).

Given a symbol a ∈ Sr(U; Rm), we will make sense of the oscillatory integral Z (4.8) ei(x−y)·ξa(x, ξ) dξ as a distribution. If r < −m then the integral converges to a continuous function ∗ −∞ m (equal to Fξ (a)(x, x − y)). Similarly if a ∈ S (U; R ) then the distribution is ∞ actually a smooth function. In this case, the action of this distribution on φ ∈ Cc (U) is 1 Z φ 7→ ei(x−y)·ξa(x, ξ)φ(x) dξ dx. (2π)m Notice that i(x−y)·ξ 2 i(x−y)·ξ (1 + ξ · Dx)e = (1 + |ξ| )e so for a ∈ Sr(U; Rm) with r < −m and for any ` ∈ N we have Z Z i(x−y)·ξ 2 −` ` i(x−y)·ξ e a(x, ξ)φ(x) dξ dx = (1 + |ξ| ) (1 + ξ · Dx) e a(x, ξ)φ(x) dξ dx Z i(x−y)·ξ ` 2 −` = e (1 − ξ · Dx) ((1 + |ξ| ) a(x, ξ)φ(x)) dξ dx

Moreover we have

` 2 −` X α (1 − ξ · Dx) ((1 + |ξ| ) a(x, ξ)φ(x)) = aα(x, ξ)Dx φ(x) |α|≤`

r−` r with each aα ∈ S (U). So for arbitrary r ∈ R and a ∈ S (U), we deﬁne the ∞ distribution (4.8) on Cc (U) by 1 Z φ 7→ ei(x−y)·ξ(1 − ξ · D )`((1 + |ξ|2)−`a(x, ξ)φ(x)) dξ dx (2π)m x where ` is any non-negative integer larger than r + m. We will continue to denote this distribution by (4.8). Alternately, we can interpret (4.8) as 1 Z φ 7→ lim ei(x−y)·ξψ(εξ)a(x, ξ)φ(x) dξ dx ε→0 (2π)m

∞ m where ψ ∈ Cc (R ) is equal to one in a neighborhood of the origin. Then we can carry out the integration by parts from the previous paragraph before taking the limit. 104 CHAPTER 4. PSEUDODIFFERENTIAL OPERATORS ON RM

For any open set U ⊆ Rm, and s ∈ R we define pseudodifferential operators on U of order r in terms of their integral kernels 1 Z Ψr(U) = K ∈ C−∞(U × U): K = ei(x−y)·ξa(x, ξ) dξ, for some a ∈ Sr(U; m) . (2π)m R The symbol a is known as the total symbol of the pseudodifferential operator. Given a ∈ Sr(U; Rm), there are several notations for the corresponding pseudodifferential operator, e.g.,

Op(a) and a(x, D).

Let us determine the singular support of K ∈ Ψr(U). The useful observation that

α i(x−y)·ξ α i(x−y)·ξ (4.9) (x − y) e = Dξ e shows that 1 Z (x − y)αK = (x − y)αei(x−y)·ξa(x, ξ) dξ (2π)m 1 Z (−1)|α| Z = (Dαei(x−y)·ξ)a(x, ξ) dξ = ei(x−y)·ξ(Dαa(x, ξ)) dξ (2π)m ξ (2π)m ξ is an absolutely convergent integral whenever a ∈ Sr(U) for r < |α|−m and moreover taking further derivatives shows that

(x − y)αK ∈ Cj(U × U) if r − |α| < −m − j so we can conclude that sing supp K ⊆ diagU×U . As with symbols, pseudodiﬀerential operators are ﬁltered by degree

r m r0 m 0 Ψ (R ) ⊆ Ψ (R ) if r < r and a special rˆoleis played by the operators of order −∞,

−∞ m \ r m Ψ (R ) = Ψ (R ). r∈R

Each a(x, D) ∈ Ψs(U) deﬁnes a continuous linear operator

∞ −∞ a(x, D): Cc (U) −→ C (U) 1 ZZ (a(x, D)φ)(x) = ei(x−y)·ξa(x, ξ)φ(y) dy dξ = F ∗(a(x, ξ)F(φ)). (2π)m 4.4. PSEUDODIFFERENTIAL OPERATORS ON RM 105

Proposition 13. For a ∈ Sr(U; Rm), the operator a(x, D) is a continuous map ∞ ∞ (4.10) a(x, D): Cc (U) −→ C (U).

For a ∈ Sr(Rm; Rm), the operator a(x, D) also extends to a linear operator m m a(x, D): S(R ) −→ S(R )

r m ∞ Proof. If a ∈ S (U; R ) and φ ∈ Cc (U) then, for every x, a(x, ξ)φb(ξ) is Schwartz function of ξ, hence the integral Z a(x, D)(φ)(x) = eix·ξa(x, ξ)φb(ξ) dξ is absolutely convergent, and one can diﬀerentiate under the integral sign and the derivatives will also be given by absolutely convergent integrals. Similarly, when a ∈ Sr(Rm), Z Z α α α ix·ξ α ix·ξ α x a(x, D)(φ) = (−1) (Dξ e )a(x, ξ)φb(ξ) dx = (−1) e (Dξ a(x, ξ)φb(ξ)) dx is absolutely convergent for every α.

The total symbol of a diﬀerential operator is a polynomial so it has an expansion k k X X β X p(x, ξ) = pβ(x)ξ = pj(x, ξ) j=0 |β|=j j=0 where each pj(x, ξ) is homogeneous

j pj(x, λξ) = λ pj(x, ξ) and a symbol of order j. It turns out we can ‘asymptotically sum’ any sequence of symbols of decreasing order.

m Proposition 14 (Asymptotic completeness). Let U be an open subset of R , {rj} rj an unbounded decreasing sequence of real numbers and aj ∈ S (U). There exists a ∈ Sr0 (U) such that, for all N ∈ N,

N−1 X rN (4.11) a − aj ∈ S (U) j=0 which we denote X a ∼ aj. 106 CHAPTER 4. PSEUDODIFFERENTIAL OPERATORS ON RM

This is analogous to Borel’s lemma to the eﬀect that any sequence of real numbers can occur as the coeﬃcients of the Taylor expansion of a smooth function at the origin.

Proof. Let us write U as the union of an increasing sequence of compact sets Kj ⊆ U, [ Ki = U, and Kj ⊆ Kj+1 for all i ∈ N. j∈N

∞ m Pick χ ∈ C (R , [0, 1]) vanishing on B1(0) and equal to one outside B2(0), pick a + sequence converging to zero, (εj) ⊆ R , and set

∞ X a(x, ξ) = χ(εjξ)aj(x, ξ). j=0

Note that on any ball {|ξ| < R} only finitely many terms of the sequence are non- zero, so the sum is a well-defined smooth function. We will see that by choosing εj decaying sufficiently quickly, we can arrange (5.2). To see that a(x, ξ) ∈ Sr0 (U) we need to show that, for each α and β there is a constant Cα,β such that

α β p 2 r0−|β| |Dx Dξ χ(εjξ)a(x, ξ)| ≤ Cα,β( 1 + |ξ| ) .

Let us estimate the individual terms in the sum

α β X α γ |β|−|µ| β−µ |Dx Dξ χ(εjξ)aj(x, ξ)| ≤ |Dx Dξ aj(x, ξ)|εj |(D χ)(εjξ)|. γ+µ=β

−1 The term with γ = β is supported in |ξ| ≤ εj and the other terms are supported −1 −1 in εj ≤ |ξ| ≤ 2εj . We can bound the former term by q p 2 rj −|β| p 2 r0−|β| −2 rj −r0 Cα,β(aj)( 1 + |ξ| ) ≤ Cα,β(aj)( 1 + |ξ| ) ( 1 + εj )

r0−rj p 2 r0−|β| ≤ Cα,β(aj)εj ( 1 + |ξ| ) and by similar reasoning we can bound the latter terms by

β−µ r0−rj p 2 r0−|β| Cα,γ(aj)kD χkL∞ εj ( 1 + |ξ| ) .

So altogether we have

r0−rj −2 |α| + |β| ≤ L =⇒ Cα,β(χaj)εj (L) ≤ j and hence

α β −2 p 2 r0−|β| |α| + |β| ≤ L =⇒ |Dx Dξ χ(εj(L)ξ)aj(x, ξ)| ≤ j ( 1 + |ξ| )

α β and |Dx Dξ a(x, ξ)| satisﬁes the bounds we need for |α|+|β| ≤ L. By diagonalization (taking εj = εj(j)) we can arrange the convergence of

p 2 −(r0−|β|) α β ( 1 + |ξ| ) |Dx Dξ a(x, ξ)| for all α, β. The same argument applies to show that

X rN χ(εjξ)aj(x, ξ) ∈ S (U) j≥N P and hence that a ∼ aj.

It is easy to see that if X a ∼ aj

−∞ P then {aj} determine a up to an element of S (U). Indeed, if b ∼ aj then

N−1 N−1 X X rN a − b = (a − aj) − (b − aj) ∈ S (U) j=0 j=0 for all N, hence a − b ∈ S−∞(U). There is a further subset of symbols that imitates polynomials a little closer. r r We say that a symbol a ∈ S (U) is a classical symbol, a ∈ Scl(U), if for each r−j j ∈ N there is a symbol ar−j ∈ S (U) homogeneous in ξ for |ξ| ≥ 1, i.e.,

r−j ar−j(x, λξ) = λ ar−j(x, ξ) whenever |ξ|, |λξ| ≥ 1, and we have ∞ X a ∼ ar−j. j=1

We refer to ar as the principal symbol of a(x, D). 108 CHAPTER 4. PSEUDODIFFERENTIAL OPERATORS ON RM

4.4.2 Amplitudes and mapping properties

In order to prove that pseudodiﬀerential operators are closed under composition and adjoints, it is convenient to allow the symbol to depend on both x and y. Remarkably, this does not lead to a larger class of operators. Let U ⊆ Rm be an open set, we say that a function b ∈ C∞(U × U × Rm) is an amplitude of order r ∈ R,

r m b ∈ S (U × U; R ) if for any multi-indices α, β, γ, there is a constant Cα,β,γ such that

α β γ r−|γ| (4.12) sup |Dx Dy Dξ b(x, y, ξ)| ≤ Cα,β,γ(1 + |ξ|) . x,y∈U

Thus an amplitude is a symbol where there are twice as many ‘space’ variables (x, y) as there are ‘cotangent’ variables ξ. An amplitude deﬁnes a distribution 1 Z φ 7→ ei(x−y)·ξb(x, y, ξ)φ(y) dξ dy. (2π)m

For fixed x this is a well-defined oscillatory integral in (y, ξ) by the discussion above. Since the estimates are uniform in x, this defines a continuous function in x. We will denote the set of distributions with amplitudes of order r by

Ψe r(U).

We clearly have Ψr(U) ⊆ Ψe r(U), and we will soon show that Ψe r(U) = Ψr(U). We start by showing this is true for operators in \ Ψe −∞(U) = Ψe r(U). r∈R Proposition 15. The following are equivalent: i) K ∈ Ψ−∞(U), 1 R i(x−y)·ξ −∞ ii) K = (2π)m e a(x, ξ) dξ for some a ∈ S (U), iii) K ∈ C∞(U × U) and for every N ∈ N and multi indices α, β,

N α β (4.13) sup |(1 + |x − y|) Dx Dy K(x, y)| < ∞. x,y∈U iv) K ∈ Ψe −∞(U). 4.4. PSEUDODIFFERENTIAL OPERATORS ON RM 109

Proof. Clearly, (ii) implies (i) and (i) implies (iv). Let us check that (iv) implies (iii). By definition, K ∈ Ψe −∞(U) means that, r m for each r ∈ R, there is an amplitude br ∈ S (U × U; R ) such that 1 Z K(x, y) = ei(x−y)·ξb (x, y, ξ) dξ. (2π)m r Let us assume that r −m so that the integral converges absolutely and we can use (4.9) and integrate by parts to find 1 Z (x − y)αDβDγK(x, y) = (−1)|α| ei(x−y)·ξ(D + iξ)β(D − iξ)γb (x, y, ξ) dξ x y (2π)m x y r which converges absolutely and uniformly in x, y provided |β| + |γ| + r < |α| − m. α β γ Thus (x − y) Dx Dy K(x, y) is uniformly bounded. Finally we show that (iii) implies (ii). If K satisfies (iii) then the function L(x, z) = K(x, x − z) satisfies N α β sup |(1 + |z|) Dx Dz L(x, z)| < ∞ for all N, α, β. Thus L(x, z) is a Schwartz function in z with bounded derivatives in x. We can write 1 Z K(x, y) = L(x, x − y) = ei(x−y)·ξ(F L)(x, ξ) dξ (2π)m/2 z −∞ m and FzL ∈ S (U; R ) so we have (ii).

At the moment, the mapping property (4.10) is not good enough to allow us to compose operators. We will need an extra condition on the support of the Schwartz ∞ kernels of pseudodiﬀerential operators to guarantee that they preserve Cc (U). Let us say that a subset R ⊆ Rm × Rm is a proper support if the left and right projections m m m R × R −→ R , restricted to R, are proper functions8. An example of a proper support is a compact set, a better example is a set R satisfying R ⊆ {(x, y, ξ): d(x, y) ≤ ε} for any ﬁxed ε > 0. We will say that a distribution K ∈ C−∞(Rm × Rm) is properly supported if supp K is a proper support. We will say that an amplitude b(x, y, ξ) ∈ C∞(Rm × Rm; Rm) is properly supported if

suppx,y b = closure of {(x, y): b(x, y, ξ) 6= 0 for some ξ} is a proper support. 8Recall that any continuous function maps compact sets to compact sets, while a function is called proper if the inverse image of a compact set is always a compact set. 110 CHAPTER 4. PSEUDODIFFERENTIAL OPERATORS ON RM

Theorem 16. Let b(x, y, ξ) ∈ C∞(U × U; Rm) be a properly supported amplitude of order r. The operator

∞ −∞ B : Cc (U) −→ C (U) Z Z (4.14) 1 i(x−y)·ξ Bφ(x) = m e b(x, y, ξ)φ(y) dy dξ (2π) Rm U in fact maps ∞ ∞ B : Cc (U) −→ Cc (U) and induces maps B : C∞(U) −→ C∞(U),B : C−∞(U) −→ C−∞(U), −∞ −∞ and B : Cc (U) −→ Cc (U). The formal adjoint of B is of the same form

∗ ∞ −∞ B : Cc (U) −→ C (U) Z Z ∗ 1 i(x−y)·ξ B φ(x) = m e b(y, x, ξ)φ(y) dy dξ. (2π) Rm U

Moreover, B ∈ Ψr(U) and its total symbol has an asymptotic expansion

|α| X i α α (4.15) a(x, ξ) ∼ (D D b(x, y, ξ)) α! ξ y x=y α where for a multi-index α = (α1, . . . , αm), α! denotes α1! ··· αm!.

Proof. First note that the description of the adjoint follows immediately from Z Z Z 1 i(x−y)·ξ hBφ, fi = m e b(x, y, ξ)φ(y)f(x) dy dξ dx (2π) Rm U U Z Z Z 1 i(x−y)·ξ = m e b(x, y, ξ)f(x)φ(y) dx dξ dy (2π) U Rm U ∞ for all φ, f ∈ Cc (U). ∞ ∞ We see that B maps Cc (U) into C (U) by the same argument as in Proposition ∞ 13. However now if φ ∈ Cc (U) and supp φ = K then we know that

−1 0 πy (K) ∩ suppx,y b = K is a compact set and it is immediate from (4.14) that supp B(φ) ⊆ K0. Thus B : ∞ ∞ Cc (U) −→ Cc (U) and by the same argument

∗ ∞ ∞ B : Cc (U) −→ Cc (U). 4.4. PSEUDODIFFERENTIAL OPERATORS ON RM 111

By duality this means that B and B∗ also induce maps C−∞(U) −→ C−∞(U). Let φ ∈ C∞(U), we know that Bφ is deﬁned, but a priori this is a distribution. Let us check that Bφ is smooth, by checking that it is smooth on any compact set K ⊆ U. We know that −1 0 πy (K) ∩ suppx,y b = K ∞ 0 is also compact, so we can choose χ ∈ Cc (U) such that χ(p) = 1 for all p ∈ K . Then, on K, we have Bφ = Bχφ so Bφ is smooth in the interior of K. This shows that B : C∞(U m) −→ C∞(U m) and since B∗ is of the same form, B∗ : C∞(U) −→ C∞(U). Then by duality both B ∗ −∞ −∞ and B induce maps Cc (U) −→ Cc (U). Let us deﬁne

|α| X i α α aj(x, ξ) = (D D b(x, y, ξ)) . α! ξ y x=y |α|=j

r−j m Note that aj ∈ S (R ) so it makes sense to deﬁne a(x, ξ) to be a symbol obtained P by asymptotically summing aj. We will be done once we show that B −a(x, D) ∈ Ψ−∞(U). Recall from Taylor’s theorem that, for each ` ∈ N, X 1 X b(x, y, ξ) = ∂αb(x, x, ξ)(y − x)α + c (x, y, ξ)(y − x)α α! y α |α|≤`−1 |α|=` Z 1 `−1 α with cα(x, y, ξ) = (1 − t) ∂y b(x, (1 − t)x + ty, ξ) dt 0

α and note the important fact that ∂y b(x, y, ξ), and cα(x, y, ξ) all satisfy (4.12). Using (4.9) and integrating by parts we ﬁnd that   Z Z 1 i(x−y)·ξ 1 i(x−y)·ξ X 1 α α m e b(x, y, ξ) dξ = m e  Dξ ∂y b(x, x, ξ) dξ (2π) m (2π) m α! R R |α|≤`−1   Z 1 i(x−y)·ξ X 1 α α + m e  Dξ ∂y cα(x, y, ξ) dξ (2π) m α! R |α|=` `−1 Z Z X 1 i(x−y)·ξ 1 i(x−y)·ξ = m e aj(x, ξ) dξ + m e R`(x, y, ξ) dξ (2π) m (2π) m j=1 R R where R`(x, y, ξ) satisﬁes (4.12) with r replaced by r − `. 112 CHAPTER 4. PSEUDODIFFERENTIAL OPERATORS ON RM

r−` By deﬁnition of a(x, ξ), we know that there is an element Q` ∈ S (U) such that `−1 X a(x, D) = aj(x, D) + Q`(x, D) j=1 and hence we know that Z 1 i(x−y)·ξ (4.16) B − a(x, D) = K(x, y) = m e (R`(x, y, ξ) − Q`(x, ξ)) dξ (2π) Rm and R`(x, y, ξ) − Q`(x, ξ) satisﬁes (4.12) with r replaced by r − `. It is clear from the proof of Proposition 15 that the inequalities (4.13) for a given N, α, β hold for K(x, y) if ` is large enough. Since we are able to choose ` as large as we like, this means that the inequalities (4.13) hold for all N, α, β, but this means that B − a(x, D) ∈ Ψ−∞(U) as required.

This theorem lets us quickly deduce that we get the same class of pseudodifferential operators using amplitudes of symbols. And it lets us show that pseudod- iﬀerential operators with properly supported amplitudes are closed under adjoints and under composition.

Corollary 17. For all r ∈ R, the sets of distributions Ψr(U) and Ψe r(U) coincide.

Proof. Let K ∈ Ψe r(U) have amplitude b(x, y, ξ). Choose χ ∈ C∞(U × U) such that χ(x, y) = 0 if d(x, y) > 1

r then we can write K as the sum of K1 ∈ Ψe (U) with amplitude χ(x, y)b(x, y, ξ) and −∞ K2 ∈ Ψe (U) with amplitude (1 − χ(x, y))b(x, y, ξ). The theorem guarantees that r −∞ K1 ∈ Ψ (U) because it is properly supported, and we already know that Ψe (U) = Ψ−∞(U).

We will say that a pseudodiﬀerential operator is properly supported if its Schwartz kernel is properly supported. It is easy to see that this occurs if and only if we can represent it using a properly supported amplitude. Corollary 18. If A ∈ Ψr(U) is properly supported then A∗ ∈ Ψr(U) is properly supported with symbol

X i|α| (4.17) σ (A∗)(x, ξ) ∼ DαDασ (A)(x, ξ). tot α! ξ x tot |α|≥0

∗ If σtot(A) is a classical symbol then so is σtot(A ) and

∗ (4.18) σr(A ) = σr(A). 4.4. PSEUDODIFFERENTIAL OPERATORS ON RM 113

Proof. For any properly supported function χ ∈ C∞(U × U) identically equal to one in a neighborhood of the diagonal containing supp KA, the amplitude

χ(x, y)σtot(A)(x, ξ) is a properly supported amplitude for A. It follows from the theorem that A∗ ∈ r r Ψe (U) = Ψ (U), with properly supported amplitude χ(y, x)σtot(A)(y, ξ), and symbol

|α| ∗ X i α α σtot(A ) ∼ (D D χ(y, x)σtot(A)(y, ξ)) α! ξ y x=y |α|≥0 which, since χ(x, y) is equal to one in a neighborhood of the diagonal, implies (4.17). This formula in turn implies (4.18). Corollary 19. If A ∈ Ψr(U),B ∈ Ψs(U) are properly supported, the composition

∞ ∞ A ◦ B : Cc (U) −→ Cc (U) is a properly supported pseudodiﬀerential operator A ◦ B ∈ Ψr+s(U) with symbol

X i|α| (4.19) σ (A ◦ B) ∼ Dασ (A)(x, ξ)Dασ (B)(x, ξ). tot α! ξ tot x tot |α|≥0

If σtot(A) and σtot(B) are classical symbols then so is σtot(A ◦ B) and

(4.20) σr+s(A ◦ B) = σr(A)σs(B).

Proof. Let a(x, ξ) = σtot(A)(x, ξ) and b(x, ξ) = σtot(B)(x, ξ). We know that 1 Z B∗u(x) = eix·ξb(x, ξ)u(ξ) dξ (2π)m b

∞ integrating against a test function φ ∈ Cc (U) gives 1 Z hBφ, ui = hφ, Bui = e−ix·ξφ(x)b(x, ξ)u(ξ) dξ dx (2π)m b and so Z Bφc (ξ) = e−iy·ξb(y, ξ)φ(y) dy.

This gives us a representation of B with an amplitude depending only on y. We can then write Z Z (A ◦ B)(φ) = A(Bφ) = eix·ξa(x, ξ)Bφc (ξ) dξ = ei(x−y)·ξa(x, ξ)b(y, ξ)φ(y) dy dξ. 114 CHAPTER 4. PSEUDODIFFERENTIAL OPERATORS ON RM

Thus A ◦ B ∈ Ψe r+s(U) with amplitude a(x, ξ)b(y, ξ). Next let us check that it is properly supported. If K ⊆ Rm is a compact set, then 0 −1 m K = πL(πR (K) ∩ supp KB) ⊆ R is compact and hence −1 −1 0 πR (K) ∩ supp KA◦B = πR (K ) ∩ supp A is compact. Similarly,

−1 −1 −1 πL (K) ∩ supp KA◦B = πL (πR(πL (K) ∩ supp KA)) ∩ supp B is compact, and so A ◦ B is properly supported. Thus, as before, we can multiply its amplitude by a cut-oﬀ function to make it a properly supported amplitude without aﬀecting formula (4.15), which yields (4.19). If A and B are classical, then formula (4.19) shows that A ◦ B is classical, and that the principal symbols satisfy (4.20).

To define pseudodifferential operators on manifolds, we need to know that an operator that looks like a pseudodifferential operator in one set of coordinates will look like a pseudodifferential operator in any coordinates.

Theorem 20. Let U and W be open subsets of Rm and let F : U −→ W be a diﬀeomorphism. Let P ∈ Ψr(U) and let

∞ ∞ F∗P : Cc (W) −→ C (W) −1 (F∗P )(φ) = P (φ ◦ F ) ◦ F

r then F∗P ∈ Ψ (W). For each multi-index α there is a function φα(x, ξ) which is a polynomial in ξ of degree at most |α|/2, with φ0 = 1, such that the symbol of (F∗P ) satisﬁes

X φα(x, ξ) (4.21) σ (F P )(F (x), ξ) ∼ (Dασ (P ))(x, (DF )t(x)ξ). tot ∗ α! ξ tot |α|≥0

If σtot(P ) is classical, so is σtot(F∗P ) and its principal symbol satisﬁes

t (4.22) σr(F∗P )(F (x), ξ) = σr(P )(x, DF (x)ξ).

Notice that, as with a symbol of a differential operator, this shows that the principal symbol of a pseudodifferential operator on U is well-defined as a section of the cotangent bundle T ∗U. 4.4. PSEUDODIFFERENTIAL OPERATORS ON RM 115

Proof. Let us denote F −1 by H. Note that Z 1 H(x) − H(y) = ∂tH(tx + (1 − t)y) dt = Φ(x, y) · (x − y) 0 where Φ is a smooth matrix-valued function. Since Φ(x, x) = limy→x Φ(x, y) = DH(x) and H is a diﬀeomorphism, the matrix Φ(x, y) is invertible in a neighborhood ∞ Z of the diagonal. Let χ ∈ Cc (Z) be identically equal to one in a neighborhood of the diagonal.

Let p = σtot(P ). We have

(F∗P )(φ)(x) = P (u ◦ F )(H(x)) Z Z 1 i(H(x)−y)·ξ = m e p(H(x), ξ)u(F (y)) dy dξ (2π) Rm U Z Z 1 i(H(x)−H(z))·ξ = m e p(H(x), ξ)u(z) det DH(z) dz dξ (2π) Rm W Z Z 1 i(x−z)·Φ(x,z)tξ = m e p(H(x), ξ)u(z) det DH(z) dz dξ. (2π) Rm W Let us briefly denote the last integrand by γ(x, z, ξ), and decompose it as χγ + (1 − χ)γ. We know that (1−χ)γ is a pseudodifferential operator whose amplitude vanishes in a neighborhood of the diagonal, and hence is an element of Ψ−∞(W). On the other hand, when integrating χγ we can make the change of variables η = Φ(x, z)tξ and find Z Z 1 i(x−z)·η t −1 det DH(z) m e χ(x, z)p(H(x), (Φ(x, z) ) η)u(z) dz dη (2π) Rm W det Φ(x, z) which is an element of Ψe r(W) with amplitude χ(x, z) p(x, z, η) = p(H(x), (Φ(x, z)t)−1η) e det DF (H(z)) det Φ(x, z) so we can apply (4.15) and establish (4.21), which in turn proves (4.22).

Finally, let us consider pseudodiﬀerential operators acting between sections of bundles. In the present context this is very simple since any vector bundle over Rm is trivial. Thus if E −→ Rm and F −→ Rm are vector bundles, a pseudodiﬀerential operator of order r acting between sections of E and sections of F,

r m A ∈ Ψ (R ; E,F ), is simply a (rank F ) × (rank E) matrix with entries in Ψr(Rm). The extension of the discussion above of composition, adjoint, and mapping properties is straightforward. 116 CHAPTER 4. PSEUDODIFFERENTIAL OPERATORS ON RM 4.5 Exercises

Exercise 1. Show that a distribution T on Rm supported at the origin is a differential operator applied to δ0. Hint: First use continuity to find a number K such that hφ, T i = 0 for all φ that vanish to order K at the origin. Next, use this to view T as a linear functional on the finite dimensional vector space consisting of the coefficients of the Taylor expansion of order K of smooth functions at the origin. Exercise 2. Show that, if P is a differential operator and T is a distribution, supp(P (T )) ⊆ supp(T ) and sing supp(P (T )) ⊆ sing supp T. Exercise 3. ∞ Let f ∈ C (R) and let δ0 be the Dirac delta distribution at the origin. Express the 3 2 distribution f∂xδ0 as a C-linear combination of the distributions δ0, ∂xδ0, ∂xδ0 and 3 ∂xδ0. Exercise 4. −∞ a) Suppose that T ∈ C (R) satisfies ∂xT = 0 prove that T is equal to a constant C ∈ C. b) Let S ∈ C−∞(R), show that there is a distribution R ∈ C−∞(R) such that ∂xR = S. ∞ R ∞ Hint: Choose ψ ∈ Cc (R) such that ψ dx = 1. Given φ ∈ Cc (M) show that ∞ R there exists φe ∈ Cc (R) such that φ = φ dx ψ + ∂xφ.e For (a), show that R hφ, T i = φ dx hψ, T i, so C = hψ, T i. For (b), show that hφ, Ri = −hφ,e Si defines a distribution that satisfies ∂xR = S. Exercise 5. Suppose that T ∈ S0(Rm) satisfies ∆T = 0 on Rm. Show that T is a polynomial. Hint: First show that F(T ) is a distribution supported at the origin. Exercise 6. Let U ⊆ Rm be an open set and K ∈ C−∞(U ×U). Recall that K defines an operator ∞ −∞ Cc (U) −→ C (U). Suppose that K satisfies

∞ ∞ −∞ −∞ K : Cc (U) −→ C (U) and K : Cc (U) −→ C (U) and that K is smooth except possibly on diagU , i.e., sing supp K ⊆ diagU . Show that

sing supp Ku ⊆ sing supp u

−∞ for all u ∈ Cc (U). 4.6. BIBLIOGRAPHY 117

Exercise 7. −∞ ∞ Let KA ∈ C (U × U) and suppose that the associated operator A : Cc (U) −→ −∞ ∞ ∞ C (U) maps Cc (U) −→ C (U). Show that: ∞ −∞ ∞ i) If KA ∈ Cc (U × U), then A extends to a map C (U) −→ Cc (U). −∞ ∞ ∗ −∞ ∞ ii) Conversely, if A : C (U) −→ Cc (U) and A : C (U) −→ Cc (U), show that ∞ KA ∈ Cc (U × U). Exercise 8. Give an example of a pseudodifferential operator A ∈ Ψ∗(Rm) (e.g., of order −∞) ∞ m and a function φ ∈ Cc (R ) such that supp A(φ) is not contained in supp φ. Exercise 9. For properly supported A ∈ Ψr(Rm),B ∈ Ψs(Rm) with classical symbols, show that the commutator [A, B] ∈ Ψr+s−1(Rm) and find its principal symbol. Exercise 10. r m Assume that A ∈ Ψ (R ) is properly supported with classical symbol. If σr(A) is m ∗ ∗ real-valued (i.e., σr(A)(x, ξ) ∈ R for all x, ξ ∈ R ). Show that [A, A ] = AA − A∗A ∈ Ψ2r−2(Rm). Exercise 11. a) Given a function ∞ m m−1 a(x, ξ) ∈ C (R × S ) uniformly bounded together with its derivatives, and a number r ∈ R show that there exists a pseudodifferential operator A ∈ Ψr(Rm) with classical symbol such that

σr(A)(x, ξ) |ξ|=1 = a(x, ξ).

r m r−1 m b) If A ∈ Ψ (R ) has classical symbol and σr(A) = 0 show that A ∈ Ψ (R ).

4.6 Bibliography

Of the books we have been using so far, both Spin geometry by Lawson and Michelson and the second volume of Partial differential equations by Taylor, treat pseudodifferential operators. For distributions, one can look at Bony, Cours d’analyse, volume one of Hormander¨ , The analysis of linear partial differential operators, and Introduction to the theory of distributions by Friedlander and Joshi. Richard Melrose has several excellent sets of lecture notes involving pseudodifferential operators at http://math.mit.edu/~rbm/Lecture_notes.html. Other good sources are the books Pseudodifferential operators by Michael Taylor and 118 CHAPTER 4. PSEUDODIFFERENTIAL OPERATORS ON RM

Pseudodifferential operators and spectral theory by Mikhail Shubin, and the lecture notes Introduction to pseudo-differential operators by Mark Joshi (available online on the arXiv). For an in-depth treatment, one can look at the four volumes of The analysis of linear partial differential operators by Hormander¨ . Chapter 5

Pseudodiﬀerential operators on manifolds

5.1 Symbol calculus and parametrices

Let M be an orientable compact Riemannian manifold, let E and F be complex vector bundles over M endowed with Hermitian metrics1 and let us denote by

HOM(E,F ) −→ M × M the vector bundle whose ﬁber over a point (ζ, ζ0) ∈ M × M is the vector space Hom(Eζ ,Fζ0 ). A distribution K ∈ C−∞(M × M; HOM(E,F )) is a classical pseudodiﬀerential operator on M of order r, acting between sections of E and sections of F,

r K ∈ Ψcl(M; E,F ) if K is smooth away from the diagonal, i.e.,

sing supp K ⊆ diagM ,

∞ and, for any coordinate chart φ : U −→ V of M and any smooth function χ ∈ Cc (V), the distribution χ(x)K(x, y)χ(y) coincides (via φ) with a pseudodiﬀerential operator on U, r rank E rank F Kφ,χ ∈ Ψ (U; R , R ) 1We assume orientability, compactness, and a choice of metrics for simplicity, but we could more invariantly work with operators on half-densities.

119 120 CHAPTER 5. PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS acting between the trivial bundles φ∗E and φ∗F, with classical symbol of order r.

Although this definition requires us to check that Kφ,χ is a pseudodifferential operator for every coordinate chart, Theorem 20 shows that it is enough to check this for a single atlas of M. This same theorem guarantees that the principal symbols of the operators Kφ,χ together define

∞ ∗ ∗ ∗ σr(K) ∈ C (T M; Hom(π E, π F )) with π : T ∗M −→ M the canonical projection. The principal symbol is homogeneous away form the zero section of T ∗M, so we do not lose any interesting information if we restrict it to the cosphere bundle

∗ ∗ S M = {v ∈ T M : g(v, v) = 1}. From now on, we will think of the symbol as a section

∞ ∗ ∗ ∗ σr(K) ∈ C (S M; Hom(π E, π F )). Let us summarize some facts about pseudodiﬀerential operators on manifolds that follow from our study of pseudodiﬀerential operators on Rm. Theorem 21. Let M be an orientable compact Riemannian manifold and let E,F, and G be complex vector bundles over M endowed with Hermitian metrics. For each r r ∈ R we have a class Ψcl(M; E,F ) of distributions satisfying r−k r Ψcl (M; E,F ) ⊆ Ψcl(M; E,F ) for all k ∈ N with a principal symbol map participating in a short exact sequence (5.1) r−1 r σr ∞ ∗ ∗ ∗ 0 −→ Ψcl (M; E,F ) −→ Ψcl(M; E,F ) −−→C (S M; Hom(π E, π F )) −→ 0.

Pseudodiﬀerential operators are asymptotically complete: r−j r For any r ∈ R and sequence Aj ∈ Ψcl (M; E,F ) there is an A ∈ Ψcl(M; E,F ) such that, for all N ∈ N, N X r−N−1 (5.2) A − Aj ∈ Ψcl (M; E,F ) j=1 which we denote X A ∼ Aj.

Pseudodiﬀerential operators act on functions and on distributions

r A ∈ Ψcl(M; E,F ) =⇒ A : C∞(M; E) −→ C∞(M; F ) and A : C−∞(M; E) −→ C−∞(M; F ), 5.1. SYMBOL CALCULUS AND PARAMETRICES 121 and they are closed under composition. Composition of pseudodiﬀerential operators is compatible with both the degree and the symbol

r s A ∈ Ψcl(M; E,F ),B ∈ Ψcl(M; F,G) =⇒ r+s A ◦ B ∈ Ψcl (M; E,G) and σr+s(A ◦ B) = σr(A) ◦ σs(B). The formal adjoint of a pseudodifferential operators is again a pseudodifferential operator of the same order and their principal symbols are mutually adjoint. Pseudodifferential operators contain differential operators

k k Diff (M; E,F ) ⊆ Ψcl(M; E,F ) and the symbol of a differential operators coincides with its symbol as a pseudodifferential operator. Pseudodifferential operators also contain smoothing operators

−∞ \ r ∞ A ∈ Ψcl (M; E,F ) = Ψcl(M; E,F ) ⇐⇒ A ∈ C (M × M; HOM(E,F )) r∈R ⇐⇒ A : C−∞(M; E) −→ C∞(M; F ) and A∗ : C−∞(M; F ) −→ C∞(M; E).

Notice that, since M is compact, we do not need to distinguish between smooth functions and those with compact support (and similarly for distributions) and that all distributions are properly supported. r We say that a pseudodiﬀerential operator A ∈ Ψcl(M; E,F ) is elliptic if its principal symbol is invertible, i.e.,

∞ ∗ ∗ σr(A) ∈ C (M; Iso(π E, π F )).

Thus an elliptic diﬀerential operator, such as a Laplace-type or Dirac-type operator, is also an elliptic pseudodiﬀerential operator. Although invertibility of the symbol can not guarantee invertibility of the operator, e.g., the Laplacian on functions has all constant functions in its null space, it turns out that invertibility of the symbol is equivalent to invertibility of the operator up to smoothing operators.

r Theorem 22. A pseudodiﬀerential operator A ∈ Ψcl(M; E,F ) is elliptic if and only if there exists a pseudodiﬀerential operator

−r −∞ −∞ (5.3) B ∈ Ψcl (M; F,E) s.t. AB − IdF ∈ Ψcl (M; F ),BA − IdE ∈ Ψcl (M; E)

If B satisﬁes (5.3) we say that B is a parametrix for A. It is easy to see that A has a unique parametrix up to smoothing operators (see exercise 2). 122 CHAPTER 5. PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS

Proof. Since the symbol of A is invertible and the sequence (5.1) is exact, we know that there is an operator

−r −1 ∞ ∗ ∗ ∗ B1 ∈ Ψcl (M; F,E), with σ−r(B) = σr(A) ∈ C (S M; Iso(π F, π E)).

The composition A ◦ B1 is a pseudodiﬀerential operator of order zero and

σ0(A ◦ B1) = σr(A) ◦ σ−r(B) = Idπ∗E 0 is the same symbol as the identity operator IdE ∈ Ψcl(M; E). Again appealing to the exactness of (5.1) we see that

−1 R1 = IdE −AB1 ∈ Ψcl (M; E).

Let us assume inductively that we have found B1,...,Bk satisfying B ∈ Ψ−r−(i−1)(M; F,E) (5.4) i cl −k and Rk = IdE −A(B1 + ... + Bk) ∈ Ψcl (M; E) −r−k and try to ﬁnd Bk+1 ∈ Ψcl (M; F,E) such that −k−1 Rk+1 = IdE −A(B1 + ... + Bk+1) ∈ Ψcl (M; E). Note that we can write this last equation as

Rk+1 = Rk − ABk+1 −k and since both Rk and ABk+1 are elements of Ψcl (M; E) exactness of (5.1) tells us that −k−1 Rk+1 ∈ Ψcl (M; E) ⇐⇒ σk(Rk − ABk+1) = 0. −r−k So it suﬃces to let Bk+1 be any element of Ψcl (M; F,E) with principal symbol −1 σ−r−k(Bk+1) = σr(A) ◦ σ−k(Rk) which we know exists by another application of exactness of (5.1). Thus inductively we can solve (5.4) for all k ∈ N. −r Next use asymptotic completeness to ﬁnd B ∈ Ψcl (M; F,E) such that X B ∼ Bi and notice that, for any N ∈ N, −N IdE −AB = IdE −A(B1 + ... + BN ) − A(B − B1 − ... − BN ) ∈ Ψcl (M; E) −∞ hence IdE −AB ∈ Ψcl (M; E) as required. 0 −r The same construction can be used to construct B ∈ Ψcl (M; F,E) such that 0 −∞ 0 −∞ IdF −B A ∈ Ψcl (M; F ) but then B − B ∈ Ψcl (M; F,E) by exercise 2 and so we −∞ also have IdF −BA ∈ Ψcl (M; F ). 5.2. HODGE DECOMPOSITION 123

An immediate corollary is elliptic regularity.

r Corollary 23. Let A ∈ Ψcl(M; E,F ) be an elliptic pseudodiﬀerential operator. If u ∈ C−∞(M; E) then

Au ∈ C∞(M; F ) =⇒ u ∈ C∞(M; E).

More generally,

sing supp Au = sing supp u for all u ∈ C−∞(M; E).

Proof. It is true for any pseudodiﬀerential operator A that

sing supp Au ⊆ sing supp u,

(see exercise 6 of the previous lecture) the surprising part is the opposite inclusion. −r −∞ Let B ∈ Ψcl (M; F,E) be a parametrix for A and R = IdE −BA ∈ Ψcl (M; E) then any u ∈ C−∞(M; E) satisﬁes

u = BAu + Ru and since Ru ∈ C∞(M; E) we have

sing supp u = sing supp BAu ⊆ sing supp Au as required.

In the next few sections we will continue to develop consequences of the existence of a parametrix.

5.2 Hodge decomposition

∗ An elliptic operator in Ψcl(M; E,F ) induces a decomposition of the sections of E into its null space, which is finite dimensional, and its orthogonal complement and of the sections of F into its image, which has finite codimension, and its orthogonal complement. This is a version of the Hodge decomposition for an arbitrary elliptic operator. We will establish this and then show that it implies the Hodge decomposition of differential forms. A map between topological vector spaces is Fredholm if its null space is finite dimensional and its range is closed and has a finite dimensional complementary space. The following theorem says that elliptic operators induce Fredholm maps between sections of vector bundles in both the smooth and distributional settings. 124 CHAPTER 5. PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS

Theorem 24. If M is a compact Riemannian manifold, E and F Hermitian vector r bundles over M, and A ∈ Ψcl(M; E,F ) an elliptic pseudodiﬀerential operator of order r ∈ R, then: i) The null space of A acting on distributions is a ﬁnite dimensional space of smooth sections of E,

ker A = {u ∈ C−∞(M; E): Au = 0} ⊆ C∞(M; E) is ﬁnite dimensional. ii) The image of A acting on smooth sections of E is a closed subspace of C∞(M; F ), and the image of A acting on distributional sections of E is a closed subspace of C−∞(M; F ). iii) We have decompositions

C∞(M; F ) = A(C∞(M; E)) ⊕ ker A∗, C−∞(M; F ) = A(C−∞(M; E)) ⊕ ker A∗ that are ‘orthogonal’ with respect to the distributional pairing on sections of F.

−r Proof. Let B ∈ Ψcl (M; F,E) be a parametrix for A, so that −∞ −∞ R = AB − IdF ∈ Ψcl (M; F ), and L = BA − IdE ∈ Ψcl (M; E).

As noted above, any u ∈ C−∞(M; E) satisfies u = BAu − Lu, hence elements ∞ of ker A satisfy u = −Lu and are in C (M; E). Moreover, if (un) is an orthonormal −∞ ∞ basis of ker A then L ∈ Ψcl (M; E) implies that (−Lun) = (un) is a C -bounded sequence and Arzela-Ascoli implies that it has a subsequence that converges in C∞ and hence in L2. Since an infinite orthonormal sequence is never L2-Cauchy, we can conclude that ker A is finite dimensional. Note that the same argument applies to A∗, since it too is an elliptic operator. Let us prove (ii) and (iii) first for smooth sections, starting with showing that A(C∞(M; E)) is a closed subset of C∞(M; F ). Let v ∈ C∞(M; F ) be a limit point of ∞ ∞ ⊥ A(C (M; E)) and choose (un) ⊆ C (M; E) ∩ (ker A) such that Aun → v. 2 If (un) has a subsequence that is bounded in L , then Arzela-Ascoli implies ∞ ∞ that (−Lun) has a subsequence that converges in C , say to w ∈ C (M; F ). Passing to this subsequence, we have

ABAun → ABv, and ABAun = Aun + ALun → v − Aw,

∞ and hence v = ABv + Aw ∈ A(C (M; E)). On the other hand, if kunkL2 → ∞, then un ubn = kunk ∞ ∞ satisfies (ubn) ⊆ C (M; E) and Aubn → 0 in C (M; F ). Arguing as before we can find ∞ 2 a subsequence of (ubn) that converges in C (M; E). The limit must have L -norm 5.2. HODGE DECOMPOSITION 125 equal to one, be in ker A, and be in (ker A)⊥, which is a contradiction. Thus we have shown that A(C∞(M; E)) is closed in C∞(M; F ). Directly from the defining property of A∗,

∗ hAu, viC∞(M;E) = hu, A viC∞(M;F ), we see that the orthogonal complement of A(C∞(M; E)) in C∞(M; F ) is ker A∗. To see that the closed subspace

A(C∞(M; E)) ⊕ ker A∗ ⊆ C∞(M; F ) is all of C∞(M; F ), it suffices to see that if φ ∈ C−∞(M; F ) vanishes on this subspace then φ = 0. Any φ that vanishes on A(C∞(M; E)), i.e., that satisfies hφ, Aui = 0 for all u ∈ C∞(M; E), satisfies A∗φ = 0 and so φ ∈ ker A∗; thus if φ also vanishes on ker A∗ then φ must be identically zero.

This also implies that (ii) and (iii) hold for distributional sections since we can identify A(C−∞(M; E)) with the annihilator of ker A∗ with respect to the distributional pairing on sections of F. Indeed, the distributional pairing

hAφ, ui = hφ, A∗ui makes it obvious that A(C−∞(M; E)) is a subset of the annihilator of ker A∗. On the other hand, given ψ ∈ C−∞(M; F ) such that ψ(u) = 0 for all u ∈ ker A∗, let us deﬁne α ∈ C−∞(M; E) as follows: given u ∈ C∞(M; E), with u = A∗v + w its decomposition with respect to

C∞(M; E) = A∗(C∞(M; F )) ⊕ ker A, let α(u) = ψ(v). To see that this is well-deﬁned, note that if u = A∗v0 + w0 then w = w0 and v − v0 ∈ ker A∗, but then ψ(v) = ψ(v0). To see that α is a distribution, i.e., is a continuous functional, we apply the open mapping theorem2 to the map A : C∞(M; E) −→ A(C∞(M; E)) and use that the range of A is closed. Thus we have deﬁned α ∈ C−∞(M; E) such that ψ = A(α), showing that the annihilator of ker A∗ coincides with A(C−∞(M; E)).

We can interpret this theorem as telling us about solutions to the equation

Au = f.

2See, e.g., Theorem 2.11 of Walter Rudin, Functional Analysis International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991. 126 CHAPTER 5. PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS

Namely, we can solve this equation precisely when f ∈ C−∞(M; F ) satisﬁes the (ﬁnitely many) conditions

∗ hf, u1i = 0, hf, u2i = 0,..., hf, uki = 0 where {u1, . . . , uk} is a basis of ker A and, in that case, there is a ﬁnite dimensional space of solutions, u0 + ker A ⊆ −∞ C (M; E) with u0 a particular solution. By elliptic regularity these solutions are smooth if and only if f is smooth. We also know that the ‘cokernel’ of A as an operator on smooth sections

coker(A) = C∞(M; F )/A(C∞(M; E)) or as an operator on distributional sections

coker(A) = C−∞(M; F )/A(C−∞(M; E)) is naturally isomorphic to ker A∗ ⊆ C∞(M; F ). The index of A as a Fredholm operator ind(A) = dim ker A − dim coker A = dim ker A − dim ker A∗ is ﬁnite and is the same when A acts on functions and when A acts on distributions. Another immediate consequence of this corollary is the Hodge decomposition.

Corollary 25. Let M be a compact Riemannian manifold, for any k ∈ N we have decompositions

C∞(M;ΛkT ∗M) = d C∞(M;Λk−1T ∗M) ⊕ δ C∞(M;Λk+1T ∗M) ⊕ Hk(M) C−∞(M;ΛkT ∗M) = d C−∞(M;Λk−1T ∗M) ⊕ δ C−∞(M;Λk+1T ∗M) ⊕ Hk(M). where the latter is a topological direct sum and the former an orthogonal direct sum. It follows that the de Rham cohomology groups coincide with the Hodge cohomology groups, and this remains true if the de Rham cohomology is computed distributionally.

Proof. Applying Theorem 24 to the Hodge Laplacian we know that

∞ k ∗ k C (M;Λ T M) = Im(∆k) ⊕ H (M) ⊆ d C∞(M;Λk−1T ∗M) + δ C∞(M;Λk+1T ∗M) ⊕ Hk(M).

However it is easy to see that d C∞(M;Λk−1T ∗M) ⊥ δ C∞(M;Λk+1T ∗M) since

2 hdω, δηiΛkT ∗M = hd ω, ηiΛ∗T ∗M = 0, which establishes the decomposition for smooth forms. 5.3. THE INDEX OF AN ELLIPTIC OPERATOR 127

The decomposition for smooth forms shows that any distribution that vanishes on d C∞(M;Λk−1T ∗M) , δ C∞(M;Λk+1T ∗M) , and Hk(M) must be identically zero. Since this is true for distributions in

d C−∞(M;Λk−1T ∗M) ∩ δ C−∞(M;Λk+1T ∗M) , we also have the decomposition for distributional forms. The decomposition clearly shows that

ker d :Ωk(M) −→ Ωk+1(M) ∼= Hk(M) Im (d :Ωk−1(M) −→ Ωk(M)) and similarly for distributions.

The Hodge decomposition also holds on the Dolbeault complex (3.10), and more generally for an elliptic complex defined as follows: Let E1,...,E` be Her- mitian vector bundles over a compact Riemannian manifold M and suppose that we 1 have differential operators Di ∈ Diff (M; Ei,Ei+1),

∞ D1 ∞ D2 D`−1 ∞ 0 −→ C (M; E1) −−−→C (M; E2) −−−→ ... −−−−→C (M; E`) −→ 0

∗ such that Di ◦ Di+1 = 0. This complex is an elliptic complex if, for all ξ ∈ S M, the sequence of leading symbols

∗ σ1(D1)(ξ) ∗ σ1(D2)(ξ) σ1(D`−1)(ξ) ∗ 0 −→ π E1 −−−−−−−→ π E2 −−−−−−−→ ... −−−−−−−−→ π E` −→ 0 is exact. In this setting the same proof as above shows that we have a natural isomorphism of ﬁnite dimensional vector spaces

ker Di ∼ ∗ ∗ = ker(DiDi−1 + Di+1Di), Im Di−1 i.e., the cohomology spaces of the complex are ﬁnite dimensional and coincide with the Hodge cohomology spaces.

5.3 The index of an elliptic operator

An elliptic operator acting on a closed manifold has a distinguished parametrix. r Given A ∈ Ψcl(M; E,F ) elliptic, let us write

C∞(M; E) = ker A ⊕ Im(A∗), C∞(M; F ) = ker A∗ ⊕ Im(A) 128 CHAPTER 5. PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS and deﬁne G : C∞(M; F ) −→ C∞(M; E) by ( Gs = 0 if s ∈ ker A∗ Gs = u if s ∈ Im(A) and u is the unique element of Im(A∗) s.t. Au = s

The operator G is known as the (L2) generalized inverse of A. It satisﬁes

GA = IdE −Pker A, AG = IdF −Pker A∗

∗ where Pker A and Pker A∗ are the orthogonal projections onto ker A and ker A , respectively. Notice that these projections are smoothing operators, since we can write

X ∗ KPker A (x, y) = φj(x)φj (y), (5.5) j ∗ with {φj} a basis of ker A and {φj } the dual basis and similarly for Pker A∗ . It turns out that G is itself a pseudodiﬀerential operator, and we can express the index of A using the trace of a smoothing operator

−∞ Tr : Ψcl (M; E) −→ C Z Tr(P ) = trE KP diag dvolg M M

r ∞ Theorem 26. Let A ∈ Ψcl(M; E,F ) be an elliptic operator and G : C (M; F ) −→ ∞ −r C (M; E) its generalized inverse, then G ∈ Ψcl (M; F,E) and

ind(A) = Tr(Pker A) − Tr(Pker A∗ ) =: Tr([A, G]).

−r Proof. Let B ∈ Ψcl (M; F,E) be a parametrix for A and

−∞ −∞ R = BA − IdE ∈ Ψcl (M; E),S = AB − IdF ∈ Ψcl (M; F ). Making use of associativity of multiplication, we have

BAG = G + RG = B − BPker A∗ , GAB = G + GS = B − Pker A

hence G = B − BPker A∗ − RG and G = B − Pker AB − GS, and substituting the latter into the former,

G = B − BPker A∗ − RB + RPker AB + RGS.

This ﬁnal term satisﬁes

RGS : C−∞(M; F ) −→ C∞(M; E), (RGS)∗ = S∗G∗R∗ : C−∞(M; E) −→ C∞(M; F ) 5.3. THE INDEX OF AN ELLIPTIC OPERATOR 129

−∞ −∞ which shows that RGS ∈ Ψcl (M; F,E), and so G − B ∈ Ψcl (M; F,E) and −r G ∈ Ψcl (M; F,E). Now note that, from (5.5),

X 2 Tr(Pker A) = kφjkL2(M;E) = dim ker A and similarly for Pker A∗ so the formula

ind(A) = Tr(Pker A) − Tr(Pker A∗ ) is immediate.

Let us establish a couple of properties of the trace of smoothing operators that will be useful for understanding the index. Proposition 27. Let M be a compact Riemannian manifold and E,F Hermitian vector bundles. i) The operators of ﬁnite rank −∞ {K ∈ Ψcl (M; E,F ) : dim Im(K) < ∞} −∞ are dense in Ψcl (M; E,F ). −∞ r ii) If P ∈ Ψcl (M; F,E) and A ∈ Ψcl(M; E,F ) then Tr(AP ) = Tr(PA).

−∞ ∞ Proof. i) Given P ∈ Ψcl (M; E,F ) so that KP ∈ C (M × M; HOM(E,F )) let us ∞ show that we can approximate KP in the C -topology by finite rank operators. Us- ing partitions of unity we can assume that KP is supported in an open set of M ×M over which E and F are trivialized, so KP is just a smooth matrix valued function on an open set of Rm × Rm. Identifying this open set with an open set in Tm × Tm with Tm the m-dimensional torus, we can expand each of the entries in a Fourier series in each variable. The truncated series then are finite rank operators that converge uniformly with all of their derivatives to KP , and restricting their support we get finite rank operators on M that approximate P.

ii) By continuity we can assume that P has ﬁnite rank and by linearity that P has rank one, say P φ = v(φ)s, for some v ∈ C∞(M; F ∗), s ∈ C∞(M; E). Then we have AP (φ) = v(φ)As, P A(φ) = v(Aφ)s Z so Tr(AP ) = v(As) dvolg = Tr(PA). 130 CHAPTER 5. PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS

Now we can derive some of the most important properties of the index of an elliptic operator. Theorem 28. Let M be a compact Riemannian manifold, E,F, and G Hermitian r s vector bundles over M, and let A ∈ Ψcl(M; E,F ) and B ∈ Ψcl(M; F,G) be elliptic. r+s i) B ◦ A ∈ Ψcl (M; E,G) is elliptic and ind(B ◦ A) = ind(B) + ind(A).

−r ii) If Q ∈ Ψcl (M; F,E) is any parametrix of A then

ind(A) = Tr(IdE −QA) − Tr(IdF −AQ) =: Tr([A, Q]).

−∞ iii) If P ∈ Ψcl (M; E,F ) then ind(A) = ind(A + P ). iv) The index is a smooth homotopy invariant of elliptic operators. v) The index of A only depends on σr(A).

Proof. i) Note that we have a short exact sequence

∗ 0 −→ ker A∗ −→ ker B∗A∗ −−−→A Im(A∗) ∩ ker B∗ −→ 0 in which all of the spaces are ﬁnite dimensional, so we can conclude that

dim ker(A∗B∗) = dim ker(B∗) + dim((ker A∗) ∩ (Im B∗)).

By the same reasoning

dim ker(BA) = dim ker(A) + dim((ker B) ∩ (Im A)) so that ind(BA) is

dim ker A + dim((ker B) ∩ (Im A)) − dim ker(B∗) − dim((ker A∗) ∩ (Im B∗)) = dim ker A + dim((ker B) ∩ (ker A∗)⊥) − dim ker(B∗) − dim((ker A∗) ∩ (ker B)⊥) = dim ker A + dim(ker B) − dim ker(B∗) − dim(ker A∗) = ind(A) + ind(B) as required. ii) From Theorem 26, we know that if G is the generalized inverse of A then

Tr([A, G]) = ind(A)

−∞ and from the proof of Theorem 26 we know that G − Q ∈ Ψcl (M; F,E). It follows that Tr([A, Q]) − Tr([A, G])

= Tr(IdE −QA) − Tr(IdF −AQ) − Tr(IdE −GA) + Tr(IdF −AG) = Tr((G − Q)A) − Tr(A(G − Q)) 5.4. PSEUDODIFFERENTIAL OPERATORS ACTING ON L2 131 which vanishes by Proposition 27. iii) follows immediately from (ii) since any parametrix of A is a parametrix of A + P. r iv) If t 7→ At ∈ Ψcl(M; E,F ) is a smooth family of elliptic pseudodiﬀeren- −r tial operators then we can construct a smooth family t 7→ Qt ∈ Ψcl (M; F,E) of parametrices and hence ind(At) = Tr([At,Qt]) is a smooth function of t. Since this is valued in the integers, it must be constant and independent of t. v) It suﬃces to note that if two operators A and A0 have the same elliptic symbol then tA + (1 − t)A0 is a smooth family of elliptic operators connecting A and A0, so by (iv) they have the same index.

5.4 Pseudodiﬀerential operators acting on L2

So far, we have been studying pseudodiﬀerential operators as maps C∞(M; E) −→ C∞(M; F ) or C−∞(M; E) −→ C−∞(M; F ), now we want to look at more reﬁned mapping properties by looking at distributions with some intermediate regularity, e.g., C∞(M; E) ⊆ L2(M; E) ⊆ C−∞(M; E).

Let us recall some definitions for linear operators acting between Hilbert spaces3, H1 and H2. It is convenient, particularly when differential operators are involved, to allow for linear operators to be defined on a subspace of H1,

P : D(P ) ⊆ H1 −→ H2. We think of a linear operator between Hilbert spaces as the pair (P, D(P )) and only omit D(P ) when it is clear from context. We will always assume that D(P ) is dense in H1. r Thus for instance any pseudodiﬀerential operator A ∈ Ψcl(M; E,F ) deﬁnes a linear operator between L2 spaces with domain C∞(M; E), A : C∞(M; E) ⊆ L2(M; E) −→ L2(M; F ).

3For simplicity, we will focus on Hilbert spaces based on L2(M; E), but pseudodiﬀerential operators also have good mapping properties on Banach spaces such as Lp(M; E), p 6= 2. 132 CHAPTER 5. PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS

Of course, with this domain A actually maps into C∞(M; F ), so we will be interested in extending A to a larger domain. An extension of a linear operator (P, D(P )) is an operator (P,e D(Pe)),

Pe : D(Pe) ⊆ H1 −→ H2 such that D(P ) ⊆ D(Pe) and Pe u = P u for all u ∈ D(P ). An operator (P, D(P )) is bounded if it sends bounded sets to bounded sets. In particular, there is a constant C such that

kP ukH2 ≤ CkukH1 for all u ∈ D(P ) and the smallest constant is called the operator norm of P and denoted

kP kH1−→H2 . Recall that an operator (P, D(P )) is continuous at a point if and only if it is continuous at every point in D(P ) if and only if it is bounded. A bounded operator has a unique extension to an operator deﬁned on all of H1, moreover this extension is bounded with the same operator norm. If an operator is bounded, we will assume that its domain is H1 and generally omit it from the notation. The set of bounded operators between H1 and H2 is denoted

B(H1, H2).

A compact operator is an operator that sends bounded sets to pre-compact sets, and is in particular necessarily bounded. We denote the space of compact operators between H1 and H2 by

K(H1, H2). An example of a compact operator is a smoothing operator. Indeed, for smoothing operators between sections of E and sections of F, the image of a bounded subset of L2(M; E) is bounded in C∞(M; F ) and hence by Arzela-Ascoli compact in L2(M; F ). Lemma 29. Let M be a Riemannian manifold and E,F Hermitian vector bun- −∞ ∞ dles. If R ∈ Ψcl (M; E,F ), then (R, C (M; E)) is a compact operator between L2(M; E) −→ L2(M; F ). Its unique extension to L2(M; E) is the restriction of R as an operator on C−∞(M; E).

Since the identity map is trivially bounded, we can use this lemma to see that −∞ any operator of the form IdE +R with R ∈ Ψcl (M; E) defines a bounded operator on L2(M; E). There is a clever way of using this to show that any pseudodifferential operator of order zero defines a bounded operator on L2. 5.4. PSEUDODIFFERENTIAL OPERATORS ACTING ON L2 133

r ∞ Proposition 30. If r ≤ 0 and P ∈ Ψcl(M; E,F ) then (P, C (M; E)) deﬁnes a bounded linear operator between L2(M,E) and L2(M,F ).

Proof. We follow H¨ormanderand reduce to the smoothing case as follows. Assume 0 −∞ we can ﬁnd Q ∈ Ψcl(M,E), C > 0, and R ∈ Ψcl (M,E) such that

P ∗P + Q∗Q = C Id +R.

Then we have

2 ∗ kP ukL2(M,F ) = hP u, P uiF = hP P u, uiE ∗ ∗ 2 ≤ hP P u, ui+hQ Qu, ui = Chu, ui+hRu, ui ≤ (C+kRkL2(M;E)→L2(M;E))kukL2(M;E), so the proposition is proved once we construct Q, C, and R.

We proceed iteratively. Let σ0(P ) denote the symbol of degree zero of P, e.g., ∗ σ0(P ) = 0 if r < 0. Choose C > 0 large enough so that C − σ0(P ) σ0(P ) is, at each ∗ 0 point of S M, a positive element of hom(E), then choose Q0 so that its symbol is a ∗ 1 0 0 ∗ positive square root of C − σ0(P )σ0(P ), and let Q0 = 2 (Q0 + (Q0) ). Note that Q0 0 is a formally self-adjoint elliptic operator with the same symbol as Q0.

Suppose we have found formally self-adjoint operators Q0,...,QN satisfying −i Qi ∈ Ψcl (M; E) and

N !2 ∗ X −1−N P P + Qi = C Id −RN ,RN ∈ Ψcl (M; E). i=0

0 −N−1 Then, if QN+1 ∈ Ψcl (M,E), we have

N !2 ∗ X 0 P P + Qi + QN+1 − C Id i=0 N ! N ! 0 X X 0 = −RN + QN+1 Qi + Qi QN+1, i=0 i=0

−2−N 0 so in order for this to be in Ψcl (M; E), QN+1 must satisfy

0 0 σ(QN+1Q0 + Q0QN+1) = σ(RN ).

We can always solve this equation because σ(Q0) is self-adjoint and positive as shown in the following lemma. 134 CHAPTER 5. PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS

Lemma 31. Let A be a positive self-adjoint k × k matrix, and let

ΦA(B) = AB + BA, then ΦA is a bijection from the space of self-adjoint matrices to itself.

Proof. We can ﬁnd a matrix S such that S∗ = S−1 and S∗AS = D is diagonal. ∗ −1 Deﬁne Ψcl(B) = S BS and note that Ψcl ΦAΨcl = ΦΨcl(A) so we may replace A with D = diag(λ1, . . . , λk). Let Est denote the matrix with entries ( 1 if s = i, t = j (Est)ij = 0 otherwise and note that ΦD(Est + Ets) = (λs + λt)(Est + Ets).

Hence the matrices {Est + Ets} with s ≤ t form an eigenbasis of ΦD on the space of self-adjoint k × k matrices. Since, by assumption, each λs > 0 it follows that ΦD is bijective.

0 So using the lemma we can find σ(QN+1) among self-adjoint matrices and then 1 0 0 ∗ define QN+1 = 2 (QN+1 + (QN+1) ), finishing the iterative step. Finally, we define Q by asymptotically summing the Qi and the result follows.

r Corollary 32. Let A ∈ Ψcl(M; E,F ), r ≤ 0, and let

kσ0(A)k∞ = sup |σ0(A)(ζ, ξ)|, S∗M

−∞ then for any ε > 0 there is a non-negative self-adjoint operator Rε ∈ Ψcl (M,E) such that, for any u ∈ L2(M,E)

2 2 2 (5.6) kAukL2(M,F ) ≤ (kσ0(A)k∞ + ε)kukL2(M,E) + hRεu, uiE.

r In particular, if r < 0 then A ∈ Ψcl(M; E,F ) deﬁnes a compact operator (A, C∞(M; E)) between L2(M; E) and L2(M; F ).

Proof. The existence of Rε satisfying the inequality (5.6) follows directly from the proof of the theorem.

If kσ0(A)k∞ = 0, we show that A is a compact operator by proving that the image of a bounded sequence is Cauchy. Let (uj) be a bounded sequence in 2 L (M; E), say kujk ≤ C for all j ∈ N. For each n ∈ N we know that (R 1 uj) has a n 5.4. PSEUDODIFFERENTIAL OPERATORS ACTING ON L2 135 convergent subsequence in L2(M; E) and so by a standard diagonalization argument 0 0 we can ﬁnd a subsequence (uj) of (uj) such that (R 1 uj) converges for all n. n 0 2 It follows that (Auj) is a Cauchy sequence in L (M; F ). Indeed, given η > 0 choose N ∈ N such that j, k > N implies

0 0 kRη(uj − uk)kL2(M;E) ≤ η.

It follows that for such j, k

0 0 2 0 0 0 0 0 0 kA(uj−uk)kL2(M;F ) ≤ ηkuj−ukkL2(M;E)+hRη(uj−uk), (uj−uk)iL2(M;E) ≤ η(2C)+ηC.

0 Hence (Auj) is Cauchy and A is compact.

The graph of a linear operator (P, D(P )) is the subspace of H1 × H2 given by {(u, P u) ∈ H1 × H2 : u ∈ D(P )}. We say that a linear operator (P, D(P )) is closed if and only if its graph is a closed subspace of H1 × H2. Any bounded operator has a closed graph, and the closed graph theorem says that an closed operator (P, D(P )) with D(P ) = H1 is bounded. Let us point out that a closed operator need not have closed domain or closed range, however the null space of a closed operator does form a closed set. It is very convenient to deal with closed operators, especially when studying the spectrum of an operator or dealing with adjoints. For instance, the spectrum of a linear operator that is not closed consists of the entire complex plane! r ∞ If A ∈ Ψcl(M; E,F ) then (A, C (M; E)) has at least two closed extensions with domains: the maximal domain comes from restricting the action of A as a continuous operator on distributions,

2 2 Dmax(A) = {u ∈ L (M; E): Au ∈ L (M; F )}, the minimal domain comes from extending the action of A as a continuous operator on smooth functions,

2 Dmin(A) = {u ∈ L (M; E): ∞ there exists (un) ⊆ C (M; E) s.t. lim un = u and (Aun) converges}.

In the latter case, we deﬁne Au = lim Aun and this is independent of the choice of sequence un as long as lim un = u and Aun converges. Since the graph of (A, Dmin(A)) is the closure of the graph of (A, C∞(M; E)), it is clearly the minimal closed extension of A and (A, Dmax(A)) is also clearly the maximal closed extension of A. 136 CHAPTER 5. PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS

There are operators for which Dmin(A) 6= Dmax(A) (see exercise 11). However these must have positive order, since we have shown that operators of order zero are 2 bounded (which implies that Dmin(A) = Dmax(A) = L (M; E)). Next we show that elliptic operators also have a unique closed extension.

Proposition 33. Let M be a compact Riemannian manifold and E,F Hermitian r vector bundles over M, and r > 0. If A ∈ Ψcl(M; E,F ) is an elliptic operator then

(5.7) Dmin(A) = Dmax(A).

−r In fact the following stronger statement is true: For any Q ∈ Ψcl (M; F,E),

2 2 (5.8) Im(Q : L (M; F ) −→ L (M; E)) ⊆ Dmin(A).

Proof. Let us start by proving (5.8). Let u ∈ L2(M; F ) and v = Qu, we need to ∞ show that v ∈ Dmin(A). Choose (un) ⊆ C (M; F ), a sequence converging to u, and note that, since Q and AQ are bounded operators on L2(M; F ), we have

vn = Qun → v, Avn = AQun → Av.

This implies v ∈ Dmin(A) as required. −r Next, to prove (5.7), let B ∈ Ψcl (M; F,E) be a parametrix for A and w ∈ Dmax(A). Then −∞ w = BAw + Rw, R ∈ Ψcl (M; E), shows that w is a sum of an section in the image of B and a smooth section, and by (5.8) both of these are in Dmin(A).

5.5 Sobolev spaces

The second part of Proposition 33 strongly suggests that the L2-domain of an elliptic pseudodiﬀerential operator of order r is independent of the actual operator involved. This is true, and the common domain is known as the Sobolev space of order r. We deﬁne these spaces, for any s ∈ R, by

s −∞ 2 s H (M; E) = {u ∈ C (M; E): Au ∈ L (M; E) for all A ∈ Ψcl(M; E)}.

We can think of Hs as the sections ‘whose ﬁrst s weak derivatives are in L2’, and if s ∈ N this is an accurate description. Notice that

s > 0 =⇒ Hs(M; E) ⊆ L2(M; E) ⊆ H−s(M; E). 5.5. SOBOLEV SPACES 137

It will be useful to have at our disposal a family of operators

s R 3 s 7→ Λs ∈ Ψcl(M; E), (5.9) ∞ ∗ ∗ σs(Λs) = idE ∈ C (S M; hom(π E)) that are elliptic, (formally) self-adjoint, strictly positive, and moreover satisfy

−1 ∞ Λ−s = Λs on C (M; E).

These are easy to construct. First deﬁne Λ0 = IdE . Next for s > 0, start with any s/2 operator A ∈ Ψcl (M; E) with σs/2(A) = idE and then deﬁne

∗ Λs = A A + IdE .

Finally, for s < 0, take Λ−s to be the generalized inverse from section 5.3 and note −1 that since Λs is (formally) self-adjoint and injective, Λ−s = Λs . (When there is E more than one bundle involved, we will denote Λs by Λs .) Using these operators, let us deﬁne an inner product on Hs(M; E) by

(5.10) Hs(M; E) × Hs(M; E) / C

(u, v) / hu, viHs(M;E) = hΛsu, ΛsviL2(M;E) and the corresponding norm

2 kukHs(M;E) = hu, uiHs(M;E). It is easy to see that this inner product gives Hs(M; E) the structure of a Hilbert space. Theorem 34. Let M be a compact Riemannian manifold and E,F Hermitian vector r bundles over M. A pseudodiﬀerential operator A ∈ Ψcl(M; E,F ) deﬁnes bounded operators s s−r A : H (M; E) −→ H (M; F ) for any s ∈ R. For a distribution u ∈ C−∞(M; E), the following are equivalent: 2 r r i) Au ∈ L (M; E) for all A ∈ Ψcl(M; E), i.e., u ∈ H (M; E) r 2 ii) There exists P ∈ Ψcl(M; E,F ) elliptic such that P u ∈ L (M; F ). ∞ iii) u is in the closure of C (M; E) with respect to the norm k·kHr(M;E).

r s Moreover, every elliptic operator P ∈ Ψcl(M; E,F ) deﬁnes a norm on H (M; E),

kukP,s = kP ukHs−r(M;F ) + kPker P ukHs(M;E) which is equivalent to the norm k·kHs(M;E). 138 CHAPTER 5. PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS

In particular, the Hilbert space structure on Hs(M; E) is independent of the choice of family Λs as another choice of family yields equivalent norms.

Proof. Showing that the map A : C∞(M; E) ⊆ Hs(M; E) −→ Hs−r(M; F ) is bounded is equivalent to showing that the map

F E ∞ 2 2 Λs−r ◦ A ◦ Λ−s : C (M; E) ⊆ L (M; E) −→ L (M; F ) is bounded. But this is an operator of order zero, and so is bounded by Proposition 30. r It follows that, for any A ∈ Ψcl(M; E,F ) and any s > 0 there is a constant C > 0 such that

kAukHs−r(M;F ) ≤ CkukHs(M;E). If A = P is elliptic, and G is the generalized inverse of P, then since it has order −r we have

kGvkHs−r(M;E) ≤ CkvkHs(M;F ). In particular, we have

kP ukHs−r(M;F ) ≥ CkGP ukHs(M;E) = ku − Pker P ukHs(M;E) hence

kukHs(M;E) ≤ C(kP ukHs−r(M;F ) + kPker P ukHs−r(M;E)), which proves that kukP,s is equivalent to kukHs(M;E), since ker P is a ﬁnite dimensional space and all norms on ﬁnite dimensional spaces are equivalent. This also establishes the equivalence of (i) and (ii). For any s ∈ R, choose s > r and note that Hs(M; E) is the maximal domain of ∞ r r Λs−r : C (M; E) ⊆ H (M; E) −→ H (M; F ) the same argument as in Proposition 33 shows that the minimal domain equals the maximal domain and this shows that C∞(M; E) is dense in Hs(M; E).

In this way we can reﬁne our understanding of regularity of sections from

C∞(M; E) ⊆ L2(M; E) ⊆ C−∞(M; E) to, for any s > 0,

C∞(M; E) ⊆ Hs(M; E) ⊆ L2(M; E) ⊆ H−s(M; E) ⊆ C−∞(M; E). 5.5. SOBOLEV SPACES 139

In fact, the natural L2-pairing which exhibits C−∞(M; E) as the dual of C∞(M; E) and L2(M; E) as its own dual puts Hs(M; E) in duality with H−s(M; E). Formally, for every s ∈ R there is a non-degenerate pairing

Hs(M; E) × H−s(M; E) / C

(u, v) / hu, viHs×H−s = hΛ−su, ΛsviL2(M;E) and so we can identify (Hs(M; E))∗ = H−s(M; E). (Notice that this is diﬀerent from the pairing (5.10) with respect to which Hs is its own dual.) Also notice that there are natural inclusions

s < t =⇒ Ht(M : E) ⊆ Hs(M; E) and indeed we can isometrically embed Ht(M; E) into Hs(M; E) by the operator Λs−t. Notice that since these are pseudodiﬀerential operators of negative order, this inclusion is compact! Put another way, if we have a sequence of distributions in Hs(M; E) that is bounded in Ht(M; E), then it has a convergent sequence in Hs(M; E). This is an L2-version of the usual Arzela-Ascoli theorem on a compact set K: a sequence of functions in C0(K) that is bounded in C1(K) has a convergent sequence in C0(K).

It is easy to see that for any ` ∈ N0 C`(M; E) ⊆ Hs(M; E) whenever s > ` interestingly there is a sort of converse embedding, known as the Sobolev embedding theorem, (see exercises 7 and 8)

s ` 1 H (M; E) ⊆ C (M; E) if ` ∈ N0, ` > s − 2 dim M. This lets us see that \ [ C∞(M; E) = Hs(M; E), and C−∞(M; E) = Hs(M; E) s∈R s∈R s so that the scale of Sobolev spaces {H (M; E)}s∈R is a complete refinement of the inclusion of smooth sections into distributional sections. With the machinery we have built so far, it is a simple matter to refine Theorem 24 to Sobolev spaces. Theorem 35. Let M be a compact Riemannian manifold, E and F Hermitian r vector bundles over M. Let P ∈ Ψcl(M; E,F ) be an elliptic operator of order r ∈ R. We have a refined elliptic regularity statement:

−∞ t t+r u ∈ C (M; E), P u ∈ H (M; F ) =⇒ u ∈ H (M; E) for any t ∈ R. 140 CHAPTER 5. PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS

Also, for any s ∈ R,P deﬁnes a Fredholm operator either as a closed unbounded operator P : Hs+r(M; E) ⊆ Hs(M; E) −→ Hs(M; F ) or as a bounded operator

P : Hs+r(M; E) −→ Hs(M; F ) and its index is independent of s. Explicitly this means that: i) The null spaces of P and P ∗ in either case are ﬁnite dimensional (in fact these spaces are themselves independent of s.) ii) We have an orthogonal decomposition

Hs(M; F ) = P (Hs+r(M; E)) ⊕ ker P ∗.

A consequence of this theorem is a reﬁned version of the Hodge decomposition. Namely, for any s ∈ R, Hs(M;ΛkT ∗M) = d Hs+1(M;Λk−1T ∗M) ⊕ δ Hs+1(M;Λk+1T ∗M) ⊕ Hk(M).

5.6 Exercises

Exercise 1. Make sure you understand how to prove all parts of Theorem 21. Exercise 2. r Let A ∈ Ψcl(M; E,F ) be an elliptic pseudodiﬀerential operator. Show that if −r −∞ BL,BR ∈ Ψcl (M; F,E) satisfy BLA − IdE ∈ Ψcl (M; E) and ABR − IdF ∈ −∞ −∞ Ψcl (M; F ) then BL − BR ∈ Ψcl (M; F,E). Exercise 3. (Schur’s Lemma) Suppose (X, µ) is a measure space and K is a measurable complex- valued function on X × X satisfying Z Z |K(x, y)| dµ(x) ≤ C1, |K(x, y)| dµ(y) ≤ C2, X X for all y and for all x respectively. Show that the prescription Z Ku(x) = K(x, y)u(y) dµ(y) X deﬁnes a bounded operator on Lp(X, µ) for each p ∈ (1, ∞) with operator norm

1/p 1/q 1 1 p p kKkL →L ≤ C1 C2 , where q satisﬁes p + q = 1. 5.6. EXERCISES 141

Hint:It suﬃces to bound |hKu, vi| for u and v with kukLp ≤ 1, kvkLq ≤ 1. Start with the expression Z Z

|hKu, vi| = K(x, y)u(y)v(x) dµ(y)dµ(x) X X and apply the inequality ap bq ab ≤ p + q with a = tu and b = t−1v where t is a positive real number to be determined. Minimize over t to get the desired bound. Exercise 4. Show that the linear operator T : C∞([0, 1]) ⊆ L2([0, 1]) −→ L2([0, 1]) given by T u = ∂xu is not bounded and not closed. Exercise 5. 2 Let H1 = L (R), H2 = R and consider the operator Z Au = u(x) dx R ∞ from H1 to H2 with domain D(A) = Cc (R). Prove that the closure of the graph of A is not the graph of a linear operator. Hint: For each n ∈ N, let

( 1 n if |x| ≤ n un(x) = 0 if |x| > n

2 Show that un → 0 in L (R) and Aun = 2 for all n. Why does this solve the exercise? Exercise 6. Let A be a linear operator A : D(A) ⊆ H −→ H. Given λ ∈ C, show that A − λ Id is closed if and only if A is closed. Exercise 7. One can also deﬁne Sobolev spaces on Rm where there is a natural choice for the operators Λs in (5.9), namely

−1 2 s/2 Λsf(x) = F (1 + |ξ| ) F(f)

s m ∞ m and then as before H (R ) is the closure of Cc (R ) with respect to the norm kfks = kΛsfkL2(Rm). 1 Let s ∈ R and k ∈ N0 such that k < s − 2 m show that there is a constant C such that α sup sup |D f| ≤ Ckfks. |α|≤k ζ∈Rm 142 CHAPTER 5. PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS

m Hint: First note that kφkL∞(Rm) ≤ kF(φ)kL1(Rm) for any φ ∈ S(R ). Next note that

α α 2 |α|/2 kF(D f)kL1(Rm) = kξ F(f)kL1(Rm) ≤ k(1 + |ξ| ) F(f)kL1(Rm) Finally, write (1 + |ξ|2)k/2 = (1 + |ξ|2)s/2(1 + |ξ|2)(k−s)/2 and use Cauchy-Schwarz. Exercise 8. Prove the Sobolev embedding theorem for integer degree Sobolev spaces: If s ∈ N0 s ` there is a continuous embedding of H (M; E) into C (M; E) for any ` ∈ N0 satisfy- 1 ing ` < s − 2 dim M. Hint: Let f ∈ Hs(M; E), use the fact that multiplication by a smooth function is a pseudodiﬀerential operator of order zero to reduce to the case where f is supported in a coordinate chart φ : U ⊆ Rm −→ M that trivializes E. Since s is a natural s number the Sobolev norm on H (M; E) is equivalent to a norm k·kD with D a differential operator, use this to show that f ◦ φ ∈ Hs(U; Crank E). Use the previous exercise to conclude that f has ` continuous derivatives. One can use a technique called ‘interpolation’ to deduce the general Sobolev embedding theorem from this integer degree version. Exercise 9. (The adjoint of a linear operator) Let H1 and H2 be C-Hilbert spaces and A : D(A) ⊆ H1 −→ H2 a linear operator. The (Hilbert space) adjoint of A, is a linear operator from H2 to H1 deﬁned on

∗ D(A) = {v ∈ H2 : ∃w ∈ H1 s.t. hAu, viH2 = hu, wiH1 for all u ∈ D(A)} by the prescription

hAu, vi = hu, A∗vi for all u ∈ D(A), v ∈ (D(A))∗.

(Note that we always assume our linear operators have dense domains, so A∗v is determined by this relation.) If H1 = H2, a linear operator (A, D(A)) is self-adjoint if D(A)∗ = D(A) and

hAu, viH1 = hu, AviH1 for all u, v ∈ D(A).

∗ ∗ i) Show that the graph of (A , D(A) ) is the orthogonal complement in H2 × H1 of the set {(−Au, u) ∈ H2 × H1 : u ∈ D(A)} so in particular (A∗, D(A)∗) is always a closed operator (even if A is not a closed operator). ∗ ii) If (A, D(A)) is a closed operator, show that (D(A)) is a dense subspace of H2 (e.g., by showing that z ⊥ D(A)∗ =⇒ z = 0) so we can deﬁne the adjoint of (A∗, D(A)∗). Prove that ((A∗)∗, (D(A)∗)∗) = (A, D(A)). 5.6. EXERCISES 143

2 2 ii) Suppose H1 = L (M; E), H2 = L (M; F ) where M is a Riemannian manifold r and E,F are Hermitian vector bundles, and A ∈ Ψcl(M; E,F ) for some r ∈ R. Let A† denote the formal adjoint of A, show that the Hilbert space adjoint of the linear operator A : C∞(M; E) ⊆ L2(M; E) −→ L2(M; F ) is the linear operator

† † 2 2 A : Dmax(A ) ⊆ L (M; F ) −→ L (M; E). Hint: Recall the deﬁnition of the action of A† on distributions. 2 r iii) Let H1 = H2 = L (M; E). Show that if A ∈ Ψcl(M; E) is elliptic and formally self-adjoint, then the unique closed extension of A as an operator on L2(M; E) is self-adjoint. Exercise 10. ∗ In this exercise we start with a formally self-adjoint operator A ∈ Ψcl(M; E) and establish a description due to von Neumann4 of its maximal domain,

Dmax(A) = Dmin(A) ⊕ ker(A − λ) ⊕ ker(A − λ), for any λ ∈ C \ R. i) Fix λ ∈ C \ R. Show that k(A − λ)uk ≥ |Im(λ)|kuk, for all u ∈ L2(M; E) \{0} and conclude that (A − λ, Dmin(A)) is a closed injective operator with closed range,

⊥ (5.11) Image((A − λ, Dmin(A))) = ker((A − λ), Dmax(A)) . ii) Show that Dmin(A), ker((A−λ), Dmax(A)), and ker((A−λ), Dmax(A)) are linearly independent subspaces of Dmax(A). iii) Finally, given w ∈ Dmax(A), use (5.11) to write

(A − λ)w = (A − λ)u + ηe for some u ∈ Dmin(A) and ηe ∈ ker((A − λ), Dmax(A)). Show that you can ﬁnd η ∈ ker((A − λ), Dmax(A)) such that (A − λ)η = η.e Finish the proof of the theorem by noting that w − u − η ∈ ker((A − λ), Dmax(A)). Exercise 11. Show that the operator A = ∂x sin(x)∂x on the circle is formally self-adjoint but that Dmin(A) 6= Dmax(A). Hint: Use exercise 10. Use Fourier series to show that the equation (A + i)u = 0 has a solution in L2. 4John von Neumann, Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren. Math. Ann. 102 (1930), no. 1, 49–131. Note that this has nothing to do with pseudodiﬀer- ential operators and works for any symmetric operator on a Hilbert space (an operator (A, D(A)) is symmetric if (A, D(A))∗ is an extension of (A, D(A))). 144 CHAPTER 5. PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS

Exercise 12. Let H1 and H2 be C-Hilbert spaces and A : D(A) ⊆ H1 −→ H2 a linear operator. Recall that A is Fredholm if: i) ker A is a ﬁnite dimensional subspace of H1 ∗ ii) ker A is a ﬁnite dimensional subspace of H2 ∗ iii) H2 has an orthogonal decomposition H2 = Im(A)⊕ker A (i.e., Im(A) is closed). Prove that an operator is Fredholm if and only if it has an inverse modulo compact operators. That is, there exists B : D(B) ⊆ H2 −→ H1 such that AB − IdH2 and

BA − IdH1 are compact operators. Hint: One can follow the arguments above very closely, replacing smoothing operators by compact operators.

5.7 Bibliography

Same as Lecture 4. Chapter 6

Spectrum of the Laplacian

6.1 Spectrum and resolvent

Let H be a complex Hilbert space and A : D(A) ⊆ H −→ H a linear operator. We divide the complex numbers into the resolvent set of A

ρ(A) = {λ ∈ C : A − λ IdH has a bounded inverse} and its complement, the spectrum of A,

Spec(A) = C \ ρ(A).

The resolvent of A is the family of bounded linear operators

ρ(A) / B(H) −1 λ / R(λ) = (A − λ IdH)

This is, in a natural sense, a holomorphic family of bounded operators1. Notice that the graph of the resolvent satisﬁes

{(u, R(λ)u): u ∈ H} = {((A − λ IdH)v, v): v ∈ H} and hence λ ∈ ρ(A) implies that the operator A − λ IdH is closed, and so by exercise 6 of the previous lecture, that A is closed. In other words if A is not closed then Spec(A) = C.

1See for instance, Walter Rudin, Functional Analysis International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991., for the meaning of a holomorphic vector-valued function.

145 146 CHAPTER 6. SPECTRUM OF THE LAPLACIAN

Since a complex number λ is in the spectrum of A if A − λ IdH fails to have a bounded inverse, there is a natural division of Spec(A) by taking into account why there is no bounded inverse: the point spectrum or eigenvalues of A consists of

Specpt(A) = {λ ∈ C : A − λ IdH is not injective }, the continuous spectrum of A consists of

Specc(A) = {λ ∈ C : A−λ IdH is injective and has dense range, but is not surjective}, everything else is called the residual spectrum,

Specres(A) = {λ ∈ C : A − λ IdH is injective but its range is not dense}.

If λ ∈ Specpt(A) then ker(A − λ IdH) is known as the λ-eigenspace and will be denoted Eλ(A), its elements are called eigenvectors. Recall that the adjoint of a bounded operator A ∈ B(H) is the unique operator A∗ ∈ B(H) satisfying

∗ hAu, viH = hu, A viH for all u, v ∈ H.

Just as on ﬁnite dimensional spaces, for any operator eigenvectors of diﬀerent eigenvalues are always linearly independent and if the operator is self-adjoint then they are also orthogonal (and the eigenvalues are real numbers). Let us recall some other elementary functional analytic facts.

Proposition 36. Let H be a complex Hilbert space and T ∈ B(H). If T = T ∗ then

kT kH→H = sup |hT u, uiH| = max |λ|. kuk=1 λ∈Spec(T )

4hT u, viH = hT (u + v), (u + v)iH − hT (u − v), (u − v)iH 2 2 2 2 ≤ |T |(ku + vkH + ku − vkH) = 2|T |(kukH + kvkH) = 4|T | and hence

kT kH→H = sup kT ukH = sup sup hT u, viH ≤ |T |. kukH=1 kukH=1 kvkH=1 6.1. SPECTRUM AND RESOLVENT 147

Next, replacing T by −T if necessary, we can assume that sup hT u, ui > kukH=1 H 0. Choose a sequence of unit vectors (un) such that

lim hT un, uni = |T | and note that

2 2 2 2 k(T − |T | Id)unkH = kT unkH + |T | − 2|T |hT un, uni ≤ 2|T | − 2|T |hT un, uni → 0 implies that (T − |T | Id) is not invertible, so |T | ∈ Spec(T ).

Finally, we need to show that if λ > kT kH→H then λ ∈ ρ(T ), i.e., that (T − 1 λ IdH) is invertible. It suﬃces to note that A = λ T has operator norm less than one and hence the series X (−1)kAk k≥0 converges in B(H) to an inverse of IdH −A.

With those preliminaries recalled, we can now establish the spectral theorem for compact operators. Theorem 37. Let H be a complex Hilbert space and K ∈ K(H) a compact operator. The spectrum of K satisﬁes

Spec(K) ⊆ Specpt(K) ∪ {0} with equality if H is infinite dimensional, in which case zero may be an eigenvalue or an element of the continuous spectrum. Each non-zero eigenspace of K is finite dimensional and the point spectrum is either finite or consists of a sequence converging to 0 ∈ C. If K is also self-adjoint there is a complete orthonormal eigenbasis of H and we can expand K in the convergent series X K = λPλ λ∈Spec(K) where Pλ is the orthogonal projection onto Eλ(K).

Proof. First notice that if a compact operator on a space V is invertible, then the unit ball of V is compact and hence V is finite dimensional. Thus if H is infinite dimensional, then zero must be in the spectrum of K. 1 1 Let λ 6= 0 then K − λ IdH = −λ(IdH − λ K) and we know that IdH − λ K is a Fredholm operator of index zero (since we can connect it to IdH along Fredholm 148 CHAPTER 6. SPECTRUM OF THE LAPLACIAN operators). Thus its null space is finite dimensional and if it is injective it is surjective so by the closed graph theorem has a bounded inverse. This shows that Spec(K)\{0} consists of eigenvalues with finite dimensional eigenspaces. Notice that, for any ε > 0 the operator K restricted to    [  span  Eλ(K)   λ∈Spec(K) |λ|>ε is both compact and invertible, hence this space must be finite dimensional. Thus if the spectrum is not finite, the eigenvalues form a sequence converging to zero. Finally, if K is self-adjoint, let H0 be the orthogonal complement of   [ span  Eλ(K) .

λ∈Specpt(K)

0 0 Notice that K preserves H , since if v ∈ Eλ(K) and w ∈ H we have

hv, KwiH = hKv, wiH = λhv, wi = 0.

Therefore K H0 is a compact self-adjoint operator with

Spec(K H0 ) = {0} and so must be the zero operator. But this means that H0 is both in the null space of K and orthogonal to the null space of K, so H0 = {0}. Similarly, notice that X K − λPλ λ∈Spec(K) |λ|>ε is a self-adjoint operator with spectrum contained in [−ε, ε] ⊆ R. It follows that its operator norm is at most ε which establishes the convergence of the series to K.

This immediately yields the spectral theorem for elliptic operators. Theorem 38. Let M be a closed manifold, E a Hermitian vector bundle over M, and r P ∈ Ψcl(M; E) an elliptic operator of positive order r ∈ R. If ρ(P ) 6= ∅ then Spec(P ) consists of a sequence of eigenvalues diverging to inﬁnity, with each eigenspace a ﬁnite dimensional space of smooth sections of E. If P is self-adjoint then ρ(P ) 6= ∅, the eigenvalues are real, the eigenspaces are orthogonal, and there is an orthonormal eigenbasis of L2(M; E). 6.2. THE HEAT KERNEL 149

−1 −r Proof. If ζ ∈ ρ(P ) then (P − ζ IdE) ∈ Ψcl (M; E) is a compact operator on L2(M; E) and we can apply the spectral theorem for compact operators. So it suﬃces to note that

−1 −1 P v = λv ⇐⇒ (P − ζ IdE) v = (λ − ζ) v.

If P is self-adjoint then Spec(P ) ⊆ R, so ρ(P ) 6= ∅.

This is the case in particular for the Laplacian of a Riemannian metric. A classical problem is to understand how much of the geometry can be recovered from the spectrum of its Laplacian. This was popularized by an article of Mark Kac entitled ‘Can you hear the shape of a drum?’.

6.2 The heat kernel

The fundamental solution of the heat equation ( (∂t + ∆)u = 0

u t=0 = u0 is a surprisingly useful tool in geometric analysis. In particular, it will help us to show that the spectrum of the Laplacian contains lots of geometric information. We will discuss an approach to the heat equation known as the Hadamard parametrix construction, following Richard Melrose. For simplicity we will work with the scalar Laplacian, but the construction extends to any Laplace-type operator with only notational changes. On any compact Riemannian manifold, the heat operator

−t∆ ∞ ∞ + e : C (M) −→ C (R × M) is the unique linear map satisfying

( −t∆ (∂t + ∆)e u0 = 0 −t∆ e u0 t=0 = u0

Its integral kernel H(t, x, y) is known as the heat kernel and it is a priori a distribution on R+ × M × M, but we will soon show that it is a smooth function for t > 0 and understand its regularity as t → 0. Once we know H(t, x, y) then we can 150 CHAPTER 6. SPECTRUM OF THE LAPLACIAN solve the heat equation, even with an inhomogeneous term, since ( (∂t + ∆)v = f(t, x) =⇒ v t=0 = v0(x) Z Z t Z v(t, x) = H(t, x, y)v0(y) dvolg(y) + H(t − s, x, y)f(s, y) dvolg(y)ds. M 0 M

It is straightforward to write e−t∆ as an operator on L2(M) by using the 2 spectral data of ∆. Indeed, if {φj} is an orthonormal basis of L (M), with ∆φj = λjφj, then we have

X −tλj H(t, x, y) = e φj(x)φj(y) for all t > 0. j However it is hard to extract geometric information from this presentation. On Euclidean space, the heat kernel is well-known (see exercise 7) 1 |x − y|2 H m (t, x, y) = exp − . R (4πt)m/2 4t

Notice that HRm is smooth in t, x, y as long as t > 0. As t → 0,HRm vanishes exponentially fast away from diagRm = {x = y} and blows-up as we approach

{0} × diagRm . m/2 Notice that t HRm is still singular as we approach this submanifold since the argument of the exponential as we approach a point (0, x, x) depends not just on x, but also on the way we approach. For instance, for any z ∈ Rm, the limit of tm/2H(t, x, y) as t → 0 along the curve (t, x + t1/2z, x) is |z| exp − . 4

To understand the singularity at {0} × diagRm we need to take into account the different ways of approaching this submanifold. We do this by introducing ‘parabolic’ polar coordinates (parabolic because the t-direction should scale differ- ently from the x − y direction). Define

+ m 2 SP = {(θ, ω) ∈ R × R : θ + |ω| = 1} and notice that any point (t, x, y) ∈ R+ ×Rm ×Rm can be written in the coordinates (r, θ, ω, x) where

t = r2θ, x = x, y = x − rω, with r = pt + |x − y|2 . 6.2. THE HEAT KERNEL 151

Indeed, (r, θ, ω, x) are the parabolic polar coordinates for R+ × Rm × Rm. In these m/2 coordinates, we can write t HRm as |ω|2 exp − 4θ and, since θ and ω never vanish simultaneously, this is now a smooth function! We say that these coordinates have resolved the singularity of the heat kernel. Taking a closer look at the parabolic polar coordinates, notice that these are coordinates only when r 6= 0. Each point in the diagonal of Rm at t = 0 corresponds to r = 0 and to a whole SP . A better way of thinking about this is that we have constructed a new manifold with corners (the heat space of Rm) m + m HR = R × SP × R with a natural map β HRm / R+ × Rm × Rm (r, θ, ω, x) / (r2θ, x, y − rω) which restricts to a diﬀeomorphism

m + m m HR \{r = 0} −→ (R × R × R ) \ ({0} × diagM ).

We refer to HRm (together with the map β) as the parabolic blow-up of R+×Rm×Rm along {0} × diagM . Now let (M, g) be a Riemannian manifold. As a ﬁrst approximation to the heat kernel of ∆g consider 1 d(x, y)2 G(t, x, y) = exp − . (4πt)m/2 4t

Analogously to the discussion above, this is a smooth function on R+ ×M 2 except at {0} × diagM , and we will resolve this singularity by means of a ‘parabolic blow-up’. What should replace the diagonal at time zero? Consider the space R+ × TM with its natural map down to M,

+ π : R × TM −→ TM −→ M. At each point p ∈ M deﬁne

+ 2 SP (p) = {(θ, ω) ∈ R × TpM : t + |ω|g = 1} and then [ SP (M) = SP (p). p∈M 152 CHAPTER 6. SPECTRUM OF THE LAPLACIAN

The map π restricts to a map π : SP (M) −→ M which is a locally trivial fiber bundle with fibers SP . Let us define the heat space of M as the disjoint union

+ 2 G HM = (R × M \{0} × diagM ) SP M,

2 endowed with a smooth structure as a manifold with corners where we attach SP M + 2 to R ×M \{0}×diagM by the exponential map of g. There is a natural ‘blow-down map’ + 2 β : HM −→ R × M + 2 which is the identity on R × M \{0} × diagM and is equal to π on SP M. This means that given a normal coordinate system for R+ × M 2 at (0, p, p), + + 2 U ⊆ R × TpM × TpM −→ R × M we get a coordinate system around β−1(0, p, p) by introducing parabolic polar coordinates as before, but now on the fibers of TpM. Thus, starting with a coordinate system induced by expp : TpM −→ M, say ζ = (ζ1, ζ2, . . . , ζm), we get a coordinate system on HM, + m (r, (θ, ω), ζ) ∈ R × SP × R with the identification of r = 0 with SP (expp(ζ)) and the identification of points with r 6= 0 with

2 + 2 (r θ, expp(rω + ζ), expp(ζ)) ∈ (R × M ) \ ({0} × diagM ).

2 Recall that the distance between expp(rω + ζ) and expp(ζ) is given by |rω|g(p) + 4 O(|rω|g(p)) so that our approximation to the heat kernel in these coordinates is ! 1 |rω|2 + O(|rω|4 ) exp − g(p) g(p) (4πr2θ)m/2 4r2θ !! 1 |ω|2 r2|ω|4 = exp − g(p) + O g(p) (4πr2θ)m/2 4θ θ and this is r−m times a smooth function, so we have again resolved the singularity. This computation shows that G approximates the Euclidean heat kernel to ∞ leading order as t → 0. That is enough to see that, for any u0 ∈ C (M), Z G(t, x, y)χ(dg(x, y))u0(y) dvolg(y) → u0(x) M 2For a careful proof, see Proposition 7.7 in Richard B. Melrose, The Atiyah-Patodi-Singer index theorem. Research Notes in Mathematics, 4. A K Peters, Ltd., Wellesley, MA, 1993. 6.2. THE HEAT KERNEL 153

where χ(dg(x, y)) is a cut-oﬀ function near the diagonal . In fact, we will proceed much as in the construction of an elliptic parametrix and ﬁnd a sequence of functions Aj such that the operator

∗ X j/2 He(r, (θ, ω), x) ∼ β (χ(dg(x, y))G(t, x, y)) t Aj(r, (θ, ω), x), j≥0

∗ ∞ satisﬁes β (∂t +∆)He(r, (θ, ω), x) = O(r ) as r → 0. Here we are using the fact that, since β is the identity away from the blown-up face, it makes sense to pull-back the diﬀerential operator ∂t + ∆; in a moment, we will compute this in local coordinates. ∗ Let us denote β (χ(dg(x, y))G(t, x, y)) by Ge(r, (θ, ω), x). We will assume inductively that we have found A0,A1,...,AN such that

N ! ∗ X j −n+N+1 β (∂t + ∆) Ge(r, (θ, ω), x) r Aj(r, (θ, ω), x) = r EN (r, (θ, ω), x) j≥0 with EN a smooth function on the blown-up space, vanishing to inﬁnite order at t = 0 away from the diagonal, and then we need to ﬁnd AN+1(r, (θ, ω), x) such that

−n+N+1 ∗ (N+1) r EN (r, (θ, ω), x) + β (∂t + ∆) r GAe N+1 vanishes to order −n + N + 2 at the blown-up boundary face {r = 0}, i.e.,

h n−N−1 ∗ (N+1) i r β (∂t + ∆) r GAe N+1 = −EN (0, (θ, ω), x). r=0

To carry out this construction, it is convenient to use a diﬀerent set of coordinates, ‘projective coordinates’ √ x − y (6.1) τ = t ,X = √ , y t in which we can compute

∗ 1 1 X ∗ β ((t∂t)f(t, x, y)) = τ∂τ − Xj∂X β (f(t, x, y)) (6.2) 2 2 j ∗ 1 ∗ β (∂xj f(t, x, y)) = τ ∂Xj β (f(t, x, y)). In these coordinates,

1 |X| β∗G(t, x, y) = exp − + O(τ 2|X|2) (4π)m/2τ m 4 154 CHAPTER 6. SPECTRUM OF THE LAPLACIAN so altogether we have to solve N + 1 − m 2 ∆ − 1 X · ∂ + e−|X| /4A = −E (0, X, y). X 2 X 2 N+1 N

Taking the Fourier transform in X, this becomes

2 1 ρ=|η| 2 (|ξ| + 2 ξ · ∂ξ + C)(f(ξ)) = h(ξ) −−−−→ (ρ∂ρ + 2ρ + C)(f(ξ)) = h(ξ) and one can check that this has a unique solution which decays rapidly as ρ → ∞ ξ and depends smoothly on ξb = |ξ| . Taking the inverse Fourier transform we get the required AN+1.

Having inductively constructed all of the Aj, now let He(r, (θ, ω), x) be a function on HM such that rmHe(r, (θ, ω), x) is smooth on HM, vanishes to inﬁnite order at t = 0 away from the diagonal, and at the blown-up boundary face has Taylor P j expansion G(r, (θ, ω), x) r Aj. (The existence of such a function follows from the classical Borel’s theorem, much like asymptotic completeness of symbols above.) This function He is our parametrix for the heat kernel. It solves the heat equation to inﬁnite order as r → 0, in that

∗ \ n ∞ \ n ∞ + 2 S = β (∂t + ∆)He ∈ r C (HM) = t C (R × M ). n∈N n∈N Next we use it to find the actual solution to the heat equation. Notice that we can think of a function F ∈ C∞(R+ × M 2) as defining an operator F C∞(M) / C∞(R+ × M) R u / M F (t, x, y)u(y) dy but we can also think of it as defining an operator

Op(F ) C∞(R+ × M) / C∞(R+ × M) R t R u / 0 M F (s, x, y)u(t − s, y) dyds and the advantage is that then it makes sense to compose operators. The composition of the operators corresponding to the functions F1 and F2 is easily seen to be an operator of the same form, but now associated to the function

Z t Z F1 ∗ F2(t, x, y) = F1(s, x, z)F2(t − s, z, y) dzds. 0 M 6.2. THE HEAT KERNEL 155

With respect to this product the identity is associated to the distribution δ(t)δ(x−y) and so we can recognize the heat equation for the heat kernel ( (∂t + ∆)H(t, x, y) = 0 if t > 0 H(0, x, y) = δ(x − y) as the equation Op((∂t + ∆)H(t, x, y)) = Id . Our parametrix satisﬁes

∗ Op(β (∂t + ∆)He) = Id + Op(S) so we need to invert Id + Op(S). This can be done explicitly by a series ! X (Id + Op(S))−1 = Op (−1)jS ∗ S ∗ ... ∗ S . j≥0 One can check that this series converges to Id + Op(Q) for some

\ n ∞ + 2 Q ∈ t C (R × M ), n∈N and hence β∗H = He + (He ∗ β∗Q). In particular we see that \ β∗H − He ∈ rnC∞(HM). n∈N Theorem 39. Let (M, g) be a compact Riemannian manifold, HM its heat space with blow-down map + 2 β : HM −→ R × M

The pull-back of H(t, x, y), the heat kernel of ∆g, satisﬁes β∗H ∈ r−mC∞(HM). The trace of the heat kernel has an asymptotic expansion as t → 0, Z −t∆ −m/2 X k/2 Tr e = H(t, x, x) dvolg ∼ t t ak M k≥0

−m/2 with a0 = (4π) Vol(M) and each of the ak are integrals over M of universal polynomials in the curvature of g and its covariant derivatives, Z ak = Uk(x) dvolg . M 156 CHAPTER 6. SPECTRUM OF THE LAPLACIAN

The coeﬃcients ak are known as the local heat invariants of (M, g). One can use a symmetry argument to show that all of the ak with k odd are identically zero, but the other ak contain important geometric information.

Proof. The only thing left to prove is the statement about the coefficients ak. Its clear from the construction above that the coefficients Aj are built up in a universal fashion from the expansion of the symbol of ∆. But as this symbol is precisely the Riemannian metric, and we know that its Taylor expansion at a point in normal coordinates has coefficients given by the curvature and its covariant derivatives, the result follows with Uk(x) = Ak(0, (1, 0), x).

6.3 Weyl’s law

The following lemma is an example of a ‘Tauberian theorem’. Lemma 40 (Karamata). If µ is a positive measure on [0, ∞), α ∈ [0, ∞), then Z ∞ e−tλ dµ(λ) ∼ at−α as t → 0 0 implies Z ` a dµ(λ) ∼ `α as ` → ∞. 0 Γ(α + 1)

Proof. We start by pointing out that Z Z α −tλ −λ t e dµ(λ) = e dµt(λ)

α −1 with µt(λ)(A) = t µ(t A), so we can rewrite the hypothesis as Z Z ∞ −λ a α−1 −λ lim e dµt(λ) = a = λ e dλ t→0 Γ(α) 0 and the conclusion as Z Z ∞ a α−1 lim χ(λ) dµt(λ) = λ χ(λ) dλ t→0 Γ(α) 0 with χ(λ) equal to the indicator function for [0, `]. But this last equality is true for sλ + ∞ + χ(λ) = e for any s ∈ R , hence by density for any χ ∈ C0 (R ) and hence for Borel measurable functions, in particular for the indicator function of [0, `]. 6.3. WEYL’S LAW 157

P If we apply this to the measure dµ(λ) = δ(λ − λi), we can conclude.

Theorem 41 (Weyl’s law). Let (M, g) be a compact Riemannian manifold and let N(Λ) be the ‘counting function of the spectrum’ of the Laplacian,

X N(Λ) = dim Eλ(∆g). λ∈Spec(∆) λ<Λ

As Λ → ∞ we have Vol(M) m/2 N(Λ) ∼ m/2 m Λ . (4π) Γ( 2 + 1)

Hence the rate of growth of the eigenvalues of the Laplacian has geometric information: the volume of M and the dimension of M. We also get geometric information from the local heat invariants. In fact it was pointed out by Melrose (for planar domains) and eventually shown in general by Gilkey, that they control the Sobolev norms of the curvature.

Theorem 42 (Gilkey3). For k ≥ 3, Z Z k k−2 2 k−2 a2k = (−1) Ak|∇ R| + Bk|∇ scalg| dvolg + Ek dvolg M M where Ak,Bk are positive constants and Ek is a universal polynomial in the curvature of g and its ﬁrst k − 3 covariant derivatives.

The explicit formulæ for the heat invariants quickly get out of hand. The ﬁrst few are given by Z m/2 m/2 1 (4π) a0 = Vol(M), (4π) a2 = scal dvolg 6 M Z m/2 1 2 2 2 (4π) a4 = 5scal − 2| Ric | + 2|R| dvolg 360 M Z m/2 1 2 2 2 2 2 (4π) a6 = (−142|∇scal| − 26|∇ Ric | − 7|∇R| + 35scal − 42scal| Ric | 45360 M 2 +42scal|R| − 26 Ricij Ricjk Ricki −20 Ricij Rick` Rikj` − 8 Ricij Rik`nRjkln

−24Rijk`RijnpRk`np) dvolg

3For a proof see Gengqiang Zhou, Compactness of isospectral compact manifolds with bounded curvatures. Paciﬁc J. Math. 181 (1997), no. 1, 187–200. 158 CHAPTER 6. SPECTRUM OF THE LAPLACIAN

There is a remarkable formula of Polterovich for the kth heat invariant4,

3n m −m/2 n X 3n + 2 1 j+n 2j a2n(x) = (4π) (−1) ∆ d(x, y) j + m 4jj!(j + n)! y=x j=0 2 however it is diﬃcult to extract from here an explicit polynomial in the curvature.

6.4 Long-time behavior of the heat kernel

We know that the heat operator e−t∆ is, for every t > 0, a bounded self-adjoint operator on L2(M). Hence its operator norm is equal to the modulus of its largest eigenvalue. In the same way

N −t∆ X −tλj −tλN+1 2 2 ke − e Pλj kL (M)→L (M) = e . j=0

Let us denote N −t∆N −t∆ X −tλj e = e − e Pλj j=0 and point out that this is the heat operator of the Laplacian restricted to the orthogonal complement of span{φ0, . . . , φN }. In particular, the semigroup property applied to its integral kernel HN yields Z −(s+t)∆N −s∆N −t∆N e = e e ⇐⇒ HN (s + t, x, y) = HN (s, x, z)HN (t, z, y) dz. M

Thus Z HN (t + 2, x, y) = HN (1, x, z1)HN (t, z1, z2)HN (1, z2, y) dz1dz2, M 2 and we have

α β |Dx Dy HN (t + 2, x, y)|

α −t∆N β −tλN+1 ≤ kDx HN (1, x, z1)kL2(M)ke kL2→L2 kDy HN (1, z2, y)kL2(M) ≤ Cα,βe .

4See Gregor Weingart, A characterization of the heat kernel coeﬃcients. arXiv.org/abs/ math/0105144 6.5. SUPPLEMENT: DETERMINANT OF THE LAPLACIAN 159

Proposition 43. Let (M, g) be a compact Riemannian manifold and {φj} an orthonormal eigenbasis of the Laplacian of g such that λj ≤ λj+1. The series

N X e−tλj φ(x)φ(y) j=0 converges uniformly to the heat kernel H(t, x, y) in C∞(M). In particular, for any t > 0 X Tr(e−t∆) = e−tλj and for large t, −t∆ −tλ1 | Tr(e − Pker ∆)| ≤ e . Remark 5. One can show much more generally that the trace we deﬁned on smoothing operators coincides with the functional analytic trace given by summing the eigenvalues.

6.5 Supplement: Determinant of the Laplacian

The asymptotic expansion of the trace of the heat kernel allows us to define the determinant of the Laplacian. This is an important invariant with applications in physics, topology, and various parts of geometric analysis. First we imitate the classical Riemann zeta function and define X ζ(s) = λ−s λ∈Spec(∆) λ6=0 where eigenvalues are repeated in the sum according to their multiplicity. Here s is a complex variable, and Weyl’s law shows that this sum converges absolutely (and 1 hence defines a holomorphic function) if Re(s) > 2 dim M. Notice that on the circle the eigenfunctions are eiθn with eigenvalue n2, so the non-zero eigenvalues are the squares of the integers, each with multiplicity two. It follows that the ζ function of the circle is

X −2s ζ(s) = 2 n = 2ζRiemann(2s) n∈N essentially equal to the Riemann zeta function! Returning to general manifolds, we can use the identity 1 Z ∞ dt µ−s = tse−tµ Γ(s) 0 t 160 CHAPTER 6. SPECTRUM OF THE LAPLACIAN valid whenever Re µ > 0 to write Z ∞ 1 s −t∆ dt ζ(s) = t Tr(e − Pker ∆) Γ(s) 0 t

1 whenever Re(s) > 2 dim M. Indeed, Proposition 43 shows that the integral Z ∞ s −t∆ dt t Tr(e − Pker ∆) 1 t converges as t → ∞, for any s ∈ C, and Theorem 39 shows that the integral Z 1 s −t∆ dt t Tr(e − Pker ∆) 0 t converges as t → 0, if Re s > m/2. Moreover, it is straightforward to show that ζ(s) is a holomorphic function of s on the half-plane Re s > m/2 where the integral converges. Another use of Theorem 39 shows that the integral

N ! Z 1 X dt ts Tr(e−t∆) − t−m/2 tk/2a k t 0 k=0

m−N−1 converges, and deﬁnes a holomorphic function, on the half plane Re s > 2 . To incorporate the projection onto the null space, it suﬃces to replace ak with ( ak + dim ker ∆ if k = m eak = ak otherwise

On the other hand, for Re s > m/2, we can explicitly integrate

Z 1 N N X dt X ak ts t−m/2 tk/2a = e ek t s + (k − m)/2 0 k=0 k=0

Thus, for any N ∈ N, we have a function Z ∞ 1 −t∆ dt ζN (s) = Tr(e − Pker ∆) Γ(s) 1 t Z 1 N ! N ! X dt X ak + ts Tr(e−t∆ − P ) − t−m/2 tk/2a + e ker ∆ ek t s + (k − m)/2 0 k=0 k=0

m−N−1 which is meromorphic on Re s > 2 and coincides with ζ(s) for Re s > m/2. Since N was arbitrary we can conclude: 6.6. EXERCISES 161

Theorem 44. Let (M, g) be a compact Riemannian manifold of dimension m. The ζ function admits a meromorphic continuation to C, with at worst simple poles located at (m − k)/2, k ∈ N and with residue

eak m−k Γ( 2 )

(k−m)/2 −t∆ where eak is the coeﬃcient of t in the short-time asymptotics of Tr(e − Pker ∆).

It follows easily that the meromorphic continuation of ζ(s) is always regular at s = 0. Notice that, if the Laplacian had only ﬁnitely many positive eigenvalues {µ1, . . . , µ`} then

hX −si X Y ∂s µj = − log µj = − log µj“ = − log det ∆”. s=0 Inspired by this computation, Ray and Singer deﬁned

−∂sζ(s) det ∆ = e s=0 and, as mentioned above, this has found applications in topology, physics, and geometric analysis.

6.6 Exercises

Exercise 1. Show that zero is an eigenvalue of infinite multiplicity for d :Ω∗(M) −→ Ω∗(M), that d has no other eigenvalue, and that there is an infinite dimensional space of forms that are not in this eigenspace. Exercise 2. Verify directly that the spectrum of the Laplacian on R is [0, ∞) and that there are no eigenvalues. Exercise 3. Compute the spectrum of the Laplacian on the flat torus R2/Z2. Compute the spectrum of the Laplacian on 1-forms on the flat torus. Exercise 4. Compute the spectrum of the sphere S2 with the round metric. Exercise 5. Show that AB and BA have the same spectrum, including multiplicity, except possibly for zero. 162 CHAPTER 6. SPECTRUM OF THE LAPLACIAN

Exercise 6. Show that Eλ(∆k) form an acyclic complex. Exercise 7. Use the Fourier transform to show that the integral kernel of the heat operator on Rm is given by 0 2 0 1 − |ζ−ζ | K −t∆ (t, ζ, ζ ) = e 4t . e (4πt)m/2 Exercise 8. Use the heat kernel to give an alternate proof that a de Rham class contains a harmonic form.

6.7 Bibliography

A great place to read about spectral theory is the book Perturbation theory for linear operators by Kato, I also like Functional analysis by Rudin. Our construction of the heat kernel is due to Richard Melrose and can be found in Chapter 7 of The Atiyah-Patodi-Singer index theorem. One should also look at Chapter 8 to see how to carry out Getzler rescaling geometrically. Chapter 7

The Atiyah-Singer index theorem

7.1 Getzler rescaling

k We have shown that any elliptic P ∈ Ψcl(M; E,F ), k ≥ 0, has a finite index as an operator on smooth functions, L2 or Sobolev spaces, or on distributions; and that its index is the same in all of these contexts. We have also found several formulæ for the index such as Calderón’sformula Ind(P ) = Tr([P,Q]) where Q is any parametrix for P, and, more significantly for our current purposes, Ind(P ) = Tr(e−tP ∗P ) − Tr(e−tP P ∗ ). For Dirac-type operators this leads to an explicit formula for the index.

Let E −→ M be a Z2-graded Dirac bundle, let γ be its involution, and ðE ∈ Diﬀ1(M; E) the associated generalized Dirac operator. The construction of the heat 2 kernel above extends easily to Laplace-type operators such as ðE,

−t − + ! 2 e ðE ðE 0 −tðE e = −t + − 0 e ðE ðE and establishes short-time asymptotic expansions − + X + − X −tðE ðE −m/2 + k/2 −tðE ðE −m/2 − k/2 Tr(e ) ∼ t ak,Et , Tr(e ) ∼ t ak,Et with each coeﬃcient given by the integral of a polynomial in the curvature of g, the curvature of ∇E, and the covariant derivatives of these curvatures, Z ± ± ak,E = Uk,E(x) dvolg . M

163 164 CHAPTER 7. THE ATIYAH-SINGER INDEX THEOREM

Hence we have a formula for the index, Z + + − (7.1) ind(ðE) = Um,E(x) − Um,E(x) dvolg . M McKean and Singer pointed out this formula and conjectured that after a ‘fantastic cancellation’, the universal polynomial

+ − Um,E(x) − Um,E(x) depends only on the curvatures and on no covariant derivatives. Note that the formula (7.1) already gives interesting information about the index of a Dirac-type operator. For instance, it vanishes in odd dimensions, it is additive for connected sums, and multiplicative for ﬁnite covers.

The underlying mechanism for the fantastic cancellation is an observation of Patodi. Recall that Cl(2k) has a unique irreducible representation S,/ so that

∼ ∼ 2k Cl(2k) = End(S/), S/ = C , and that S/ has a natural involution given, up to a constant, by Cliﬀord multiplication by an oriented orthonormal basis of C2k,

Γ = ikθ1 ····· θ2k, known as the chirality operator. The identiﬁcation with End(S/) means that there is a natural trace tr : Cl(2k) −→ C, and composing with the chirality operator Γ a natural ‘supertrace’

str : Cl(2k) / C . ω / tr(Γω)

+ − This is called a supertrace because the Z2-grading Cl = Cl ⊕ Cl allows us to deﬁne graded commutators, or supercommutators, by

|u||v| [u, v]s = uv − (−1) vu, and str vanishes on graded commutators.

Lemma 45 (Patodi). The supertrace vanishes on Cl(2k−1) and satisﬁes str(Γ) = 2k. 7.1. GETZLER RESCALING 165

Proof. Let m = 2k. For any proper subset J ⊆ {1, . . . , m} with, say, ` ∈ {1, . . . , n}\ J, we have J Y j 1 ` ` J cl θ = cl θ = − 2 cl θ , cl θ cl θ s . j∈J Thus Cl(2k−1) is made up entirely of graded commutators and any supertrace must vanish on Cl(2k−1). For the particular supertrace above we have

2 2k str(Γ) = tr(Γ ) = tr(Id) = dim C .

This is especially signiﬁcant because we can rewrite the short-time asymptotic expansions considered above as follows. First recall that Lidskii’s theorem says that

− + Z −tðE ðE Tr(e ) = tr K −t − + dvolg e ðE ðE diag M where

∞ ∞ ∞ K −t − + ∈ C (M; hom(E)), tr : C (M; hom(E)) −→ C (M). e ðE ðE diag Hence if we deﬁne

−t 2 −t − + −t + − Str(e ðE ) := Tr(e ðE ðE ) − Tr(e ðE ðE ), then we can write

2 Z −tðE Str(e ) = str K −t 2 dvolg . e ðE diag M

Now the McKean-Singer formula shows that this integral is independent of t, but the integrand may well include time-dependent divergences. For example, a priori the integrand has a singular expansion as t → 0, ! −m/2 X k −m/2 X k str K −t 2 ∼ t str Ukt = t (str Uk) t e ðE diag k≥0 k≥0 However, the coeﬃcients satisfy

∞ ∞ ∗ 0 Uk ∈ C (M; hom(E)) = C (M; Cl(T M) ⊗ hom (E)) and we will show that in fact

∞ (k) ∗ 0 Uk ∈ C (M; Cl (T M) ⊗ hom (E)) 166 CHAPTER 7. THE ATIYAH-SINGER INDEX THEOREM

so that Patodi’s lemma implies that k < m =⇒ str Uk = 0 and the expansion is not singular. In the process of proving this we will keep careful track of the leading order term in the expansion in terms of the Cliﬀord ﬁltration and this will ultimately yield the formula for the index.

Recall that the integral kernel of an operator on sections of E is an element of the ‘large Hom-bundle’,

∗ Hom(E) = E E, which satisﬁes Hom(E) diag = hom(E).

−t 2 In particular, e ðE is a section of this bundle pulled-back to HM along the blow- down map β : HM −→ M 2 × R+. Notice that

∗ β Hom(E) = hom(E) SP (M)

∗ 0 since β(SP (M)) = diagM , and also that hom(E) = Cl(T M) ⊗ hom (E) since E is a Dirac bundle. Thus it makes sense to talk about the Clifford degree of a section of ∗ β Hom(E) at SP (M). With a little care we can also talk about the Clifford degree of the coefficients of the Taylor expansion of such a section. Note that the connection on E induces a connection on Hom(E) −→ M 2 which we can combine with the flat connection on R+ to obtain a connection on Hom(E) −→ M 2×R+ which we then pull-back to a connection ∇ on β∗ Hom(E) −→ HM. Now fix a transverse vector field N to SP (M) and a boundary defining function ρ such that Nρ = 1 and note that for any section u ∈ C∞(HM; Hom(E)) we have

X j u ∼ ujρ as ρ → 0 (7.2) j≥0 ∞ 1 j with uj ∈ C (SP (M); hom(E)), uj = ∇N u . j! SP (M) Lemma 46 (Melrose). Let D be the space of smooth sections of β∗ Hom(E) whose th j term at SP (M) has Cliﬀord degree at most j,

∞ ∞ (j) ∗ 0 D = {u ∈ C (M; Hom(E)) : uj ∈ C (M; Cl (T M) ⊗ hom (E))}.

There is a smooth bundle HomG(E) −→ HM with a natural bundle map iG : ∗ HomG(E) −→ β Hom(E) that is an isomorphism away from SP (M) and such that

∞ ∗ C (HM; HomG(E)) = iGD.

We call HomG(E) the rescaled bundle corresponding to the ﬁltration by Clif- ford degree. This is an example of a general construction of Melrose that takes a 7.1. GETZLER RESCALING 167

ﬁltration of a bundle near a boundary hypersurface and constructs a new rescaled bundle. This turns out to be extreme useful when working on non-compact or singular spaces, as well as in the proof of the index theorem below. Let us start with a ‘simple rescaling’: Lemma 47. Let X be manifold with boundary, E a vector bundle over X, and F a sub-bundle of E ∂X . The space of sections with boundary values in F,

∞ ∞ D = {u ∈ C (X; E): u ∂X ∈ C (∂X; F )} is the space of all sections of a vector bundle F E. That is, there is a vector bundle F E −→ X, and a bundle map F iF : E −→ E which is an isomorphism over the interior and such that

∞ F ∞ iF ◦ : C (X; E) −→ C (X; E) has image D. The restriction of F E to the boundary is the graded bundle

F ∗ E ∂X = F ⊕ N ∂X ⊗ (E ∂X /F ) where in terms of a boundary deﬁning function x, the restriction map is given by u − u ∂X ∞ ∗ D 3 u 7→ (u ∂X , dx ⊗ x ∂X ) ∈ C (∂X; F ⊕ N ∂X ⊗ (E ∂X /F ) ). (The factor of N ∗∂X makes this independent of the choice of x.)

Proof. Note that D is a C∞(X) module, in that multiplication by a smooth function ∞ on X preserves the set D. Given p ∈ M, let Ip = {f ∈ C (X): f(p) = 0} be the ideal of functions that vanish at p, and let

F F G F Ep = D/(Ip ·D), E = Ep. p∈X We can similarly describe the ﬁbers of E,

∞ ∞ Ep = C (X; E)/(Ip ·C (X; E)),

F so there is a natural map iF : Ep −→ Ep which takes a section in D and evaluates it at p.

Over the interior this map is an isomorphism, so suppose p ∈ ∂X. Let u1, . . . , uN be a local basis of smooth sections of E near p in X such that u1, . . . , uk is a basis 168 CHAPTER 7. THE ATIYAH-SINGER INDEX THEOREM for F near p in ∂X. This is always possible since we can take a basis of F near p, extend them smoothly to sections of E near p and then complete them to a basis of E. Any element of D is locally of the form

k N X X u = fjuj + xfjuj j=1 j=k+1 where x is a boundary deﬁning function for ∂X. Thus u1, . . . , uk, xuk+1, . . . , xuN is F F a local basis of E and the coeﬃcients f1, . . . , fN give a local trivialization of E. 0 0 0 0 If u1, . . . , uN is any other local basis of E near p such that u1, . . . , uk is a local basis for F, then we must have

( 0 Pk PN uj = `=1 ajù` + x `=k+1 ajù`, for j ≤ k 0 PN uj = `=1 ajù` for j > k.

0 0 0 0 It follows that each of {u1, . . . , uk, xuk+1, . . . , xuN } can be written as a smooth linear combination of {u1, . . . , uk, xuk+1, . . . , xuN } so this induces a smooth transformation F among the coeﬃcients fj. Thus E inherits a natural smooth bundle structure from ∞ F E and iF is a bundle map. Clearly the image iF (C (X; E)) is D. Finally consider the restriction of F E to the boundary. Reasoning as above F we can identify the sections of E ∂X with D/(xD), so it suﬃces to note that the restriction map in the statement of the lemma is surjective and has null space xD.

The general case is similar.

Lemma 48. Let X be manifold with boundary, E a vector bundle over X, and

k k−1 0 F ⊆ F ⊆ · · · ⊆ F ⊆ E ∂X a ﬁnite sequence of subbundles over ∂X. Assume that this ﬁltration has an extension to a neighborhood U of ∂X,

k k−1 0 Fe ⊆ Fe ⊆ · · · ⊆ Fe ⊆ E U . There is a vector bundle F E −→ X, and a bundle map

F iF : E −→ E which is an isomorphism over the interior and such that

∞ F ∞ iF ◦ : C (X; E) −→ C (X; E) 7.1. GETZLER RESCALING 169 has image

( k ) X D = u ∈ C∞(X; E): u ∈ xk−jC∞(U; Fej) + xk+1C∞(X; E) . j=0

The restriction of F E to the boundary is the graded bundle associated to the ﬁltration weighted by appropriate powers of the conormal bundle

F k ∗ k−1 k ∗ k+1 0 E ∂X = F ⊕ N ∂X ⊗ (F /F ) ⊕ · · · ⊕ (N ∂X) ⊗ (E ∂X /F ) where in terms of a boundary deﬁning function x, the restriction map is given by

k X k−j k+1 0 k+1 0 D 3 u = x uj + x u 7→ [uk ∂X ], dx ⊗ [uk−1 ∂X ],..., (dx) ⊗ [u ] j=0 k ! M ∈ C∞ ∂X; (N ∗∂X)k−j ⊗ (F j/F j+1) j=−1

(The factor of N ∗∂X makes this independent of the choice of x; by convention k+1 −1 F = {0},F = E ∂X .)

Proof. We call k the length of the filtration; note that simple rescaling corresponds to filtrations of length 0. Assume inductively that we have proved the theorem for filtrations of length k − 1, thus there is a bundle GE −→ X and a bundle map G iG : E −→ E such that the image of the induced map on sections is

( k−1 ) ∞ X k−j ∞ j k ∞ DG = u ∈ C (X; E): u ∈ x C (U; Fe ) + x C (X; E) . j=0

Moreover

G k−1 ∗ k−2 k−1 ∗ k 0 E ∂X = F ⊕ N ∂X ⊗ (F /F ) ⊕ · · · ⊕ (N ∂X) ⊗ (E ∂X /F ) has a natural sub-bundle

k k G F = F ⊕ {0} ⊕ ... ⊕ {0} ⊆ E ∂X .

Applying Lemma 47 to GE we ﬁnd a bundle F E with bundle maps

F G iF : E −→ E −→ E 170 CHAPTER 7. THE ATIYAH-SINGER INDEX THEOREM

∞ F ∞ such that iF ◦ : C (X; E) −→ C (X; E) has image

∞ G ∞ k iG u ∈ C (X; E): u ∂X ∈ C (∂X; F ⊕ {0}) n o = u ∈ C∞(X; E): u ∈ C∞(U; Fek) + xC∞(X; GE) = D and with restriction to ∂X given by

F k ∗ G k E ∂X = F ⊕ (N ∂X) ⊗ E ∂X /F k ∗ k−1 ∗ k 0 k = F ⊕ (N ∂X) ⊗ F ⊕ · · · ⊕ (N ∂X) ⊗ (E ∂X /F ) /F k ∗ k−1 k ∗ k+1 0 = F ⊕ N ∂X ⊗ (F /F ) ⊕ · · · ⊕ (N ∂X) ⊗ (E ∂X /F ) .

This rescaling procedure will yield the bundle HomG(E). Lemma 48 assumes that the filtration has an extension off of the boundary hypersurface, which our prescription (7.2) carried out implicitly via the transverse vector field N and the connection ∇. Choosing N carefully will make computations easier. In (7.2) we expand the sections in a Taylor series using a boundary defining function ρ with the property that Nρ = 1. We have already established that the heat kernel vanishes√ to infinite order as t → 0 except along SP (M) where it has an expansion in t . This suggests that we require √ (7.3) N( t ) = 1 √ √ and expand sections in a Taylor series with respect to t . Note that t is not actually a boundary defining function for SP (M), but the infinite order vanishing of the heat kernel as t → 0 away from SP (M) lets us proceed as if it were. √ This does not yet determine N. Next consider the vector field t N which is tangent to SP (M) and let us impose that it is tangent to the fibers of β : SP (M) −→ √ diagM . There is a natural choice for ( t N) acting on sections that vanish to SP (M) ◦ infinite order at ∂SP (M) : the radial vector field on the fibers of π : SP (M) −→ M, i.e., on the tangent bundle of M, √ (7.4) t N = R. SP (M)

Recall that R is the vector ﬁeld given in any local coordinates X1,...,Xm by X R = Xi ∂Xi . 7.2. ATIYAH-SINGER FOR DIRAC-TYPE OPERATORS 171

One reason that this is a natural√ choice is that in local projective coordinates (like we used in (6.1)), with τ = t , we have

∗ β N = ∂τ . It is worth mentioning again that although any N satisfying (7.3), (7.4) is singular as t → 0 away from SP (M), the inﬁnite order vanishing of the heat kernel allows us to disregard this.

Let HomG(E) be defined as in (7.2) with respect to any vector field N satisfying (7.3) and (7.4). Our claim about the Clifford degree of the terms in the short-time −t 2 expansion of Str(e ðE ) will follow if we can refine the statement

−m/2 ∞ ∗ K −t 2 ∈ ρ C (HM; β Hom(E)) (7.5) e ðE −m/2 ∞ to K −t 2 ∈ ρ C (HM; HomG(E)). e ðE

Moreover note that the restriction of a section of HomG(E) to SP (M) will be an element of

m ∼ M (j) (j−1) 0 DρD = Cl Cl ⊗ hom (E) j=0 m M ∼= ΛjT ∗M ⊗ hom0(E) = Λ∗T ∗M ⊗ hom0(E) j=0 we will write this rescaled restriction as

∞ ∞ ∗ ∗ 0 NG : C (HM; HomG(E)) −→ C (SP (M); Λ T M ⊗ hom (E)). By Patodi’s observation the supertrace will only depend on the part of form-degree m; in fact, once we show (7.5) we will have

2 ∞ + m/2 0 −tðE str K −t 2 ∈ C ( ), str K −t 2 = 2 str (Evm(NG(e ))). e ðE diag R e ðE diag t=0

7.2 The Atiyah-Singer index theorem for Dirac- type operators

We now proceed to show (7.5). First, let us point out 172 CHAPTER 7. THE ATIYAH-SINGER INDEX THEOREM

Lemma 49. If V ∈ C∞(M; TM) then √ ∗ ∞ ∞ βH,L( t ∇V ): C (HM; HomG(E)) −→ C (HM; HomG(E)). Moreover, the rescaled restriction satisfies √ ∗ 1 ∧ NG(βH,L( t ∇V )u) = −iσ(V ) + 4 R (V, R) NG(u) where R is the radial vector field on the fibers of TM, and R∧(V, R) = e(R(V ∧ R)) is the curvature operator acting by exterior multiplication. √ ∗ Proof. First, the effect of βH,L( t ∇V ) on the (usual) restriction of a section of ∗ β Hom(E) to SP (M), is easily seen (e.g., by using projective coordinates as in (6.1)) to be that of −iσ(V ) thought of as a constant coefficient differential operator on the interior of the fibers SP (M) −→ M, which we can identify with the tangent bundle of M. Next note that √ j ∗ j √ ∇ ( t β ∇V ) = ∇ ∇ ∗ N H,L N t βH,LV j √ √ j X `−1 ∗ √ j−` = ∇ ∗ ∇ + ∇ β RHom(E)(N, t V ) + ∇ ∇ . t βH,LV N N [N, t V ] N `=1 The first term on the right will again act by the symbol of V, although now the th action is√ on the j term in the Taylor expansion at SP (M). For the last term note that [N, t V ] is a vector field tangent to SP (M). Indeed, if in local coordinates P V = aj(x)∂xj , then in the corresponding projective coordinates (6.1) we have, by (6.2), X X [N, τV ] = [∂τ , aj(y + τX)∂Xj ] = (Raj)(y + τX)∂Xj . √ Thus the terms above with [N, t V ] contribute only√ terms of Clifford degree at most√j − 1. Finally, the curvature term will vanish at t = 0 if no derivative hits the t factor; when a derivative does hit this factor, recall that

RHom(E)(A, B) = RE(πL∗A, πL∗B) ◦ − ◦ RE(πR∗A, πR∗B) 1 0 0 0 and RE(Y,Z) = 4 cl (R(Y,Z)) + RE(Y,Z) with RE ∈ hom (E). This shows that the curvature term above will preserve the Cliﬀord ﬁltration and will contribute to the top order term only through

1 4 cl (R(πL∗N,V )) , which yields the formula above since the action of Cliﬀord multiplication on the rescaled restriction is through the exterior product and πL∗N = R. 7.2. ATIYAH-SINGER FOR DIRAC-TYPE OPERATORS 173

Similarly, it is easy to see that t∂t will act on sections of HomG(E),

∗ ∞ ∞ β ∇t∂t : C (HM; HomG(E)) −→ C (HM; HomG(E)).

To work out the eﬀect of t∂t on the rescaled restriction to SP (M), note that s s t∂t (t a) = t (s + t∂t)(a), hence the action will have to incorporate the form degree. Indeed it is easy to see −m ∞ that, for any u ∈ ρ C (HM; HomG(E)),

1 Evp (NG(t∂tu)) = − 2 (R + m + p)Evp(NG(u)). To abbreviate, we introduce the ‘number operator’

∗ ∗ N :Λ T M −→ N, N ΛkT ∗M = k, so that 1 NG(t∂tu) = − 2 (R + m + N)NG(u). With this lemma and the Lichnerowicz formula, it is easy to see that the square of a Dirac-type operator will act on sections of the rescaled Hom bundle.

Proposition 50. If E −→ M is a Dirac bundle with Dirac-type operator ðE, then 2 ∞ ∞ tðE : C (HM; HomG(E)) −→ C (HM; HomG(E)) 2 X 1 ∧ 2 1 0 NG(tðEu) = − (∂j + 4 R (∂j,Y )) − 2 e(RE) NG(u),

0 P 0 i j with e(RE) = i,j RE(ei, ej)e(θ )e(θ ).

Proof. The Lichnerowicz formula shows that

2 E ∗ E 1 1 0 0 X 0 i j ðE = (∇ ) ∇ + 4 scal − 2 cl (RE) , with cl (RE) = RE(ei, ej)cl θ cl θ i,j and the result follows.

2 This formula for the action of ðE on the rescaled restriction suggests that we look for the heat kernel of the operator

X 1 ∧ 2 1 0 (7.6) HE = − (∂j + 4 R (∂j,Y )) − 2 e(RE) on differential forms on Rm. To get a flavor of the solution in a simpler case, consider first the harmonic oscillator acting on functions on the real line

2 2 P1 = −∂x + x . 174 CHAPTER 7. THE ATIYAH-SINGER INDEX THEOREM

Mehler’s formula for its heat kernel is 1 1 2 2 Ke−tP1 (t, x, y) = √ exp − (x cosh 2t − 2xy + y cosh 2t) . 2π sinh 2t 2 sinh 2t

2 2 2 Scaling yields the heat kernel for −∂x + a x for any a > 0, and then multiplication of the one-dimensional heat kernels yields formulæ for the higher-dimensional heat kernels. This means that we can ﬁnd the heat kernel for

P2 = ∆ + x · Bx whenever B is a diagonal, positive matrix. We can also ﬁnd the heat kernel if B is just positive and symmetric, since then we can diagonalize using orthogonal matrices and these will commute with ∆. Next consider an antisymmetric matrix A and

2 X X j X j i ∗ P3 = − ∂xi − Aijx = ∆ + Aij(x ∂i − x ∂j) + x · A Ax.

P j i The vector ﬁeld V = Aij(x ∂i − x ∂j) is a sum of inﬁnitesimal rotations so it will commute with the Laplacian and will vanish when applied to x · A∗Ax. Hence

−tP3 1 −tP2 1 e = exp(− 2 tV )e exp(− 2 tV )

∗ where in P2 we set B = A A. Finally, we are interested in the heat kernel of (7.6) where the matrix A is valued in diﬀerential forms. Fortunately, the exterior product of diﬀerential forms is nilpotent so one can use these same formulæ to derive the heat kernel we are looking for. The formula is derived in, e.g., [Melrose, Chapter 8].

Proposition 51 (Mehler, Getzler). The heat kernel of the operator HE on diﬀer- m ential forms on R given by (7.6) has integral kernel Ke−tHE (t, x, y) equal to pdet α(t) 1 exp − (x · β(t)x − 2y · α(t)x + y · β(t)y) exp( t e(R0 )) (4πt)m/2 4t 2 E − 1 itR∧ − 1 itR∧ where α(t) = 2 , β(t) = 2 1 ∧ 1 ∧ sinh(− 2 itR ) tanh(− 2 itR )

Using this explicit formula it is straightforward to modify the heat kernel construction to show that the heat kernel is a section of HomG(E) and to compute its rescaled restriction. Theorem 52. For any Dirac bundle E −→ M we have

−m ∞ K −t 2 ∈ ρ C (HM; HomG(E)), e ðE p −t 2 det α(1) 1 1 0 N (e ðE ) = exp − (Z · β(1)Z) exp( e(R )). G (4π)m/2 4 2 E 7.2. ATIYAH-SINGER FOR DIRAC-TYPE OPERATORS 175

It follows that ∞ str K −t 2 ∈ C ([0, ∞) × M) e ðE diag and, if M is even dimensional,

(7.7) str K −t 2 e ðE diag t=0 R∧/4πi e(R0 ) = Ev det1/2 str0 exp − E m sinh(R∧/4πi) 2πi and hence Z ∧ 0 + 1/2 R /4πi 0 e(RE) (7.8) Ind(ðE) = det ∧ str exp − . M sinh(R /4πi) 2πi

The equation (7.7) is known as the local index theorem, while (7.8) is the Atiyah-Singer index formula for Dirac-type operators.

Proof. Our construction of the heat kernel above showed that

−m ∞ ∗ K −t 2 ∈ ρ C (HM; β Hom(E)) e ðE and we show that the bundle can be replaced by HomG(E) by repeating the construction of the heat kernel but always working with sections of HomG(E). We have 2 already computed the action of t∂t + tðE on sections of HomG(E), and this suggests that our construction of the heat kernel should begin with the heat kernel of this model action. However this is obtained by rescaling the heat kernel of the generalized harmonic oscillator (7.6) and so yields precisely

pdet α(1) 1 exp − (Z · β(1)Z) exp( 1 e(R0 )), (4π)m/2 4 2 E with Z the fiber variable on SP (M). The construction of the heat kernel now pro- ceeds exactly as before, we extend this model solution off of SP (M) to a section of −m ∞ 2 ρ C (HM; HomG(E)), act on it by t∂t +tðE and then restrict the leading term to SP (M) to set up the next model problem. This process yields the Taylor expansion −m ∞ of the putative heat kernel, as a section of ρ C (HM; HomG(E)) and we use the Volterra calculus to compare it to the actual heat kernel. −m ∞ Once we know that the heat kernel is a section of ρ C (HM; HomG(E)) −m/2 P k/2 then the Taylor expansion of its restriction to the diagonal t Ukt has the property that the Clifford degree of each Uk is at most k. Hence when we take the supertrace all of the singular terms vanish thanks to Patodi’s observation Lemma 45, and we see that the supertrace is a smooth function on [0, ∞) × M. 176 CHAPTER 7. THE ATIYAH-SINGER INDEX THEOREM

The closure of the submanifold (0, ∞) × diagM ⊆ HM intersects SP (M) at Z = 0, so the restriction of the heat kernel to the diagonal has rescaled restriction to SP (M) equal to pdet α(1) exp( 1 e(R0 )). (4π)m/2 2 E

The Z2-grading γ of E decomposes into cl (Γ) ⊗ γb with γb ∈ EndCl(T ∗M)(E), and another application of Patodi’s observation proves (7.7) with

0 str = tr ◦γb on EndCl(T ∗M)(E). Finally by the McKean-Singer observation we have

2 Z Z + −tðE Ind( ) = Str(e ) = str K −t 2 dvolg = str K −t 2 dvolg . ðE e ðE diag e ðE diag t=0 M M

7.3 Chern-Weil theory

The advantage of the heat equation proof of the index theorem is that it leads to an explicit formula for the index of a Dirac-type operator. It is perhaps not obvious that the resulting formula is topological, although of course this follows from the stability properties of the index. In this section we brieﬂy describe the topological context for the index formula, namely the theory of characteristic classes, which takes a vector bundle with some structure (e.g., an orientation or a complex structure) and deﬁnes cohomology classes.

Given a rank n R-vector bundle E −→ M and a connection ∇E, recall (from section 2.6) that the curvature of ∇E is a two-form valued in End(E), RE ∈ Ω2(M; End(E)).

In a local trivialization of E, say by a local frame {sj}, the connection is described by a matrix of one-forms E b ∇ sa = ωasb, b b i or in terms of the Christoﬀel symbols, ωa = Γiadx . In this frame we have E b b b c (R )a = dωa − ωc ∧ ωa, which we can express in matrix notation as RE = dω − ω ∧ ω. Taking the exterior derivative of both sides, we ﬁnd the ‘Bianchi identity’ dRE = −dω ∧ ω + ω ∧ dω = −RE ∧ ω + ω ∧ RE = [ω, RE] 7.3. CHERN-WEIL THEORY 177

Now let P : Mn(R) −→ R be an invariant polynomial, i.e., a polynomial in the entries of the matrix such that

−1 P (A) = P (T AT ) for all T ∈ GLn(R). Because of this invariance, it makes sense to evaluate P on sections of End(E) and in particular we get P (RE) ∈ Ωeven(M). Since diﬀerential forms of degree greater than dim M are automatically zero, it makes equally good sense to evaluate any invariant power series on RE.

Lemma 53. If P is an invariant power series and RE is the curvature of the connection ∇E, then P (RE) is a closed diﬀerential form and its cohomology class is independent of the connection.

Proof. By linearity, it suffices to prove this for P homogeneous, say of order q. For any q matrices, A1,...,Aq, define p(A1,...,Aq) to be the coefficient of t1t2 ··· tq in P (t1A1 + ... + tqAq). Thus p(A1,...,Aq) is a symmetric multilinear function satisfying p(A, . . . , A) = q!P (A) and moreover the invariance of P implies that

−1 −1 p(T A1T,...,T AqT ) = p(A1,...,Aq) for all T ∈ GLn(R).

For any one-parameter family of matrices T (s) in GLn(R), with T (0) = Id, we have

q −1 −1 X 0 0 = ∂s s=0p(T A1T,...,T AqT ) = p(A1,..., [Ai,T (0)],...,Aq) = 0 i=1

0 and note that T (0) is an arbitrary element of Mn(R). We can apply this identity to see that P (RE) is closed,

dP (RE) = q!dp(RE,...,RE) = q(q!)p(dRE,RE,...,RE) = q(q!)p([ω, RE],RE,...,RE) = 0.

Next assume that ∇e E is another connection on E. We know that their diﬀer- ence is an End(E) valued one-form,

∇e E − ∇E = α ∈ Ω1(M; End(E)).

E E Deﬁne ∇t = ∇ + tα, and note that this is a connection on E with curvature given locally by

E E 1 2 Rt = d(ω + tα) − (ω + tα) ∧ (ω + tα) = R + t(dα − [ω, α]) − 2 t [α, α]. 178 CHAPTER 7. THE ATIYAH-SINGER INDEX THEOREM

A computation similar to that above yields

E E E ∂tP (Rt ) = q(q!)dp(α, Rt ,...,Rt ) and so

Z 1 Z 1 E E E E E E P (R1 )−P (R0 ) = q(q!) dp(α, Rt ,...,Rt ) dt = d q(q!) p(α, Rt ,...,Rt ) dt . 0 0

E (Another way of thinking about this is that the connections ∇t can be amalgamated into a connection ∇ on the pull-back of E to [0, 1] × M, then the fact that P (∇) is E closed implies that ∂tP (Rt ) is exact.)

We will denote by [P (E)] the cohomology class of P (RE). Recall that the symmetric functions on matrices of size n are functions of the elementary symmetric polynomials τj deﬁned by

n n−1 det(t Id +A) = t + τ1(A)t + ... + τn−1(A)t + τn(A), so, e.g., τ1(A) = Tr(A), and τn(A) = Det(A). Hence the cohomology class [P (E)] is a a polynomial in the cohomology classes

2i [τi(E)] ∈ H (M).

Moreover, notice that if we choose a metric on E and a compatible connection E E b ∇ , then in a local orthonormal frame {sa} for E the matrix (R )a will be skew- E symmetric and hence τi(R ) = 0 if i is odd. The non-vanishing classes

4i [τ2i(E)] ∈ H (M) are called the Pontryagin classes of E and they generate all of the characteristic classes of E. If we restrict the class of matrices we consider then there are sometimes more examples of invariant polynomials. For instance, if we assume that E is an oriented bundle then we can assume that RE is valued in skew-symmetric matrices and, if P is a polynomial on skew-symmetric matrices, we can deﬁne P (RE) as long as

−1 P (T AT ) = P (A) for all T ∈ SOn(R).

This yields one new invariant, the Pfaﬃan characterized (up to sign) by the equation

Pﬀ(A)2 = Det(A). 7.3. CHERN-WEIL THEORY 179

Given a skew-symmetric matrix A = (aij), let

X i j 2 n α = aij dx ∧ dx ∈ Ω (R ) iChern class of E. If F is a real vector bundle and E = F ⊗ C, then 4i [τ2i(F )] = [c2i(E)] ∈ H (M) and, in this case, [c2i+1(E)] = 0, but not in general. We can specify an invariant polynomial by specifying its value on a diagonal or block-diagonal matrix. Thus for instance, if A is skew-adjoint, it can be diagonalized P µj over C with eigenvalues µj. We deﬁne the Chern polynomial Ch(A) = e and the Todd polynomial Td(A) = Q µj and then deﬁne, for any complex Hermitian (1−e−µj ) vector bundle E, Chern character of E = Ch(RE), Todd genus of E = Td(RE).

On the other hand, if B is skew-symmetric real valued matrix then, over R, we can write it in block-diagonal form where each block is either of the form 0 −µj µj 0 or is the 1-by-1 zero matrix. For these matrices we deﬁne the Hirzebruch L- Q µj Q µj polynomial by L(B) = and the A-roof genus by Ab(B) = µj tanh(µj ) 2 sinh( ) 2 (note that, as these are even functions, the sign ambiguity in the deﬁnition of µj is unimportant) and then, for any real vector bundle, L-polynomial of E = L(RE),A-roof genus of E = Ab(RE). 180 CHAPTER 7. THE ATIYAH-SINGER INDEX THEOREM 7.4 Examples

+ We have shown that the index of a Dirac-type operator ðE associated to a Dirac bundle E −→ M is obtained as the integral of ! ! 1/2 R/e 4πi E Evm det tr γb exp(iRb /2π) . sinh(R/e 4πi)

E The diﬀerential form tr γb exp(iRb /2π) is known as the relative Chern form of E, it is closed and its cohomology class is independent of the choice of Cliﬀord connection on E and will be denoted Ch0(E). Thus the index theorem in this case is Z + 0 ind(ðE) = Ab(TM) Ch (E). M Let us consider how this specializes to the various examples of Dirac-type operators mentioned above.

For the Gauss-Bonnet operator, recall that we have

even even odd ðdR :Ω (M) −→ Ω (M) even and that the index of ðdR is equal to the Euler characteristic of M. The index theorem in this case is the Chern-Gauss-Bonnet theorem Z even χ(M) = ind(ðdR ) = Pﬀ(TM). M

For the signature operator, recall that we use the Hodge star to split

∗ ∗ ∗ ∗ ∗ ∗ ΛCT M = Λ+T M ⊕ Λ−T M ∗ ∗ and then the restriction of the de Rham operator to Λ+T M is the signature operator + ∗ ∗ ∗ ∗ ðsign :Λ+T M −→ Λ−T M. + The index of ðsign is the signature of the manifold M. The index theorem in this case is the Hirzebruch signature theorem Z + sign(M) = ind(ðsign) = L(M). M

For the ∂ operator on a K¨ahler manifold, recall that we have a splitting

k ∗ M p,q Λ T M ⊗ C = Λ M p+q=k 7.5. ATIYAH-SINGER FOR ELLIPTIC OPERATORS 181 with respect to which d splits into

d = ∂ + ∂ :Ωp,qM −→ Ωp+1,qM ⊕ Ωp,q+1(M) and then operator √ + ∗ 0,even 0,odd ðRR = 2 (∂ + ∂ ):Ω M −→ Ω (M) is a generalized Dirac operator whose index is the Euler characteristic of the Dol- beault complex. The index theorem in this case is the Grothendieck-Riemann-Roch theorem Z + ind(ðRR) = Td(TM ⊗ C). M

7.5 The Atiyah-Singer index theorem for elliptic operators

Having determined the index of a Dirac-type operator, we now discuss the formula for an arbitrary elliptic pseudodifferential operator. The key is to use topological constructions to reduce the general case to the particular one. The correct topological language to carry this out is K-theory. On any compact space X, the isomorphism classes of complex vector bundles form an Abelian semigroup under Whitney sum, Vect(X), and we denote the corresponding Grothendieck group by K(X). This is the (even) K-theory group of X. Every element of K(X) is the formal difference of two isomorphism classes of complex vector bundles on X. We denote the class corresponding to a vector bundle E by [E]. Whitney sum and tensor product are both well-defined on K(X) and endow it with the structure of a commutative ring. By construction, any semigroup homomorphism between Vect(X) and a group extends to a group homomorphism between K(X) and the group. If X is a point then K(X) is naturally identified, as a commutative ring, with Z by sending a vector space to its dimension.

For X locally compact, we deﬁne Kc(X), the compactly supported K- theory of X by means of triples (E, F, α) over X, where E and F are vector bundles over X and α : E −→ F is a bundle morphism that is an isomorphism outside of a compact subset of X. Direct sum and homotopy extend from vector bundles over X to triples over X in the natural way:

(E0,F0, α0) + (E1,F1, α1) = (E0 ⊕ E1,F0 ⊕ F1, α0 ⊕ α1), 182 CHAPTER 7. THE ATIYAH-SINGER INDEX THEOREM and, given a triple (E,e F,e αe) over X × [0, 1], we say that

(Ee X×{0}, Fe X×{0}, α X×{0}) is homotopic to (Ee X×{1}, Fe X×{1}, α X×{1}). A triple (E, F, α) over X is acyclic if α is an isomorphism over all of X. We will say that two triples are equivalent,

(E0,F0, α0) ∼ (E1,F1, α1) if there are acyclic triples (G0,H0, β0), (G1,H1, β1) such that

(E0,F0, α0) + (G0,H0, β0) is homotopic to (E1,F1, α1) + (G1,H1, β1).

The equivalence classes of triples will be denoted Kc(X).

Lemma 54. For any locally compact X,Kc(X) is an Abelian group with respect to direct sum. If X is compact then Kc(X) = K(X).

Proof. Direct sum of complexes makes Kc(X) into a semigroup in which the neutral element is the equivalence class of acyclic triples. Given a triple (E, F, α) over X, choose Hermitian bundle metrics on E and F and let α∗ denote the adjoint of α. We claim that (F, E, α∗) represents an inverse to (E, F, α) in Kc(X). Indeed, 0 α∗ i α∗ (E, F, α) + (F, E, α∗) ∼ E ⊕ F,E ⊕ F, ∼ E ⊕ F,E ⊕ F, α 0 α i

0 α∗ i α∗ and, since is self-adjoint, is invertible on all of X. This shows α 0 α i that Kc(X) is an Abelian group. If X is compact, then triples over X consist of pairs of bundles (E,F ) since we can always take α = 0. The equivalence relation is that (E,F ) ∼ (E0,F 0) if and only if there are bundles G, H over X such that E ⊕ G ∼= E0 ⊕ H and F ⊕ G ∼= F 0 ⊕ H. ∼ The map (E,F ) 7→ [E] − [F ] induces an isomorphism Kc(X) = K(X).

r An elliptic pseudodiﬀerential operator P ∈ Ψcl(M; E,F ) determines a class in the compactly supported K-theory of T ∗M,

∗ ∗ ∗ k(P ) = [π E, π F, σ(P )] ∈ Kc(T M), where π : T ∗M −→ M is the vector bundle projection and σ(P ) is the principal ∗ symbol of P. Moreover it is easy to see that every class in Kc(T M) corresponds to an elliptic operator in this way, and that the index deﬁnes a homomorphism

∗ ind : Kc(T M) −→ Z. 7.5. ATIYAH-SINGER FOR ELLIPTIC OPERATORS 183

The Atiyah-Singer index theorem consists in identifying this homomorphism.

Let M be a compact manifold and πV : V −→ M a vector bundle over M. Given a triple (E, F, α) over V and a vector bundle G −→ M, we obtain a new triple

∗ ∗ (E, F, α) ⊗ G = (E ⊗ π G, F ⊗ π G, α ⊗ idπ∗G).

This extends to give Kc(V ) the structure of a K(M)-module with the product

Kc(V ) × K(M) / Kc(V ) ((E, F, α), [G] − [H]) / (E, F, α) ⊗ G − (E, F, α) ⊗ H

A remarkable fact is that whenever V is a complex vector bundle, Kc(V ) is a ﬁnitely generated rank one module over K(M). This is called Bott periodicity if V is a trivial bundle, ` ∼ Kc(C × M) = K(M), and the K-theory Thom isomorphism if V is a non-trivial bundle. It can be shown to follow from the ‘periodicity’ of complex Cliﬀord algebras1, ∼ Cl(` + 2) = Cl(`) ⊗ Cl(2), which we established in Proposition 8 (page 67).

There is an explicit generator for Kc(V ) as a K(M)-module. Endow V with a Hermitian metric and consider the triple

∗ ev ∗ odd µV = [πV Λ V, πV Λ , α] ∈ Kc(V )

∞ ∗ ev ∗ odd where α ∈ C (V ; hom(πV Λ V, πV Λ )) is given by

α(v) = ev − iv∗ and v∗ is the dual vector to v with respect to the Hermitian metric on V. (Note that α is a Cliﬀord multiplication.) Theorem 55 (K-theory Thom isomorphism). Let V −→ M be a Hermitian vector bundle over a compact manifold and µV ∈ Kc(V ) as above. The map

Thom : K(M) / Kc(V )

L / µV ⊗ L is an isomorphism.

1See, e.g., Michael Atiyah and Isadore Singer, Index theory for skew-adjoint Fredholm operators, Inst. Hautes Etudes Sci. Publ. Math. (1969), no. 37, 5–26. 184 CHAPTER 7. THE ATIYAH-SINGER INDEX THEOREM

It turns out that K-theory is a generalized cohomology theory and, using an appropriate Mayer-Vietoris sequence, the Thom isomorphism theorem can be proved by checking it in trivializing coordinate charts. In such a chart, the manifold M is spin and we can rewrite µV in terms of the spin bundle (3.11) and Clifford multi- + − plication as [S/ , S/ , cl ]. The upshot is that we obtain a generator of Kc(V ) as a K(M)-module from any element that locally looks like the spin bundle and Clifford multiplication. This is why Dirac-type operators suffice to prove the general index theorem.

Theorem 56. Let M be a closed oriented even-dimensional Riemannian manifold and let ∗ ∗ ∗ ∗ ∗ k(ðsign) = [Λ+T M, Λ−T M, σ(d + δ)] ∈ Kc(T M) be the K-theory class of the signature operator. The map

∗ K(M) ⊗ Q / Kc(T M) ⊗ Q L / k(ðsign) ⊗ L is an isomorphism.

Proof. If M is a spin manifold with trivial spin bundle (or after localizing to a coordinate chart), we have Λ∗T ∗M = S/ ⊗ S/∗ and

∗ ∗ ± ∗ Λ±T M = S/ ⊗ S/ .

Thus

+ ∗ − ∗ + − ∗ ∗ + − k(ðsign) = [S/ ⊗ S/ , S/ ⊗ S/ , cl ] = [S/ , S/ , cl ] ⊗ [S/ ] = (dim S/ )[S/ , S/ , cl ].

∗ Hence, after tensoring with Q, k(ðsign) locally generates Kc(T M) and a Mayer- Vietoris argument implies that it generates globally as well.

This reduces the index formula for arbitrary elliptic operators to the index formula for twisted signature operators.

Theorem 57 (Atiyah-Singer index theorem). Let Td(M) denote the Todd class of the complexiﬁed tangent bundle of M. The index of an elliptic operator P ∈ ∗ Ψcl(M; E,F ) is equal to Z ∗ Ind P = Ch(k(P ))π Td(TM ⊗ C), T ∗M where T ∗M is given its natural orientation. 7.6. EXERCISES 185

As discussed above, it suffices to show that when P is a twisted signature operator, the right hand side gives its index as computed via the heat equation. Thus it comes down to a computation in characteristic classes2 . Finally, let us rewrite the index formula so as to make it a more explicit integral involving the symbol of the operator on the cosphere bundle. Given an ∗ E elliptic pseudodifferential operator P ∈ Ψcl(M; E,F ), choose a connection ∇ with F curvature −2πiωE and a connection ∇ with curvature −2πiωF . Define

−1 1 −1 2 ω(t) = (1 − t)ωE + tσ(P ) ωF σ(P ) + 2πi t(1 − t)(σ (P )∇σ(P )) and then set Z 1 1 −1 ω(t) odd ∗ Ch(f σ(P )) = − Tr(σ (P )∇σ(P )e ) dt ∈ Ω (S M) 2πi 0 This is the ‘odd Chern form’ deﬁned by the symbol. The Atiyah-Singer index theorem is the formula Z ind(P ) = Ch(f σ(P )) Td(TM ⊗ C). S∗M

7.6 Exercises

Exercise 1. There is a ‘McKean-Singer’ formula for the index in terms of the resolvent. Let k P ∈ Ψcl(M; E,F ), k > 0, be an elliptic operator, and let λ ∈ C \{0} be in the resolvent set of P ∗P and ` ∈ N be large enough so that 2k` > dim M. Show that Tr((P ∗P − λ)−`) and Tr((PP ∗ − λ)−`) are both ﬁnite and that

Ind(P ) = (−λ)`(Tr((P ∗P − λ)−`) − Tr((PP ∗ − λ)−`)).

Exercise 2. Vector bundles are frequently constructed by a ‘clutching construction’. Let X = X1∪X2 and A = X1∩X2, all spaces being compact. Given vector bundles E1 −→ X1, E2 −→ X2 and a bundle isomorphism

φ : E1 A −→ E2 A, let E1 ∪φ E2 be the quotient of the disjoint union E1 tE2 by the equivalence relation

E1 A 3 e ∼ φ(e) ∈ E2 A. 2For a proof see, e.g., Theorem 6.6 in Michael Atiyah and Isadore Singer, The Index of Elliptic Operators: III. Annals of Mathematics, Second Series, 87, no. 3 (1968): 546-604. 186 CHAPTER 7. THE ATIYAH-SINGER INDEX THEOREM

i) Show that E1 ∪φ E2 is a vector bundle over X. ii) Show that every vector bundle over X is obtained in this way from vector bundles over X1 and X2. iii) Show that the isomorphism class of E1 ∪φ E2 only depends on the homotopy class of the isomorphism φ : E1 A −→ E2 A. ∗ ∗ (Hint: Let π : X × [0, 1] −→ X. Relate E1 ∪φt E2 to a vector bundle π E1 ∪Φ π E2 over X × [0, 1].) Exercise 3. Let M be a manifold and

0 −→ E0 −→ E −→ E00 −→ 0 an exact sequence of vector bundles and vector bundle maps. Show that there is a vector bundle isomorphism E ∼= E0 ⊕ E00. (Hint: Endow E with a bundle metric and show E00 ∼= (E0)⊥.) Exercise 4. A subspace V ⊆ C∞(M; E) is ample if

Φ: M × V / E (x, s) / s(x) is surjective. Show that if M is a closed manifold, then C∞(M; E) contains a ﬁnite dimensional ample subspace. Exercise 5. Let M be a compact manifold. i) Show that for every vector bundle, E −→ M, there is a vector bundle F −→ M such that E ⊕ F is trivial. ii) Show that every element of K(M) can be represented as the formal diﬀerence of a vector bundle and a trivial vector bundle. iii) Show that two vector bundles E and F over M determine the same element of K(M) if and only if there is a trivial vector bundle Cn over M such that n ∼ n E ⊕ C = F ⊕ C . (We say that E and F are stably equivalent.) (Hint: Use exercises 3 and 4.)

Exercise 6. Let E −→ S1 be the non-trivial R-bundle over the circle (the M¨obiusband). Show that the Whitney sum E ⊕ E is isomorphic to R2 −→ S1, the trivial R2-bundle over the circle. Exercise 7. ∞ ∞ Let M be a compact manifold. Let p ∈ C (M; Mn(C)) = Mn(C (M)) be a 7.7. BIBLIOGRAPHY 187 smooth family of n × n matrices valued in orthogonal projections, i.e., satisfying p2 = p = p∗. Show that the image of p is a subbundle of Cn, the trivial Cn bundle over M. Moreover every subbundle of Cn arises in this way. (By exercise 5 it follows that every vector bundle over X arises in this way.) Exercise 8. Let M be a compact manifold and E −→ M a vector bundle. From exercise 7 we know that there exists a smooth family of n × n orthogonal projections p ∈ ∞ ∼ ∞ C (M; Mn(C)) such that Image(p) = E. If q ∈ C (M; M`(C)) is another smooth family of orthogonal projections such that Image(q) ∼= E, show that there exists a smooth matrix-valued functions u such that

p = uu∗ and q = u∗u.

We say that p and q are Murray-von Neumann equivalent. (Hint: Construct u : C∞(M; C`) −→ C∞(M; Cn) using ﬁrst a projection and then an inclusion.) Exercise 9. Let M be a compact manifold. Let V (C∞(M)) be the Murray-von Neumann equivalence classes of smooth families matrices valued in orthogonal projections,

∞ ∞ 2 ∗ V (C (M)) = {[p]: p ∈ Mn(C (M)) for some n ∈ N and p = p = p }. Show that the direct sum p 0 p ∈ M (C∞(M)), q ∈ M (C∞(M)) =⇒ p ⊕ q = ∈ M (C∞(M)). n ` 0 q n+` induces the structure of Abelian semigroup on V (C∞(M)). Show that this semigroup is isomorphic to Vect(M) and hence the Grothendieck group is K(M). (This is the C∗-algebra approach to K(M).)

7.7 Bibliography

For a wonderful early exposition of the heat equation proof of the index theorem, see the article of Atiyah, Bott, and Patodi, On the Heat Equation and the Index Theorem. Inventiones mathematicae 19 (1973): 279–330. The geometric method of carrying out Getzler rescaling is due to Richard Melrose and can be found in Chapter 8 of his book The Atiyah-Patodi-Singer index theorem. Our exposition beneﬁtted from that of Chris Kottke in his lecture notes on Linear Analysis on Manifolds, available on his webpage, http://ckottke.ncf.edu/neu/7376_sp16/.