Math 524 { Linear Analysis on Manifolds Spring 2012 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 . 19 1.12 Differential operators . 21 1.13 Exercises . 22 1.14 Bibliography . 24 2 A brief introduction to Riemannian geometry 25 2.1 Riemannian metrics . 25 2.2 Parallel translation . 28 2.3 Levi-Civita connection . 32 2.4 Associated bundles . 33 i ii CONTENTS 2.5 E-valued forms . 34 2.6 Curvature . 36 2.7 Geodesics . 39 2.8 Complete manifolds of bounded geometry . 42 2.9 Riemannian volume and formal adjoint . 42 2.10 Hodge star and Hodge Laplacian . 44 2.11 Exercises . 47 2.12 Bibliography . 50 3 Laplace-type and Dirac-type operators 53 3.1 Principal symbol of a differential operator . 53 3.2 Laplace-type operators . 56 3.3 Dirac-type operators . 57 3.4 Clifford algebras . 59 3.5 Hermitian vector bundles . 63 3.6 Examples . 68 3.6.1 The Gauss-Bonnet operator . 68 3.6.2 The signature operator . 70 3.6.3 The Dirac operator . 71 3.6.4 Twisted Clifford modules . 73 3.6.5 The @ operator . 73 3.7 Exercises . 76 3.8 Bibliography . 78 4 Pseudodifferential operators on Rm 79 4.1 Distributions . 79 4.2 Examples . 84 4.2.1 Fundamental solutions . 84 4.2.2 Principal values and finite parts . 85 4.3 Tempered distributions . 86 CONTENTS 1 4.3.1 Convolution . 86 4.3.2 Fourier transform . 88 4.4 Pseudodifferential operators on Rm ................... 92 4.4.1 Symbols and oscillatory integrals . 92 4.4.2 Amplitudes and mapping properties . 98 4.5 Exercises . 106 4.6 Bibliography . 107 5 Pseudodifferential operators on manifolds 109 5.1 Symbol calculus and parametrices . 109 5.2 Hodge decomposition . 114 5.3 The index of an elliptic operator . 118 5.4 Pseudodifferential operators acting on L2 . 122 5.5 Sobolev spaces . 127 5.6 Exercises . 131 5.7 Bibliography . 134 6 Spectrum of the Laplacian 135 6.1 Spectrum and resolvent . 135 6.2 The heat kernel . 139 6.3 Weyl's law . 146 6.4 Long-time behavior of the heat kernel . 148 6.5 Determinant of the Laplacian . 149 6.6 The Atiyah-Singer index theorem . 151 6.6.1 Chern-Weil theory . 151 6.6.2 The index theorem . 155 6.7 Examples . 157 6.8 Exercises . 159 6.9 Bibliography . 159 2 CONTENTS Lecture 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 fVαg together with homeomorphisms φα : Uα −! Vα m for some open subsets fUαg of R : The integer m is known as the dimension of M and the maps φα : Uα −! Vα (or sometimes just the sets fVαg) are known as coordinate charts. An atlas is smooth if whenever Vαβ := Vα \Vβ 6= ; the map −1 −1 −1 (1.1) αβ := φβ ◦ φα : φα (Vαβ) −! φβ (Vαβ) −1 −1 m is a smooth diffeomorphism between the open sets φα (Vαβ); φβ (Vαβ) ⊆ R : (For 1 k us smooth will always mean C :) If the maps αβ 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 a subcover in which each point is covered finitely many times. 3 4 LECTURE 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− = f(x1; : : : ; xn) 2 R : x1 ≤ 0g: 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 C1(M; M 0) and when M 0 = R; simply by C1(M): Notice that a map F 2 C1(M; M 0) naturally induces a pull-back map F ∗ : C1(M 0) −! C1(M);F ∗(h)(p) = h(F (p)) for all h 2 C1(M 0); p 2 M: (So we can think of C1 as a contravariant functor on smooth manifolds.) 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. 1.2. PARTITIONS OF UNITY 5 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 X: That is, at each point p 2 X there is a coordinate chart of M; φ : U −! V with q 2 V; whose restriction to U \ Rk × f0g maps into X; φ : U\ k × f0g −! V \ X: U\Rk×{0g 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 !i of smooth functions !i : M −! R such that: i) At each p 2 M only finitely many !i(p) are non-zero ii) Each !i is supported inside some Vα iii) The sum of the !i is the constant function equal to one at each point p 2 M: One remarkable fact that follows easily from the existence of partitions of unity is that any compact smooth manifold can be identified with a smooth submanifold of RN for some N: This is easy to show using a 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 2 M in terms of the curves through that point. 6 LECTURE 1. DIFFERENTIAL OPERATORS ON MANIFOLDS 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 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 2 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 = fv 2 R : v · p = 0g: 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.