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Home , Christoffel symbols, Exterior derivative, Musical isomorphism, Riemann curvature tensor

LAPLACIANS IN GEOMETRIC ANALYSIS

Syafiq Johar syafi[email protected]

Contents

1 Trace Laplacian 1 1.1 Connections on Vector Bundles ...... 1 1.2 Local and Explicit Expressions ...... 2 1.3 Second Covariant ...... 3 1.4 on Vector Bundles ...... 4 1.5 Trace Laplacian ...... 5

2 Harmonic Functions 6 2.1 and Operators ...... 7 2.2 Laplace-Beltrami Operator ...... 7 2.3 Harmonic Functions ...... 8 2.4 Harmonic Maps ...... 8

3 Hodge Laplacian 9 3.1 Exterior ...... 9 3.2 Hodge Duals ...... 10 3.3 Hodge Laplacian ...... 12

4 Hodge Decomposition 13 4.1 ...... 13 4.2 Hodge Decomposition Theorem ...... 14

5 Weitzenb¨ock and B¨ochner Formulas 15 5.1 Weitzenb¨ock Formula ...... 15 5.1.1 0-forms ...... 15 5.1.2 k-forms ...... 15 5.2 B¨ochner Formula ...... 17 1 Trace Laplacian

In this section, we are going to present a notion of Laplacian that is regularly used in differential geometry, namely the trace Laplacian (also called the rough Laplacian or Laplacian). We recall the definition of connection on vector bundles which allows us to take the of vector bundles.

1.1 Connections on Vector Bundles

Definition 1.1 (Connection). Let M be a differentiable and E a vector bundle over M. A connection or at a point p ∈ M is a map D : Γ(E) → Γ(T ∗M ⊗ E) ∞ with the properties for any V,W ∈ TpM, σ, τ ∈ Γ(E) and f ∈ C (M), we have that DV σ ∈ Ep with the following properties:

1. D is tensorial over TpM i.e.

DfV +W σ = fDV σ + DW σ.

2. D is R-linear in Γ(E) i.e. DV (σ + τ) = DV σ + DV τ.

3. D satisfies the over Γ(E) i.e. if f ∈ C∞(M), we have:

DV (fσ) = df(V ) · σ + fDV σ.

Remark 1.2. Property (1) tells us that (DV σ)p depends on the value of V at p only. Properties

(2) and (3) tell us that (DV σ)p depends on the value of σ in a neighbourhood of p. Thus, the connection D gives us a “directional derivative” of the field σ ∈ Γ(E) at the point p at the direction V ∈ TpM. More technically, the covariant derivative is the infinitesimal difference between the of the section σ near p on a curve γ :(−ε, ε) → M such that γ(0) = p and γ0(0) = X with the vector σ at p. Due to this, a connection is not necessarily unique on a topological manifold M and it depends on the metric the manifold M is endowed with (since different metrics have different parallel transport associated to it). We can extend this definition to the directional derivative over a section V ∈ Γ(TU) on an open set U ⊂ M by doing the derivation pointwise and thus DV σ ∈ Γ(E|U ) is a section of E on U.

Remark 1.3. Given connections DE and DF on vector bundles E and F respectively, we can define a connection DE⊗F on the tensor bundle of E ⊗ F by:

E⊗F E F DX (σ ⊗ µ) = (DX σ) ⊗ µ + σ ⊗ (DX µ). Similarly, a connection DE on vector bundle E induces a connection DE∗ on E∗ by using the product rule and connection on E. Explicitly, if σ ∈ E and s ∈ E∗, we have the following:

E∗ E X(s(σ)) = (DX s)(σ) + s(DX σ).

1 Definition 1.4 (Compatible connection). A connection D on a vector bundle E is compatible with the metric h·, ·i on E if for any X ∈ Γ(TM) and σ, µ ∈ Γ(E), we have:

DX hσ, µi = Xhσ, µi = hDX σ, µi + hσ, DX µi.

An obvious example of the connection is the Levi-Civita connection in which we take the vector bundle E to be the bundle TM and the inner product to be the Riemannian metric g. We have the following fundamental theorem of :

Theorem 1.5 (Fundamental theorem of Riemannian geometry). On each (M, g), there exists a unique connection ∇ on TM which is compatible with the metric (i.e.

∇g ≡ 0) and torsion-free (i.e. ∇X Y − ∇Y X = [X,Y ] for any vector fields X and Y ). This connection is called the Levi-Civita connection.

1.2 Local and Explicit Expressions

Of course, the above notations are nice but computation wise, it can be quite limited. Therefore, we can introduce a to make things more explicit.

Definition 1.6. A moving frame for a vector bundle is a collection of n vector fields {v1(x), v2(x), . . . , vn(x)} ⊂

Γ(TM) such that at each x ∈ M, this set forms a for TxM. There are many useful frames that can be used in computation. For example, if we have a local coordinate patch U ⊂ M given by (x1, x2, . . . , xn), we can define a coordinate frame  ∂ ∂ ∂ ∂x1 , ∂x2 ,..., ∂xn which forms a basis for TxM for all x ∈ U. For example of a local compu- ∂ ∂  tation, we have, g ∂xi , ∂xj = gij. Another useful choice of frames is the orthonormal frame which is defined independently of coordinates. The orthonormal frame is the collection {e1(x), e2(x), . . . , en(x)} such that g(ei, ej) = δij. Note here that the orthonormal frame does not define a on

M locally (since, in general, [ei, ej] 6= 0 but the Lie bracket of coordinate vectors vanish i.e  ∂ ∂  ∂xi , ∂xj = 0 for all i, j). Definition 1.7. A dual to a vector bundle frame {v1(x), v2(x), . . . , vn(x)} is a collection of n ∗ j i covector fields {v1(x), v2(x), . . . , vn(x)} ⊂ Γ(T M) such that at each x ∈ M, vi(v ) = δj. In most local computations, we shall use the coordinate frame. However, in some computa- tions, it is more convenient to use the orthonormal coordinate frame (for example, in the later parts of this note). Connections on the tangent bundle TM is particularly useful in studying the of the manifold. For such connections ∇ on TM, we define the Christoffel symbols which, by and tensorial properties of connections, allow us to write the connection expression explicitly in local coordinates.

n k Definition 1.8 (Christoffel symbols). In local coordinates {xi}i=1, the Christoffel symbol Γij for a connection on TM is given by

∂ k ∂ ∇ ∂ j = Γij k . ∂xi ∂x ∂x

2 ∂ k Remark 1.9 (Notation). We write ∂xi as ∂i so the above is simply ∇∂i ∂j = Γij∂k. Also, the Levi-Civita connection on TM induces a connection on T ∗M (which we also denote as ∇). By Remark 1.3, we can show:

k k j k k j ∇ ∂ dx = −Γijdx i.e. ∇∂i dx = −Γijdx . ∂xi Remark 1.10. The Christoffel symbols for the Levi-Civita connection satisfy the following properties:

1. In local coordinates, the Christoffel symbols can be written explicitly in terms of the metric g as follows: 1 Γk = gkm (∂ g + ∂ g − ∂ g ) . ij 2 j mi i mj m ij

k k 2. It is symmetric i.e. Γij = Γji.

3. The following contraction identities are useful later:

ij k 1 p ki i p g Γ = − ∂i( |g|g ) and Γ = ∂klog( |g|), ij p|g| ki

where |g| = |det(gij)|.

1.3 Second Covariant Derivative

A connection DE on E together with a Levi-Civita connection on T ∗M induces a connection DT ∗M⊗E on T ∗M ⊗E. Therefore, we can compose the two connections to give a second covariant derivative: E T ∗M⊗E Γ(E) −−→D Γ(T ∗M ⊗ E) −−−−−−→D Γ(T ∗M ⊗ T ∗M ⊗ E).

Definition 1.11 (Second covariant derivative). We define the second covariant derivative on a vector bundle E as the map D2 : Γ(E) → Γ(T ∗M ⊗ T ∗M ⊗ E) given by

2 2 DXY σ = (D σ)(X,Y ) = (DX Dσ)(Y ) = DX (DY σ) − D∇X Y σ, where ∇ is the Levi-Civita connection. This expression is C∞(M)-linear over X and Y .

Unlike the second order derivative in , the second covariant derivatives generally do not commute. In other words, we get the following:

2 2 D σ(X,Y ) − D σ(Y,X) = DX (DY σ) − DY (DX σ) − D[X,Y ]σ.

The expression above measures the failure for the second covariant derivative to commute. We can easily show that this quantity is in fact a tensor. This quantity is called the curvature of the connection, which is a central object of study in Riemannian geometry.

3 1.4 Curvatures on Vector Bundles

Definition 1.12 (Curvature). The curvature of the connection D on the vector bundle E is the tensor R ∈ Γ(T ∗M ⊗ T ∗M ⊗ E∗ ⊗ E) given by, for any X,Y ∈ Γ(TM) and σ ∈ Γ(E):

R(X,Y )σ = D2σ(X,Y ) − D2σ(Y,X)

= DX (DY σ) − DY (DX σ) − D[X,Y ]σ ∈ Γ(E). (1)

Definition 1.13 ( curvature). When D is the Levi-Civita connection ∇ of Riemannian metric g (and hence E is TM), the quantity R in Definition 1.12 is called the Riemann curvature. i n In local coordinates {x }i=1, we can write the Riemann curvature as: ∂ R = R ldxi ⊗ dxj ⊗ dxk ⊗ . ijk ∂xl We also have the , also denoted R, which is defined as:

R(X,Y,Z,W ) = g(R(X,Y )Z,W ), for all W, X, Y, Z ∈ Γ(TM). In coordinates, the tensor can be written as:

i j k l R = Rijkldx ⊗ dx ⊗ dx ⊗ dx ,

m where Rijkl = glmRijk .

Remark 1.14. In literature, there is much variation in the . In this notes, we stick to the above convention in (1) same as John Lee’s book.

Theorem 1.15 (Symmetries of Riemann curvature tensor). The Riemann curvature tensor has the following important symmetry properties:

1. Antisymmetry in the first two arguments: Rijkl = −Rjikl.

2. Antisymmetry in the last two arguments: Rijkl = −Rijlk.

3. Symmetry between the first pair and second pair of arguments: Rijkl = Rklij.

4. First Bianchi identity: Rijkl + Riklj + Riljk = 0.

5. Second Bianchi identity: ∇∂m Rijkl + ∇∂l Rijmk + ∇∂k Rijlm = 0.

Remark 1.16 (More notations). Sometimes, the colon and semicolon notations are used for i partial and covariant derivatives respectively. If X = X ∂i ∈ Γ(TM), we write:

i i i i i k i i k i ∇∂j X = X ;j∂i where X ;j = ∇jX = ∇∂j X = ∂jX + X Γjk = X ,j + X Γjk.

i i (Warning: ∇ is for and vectors and ∂ are for scalars so ∇jX 6= ∂jX ) Similarly can be applied to derivatives on scalar and other tensors. Thus, the second Bianchi identity can be written as:

Rijkl;m + Rijmk;l + Rijlm;k = 0.

4 Furthermore, we also have the anti-symmetrisation of tensor indices denoted with the square brackets [··· ]. Suppose that X is a p tensor, then we have the following: X A[i1i2···iq]iq+1···ip = sgn(σ)Aiσ(1)iσ(2)···iσ(q)iq+1···ip . σ∈S(q)

Using this convention and by virtue of the antisymmetry in the last two arguments of the Riemann curvature tensor, the Bianchi identities can be written compactly as:

Ri[jkl] = 0 and Rij[kl;m] = 0.

The curvature tensor is a 4-tensor, which is very awkward to deal with. There is a way of simplifying the curvature tensor in a meaningful way by contracting the tensor with itself to get a 2-tensor called the tensor. We shall need the Ricci curvature tensor in Section 5. First, we define what the Ricci curvature tensor is:

Definition 1.17 (Ricci and ). The trace of the Riemann curvature tensor over the second and third (or first and fourth, by the symmetries of the Riemann curvature tensor) indices is called the Ricci curvature tensor. In coordinates, the 2-tensor can be written as:

k k pq Ricij = Rkij = Ri kj = g Ripqj.

The scalar curvature is a 0-tensor which is obtained by further tracing the Ricci curvature with g: ij Scal = g Ricij.

Remark 1.18. The tracing over the second and fourth indices of the Riemann curvature tensor to get the Ricci tensor is valid for our convention of the Riemann curvature tensor. If the curvature tensor is defined in the opposite sign, we need to trace over different indices. This is to make sure that the definition of Ricci curvature is uniform in any convention of the Riemann curvature tensor.

1.5 Trace Laplacian

Definition 1.19 (Trace Laplacian). The trace Laplacian ∆˜ on the vector bundle E is given by n the minus of the trace of the second covariant derivative. In local coordinates {xi}i=1, we have the following expression:

n 2 X ij ∆˜ σ = −trD σ = − g (D∂ (D∂ σ) − D σ). i j (∇∂i ∂j ) i,j=1

Another notion of this Laplacian is through the adjoint D∗ of the covariant derivative D. We define the adjoint of a connection D on a vector bundle E as follows:

Definition 1.20 (Adjoint of connection). Let E be a vector bundle on the Riemannian manifold (M, g) with a connection D : Γ(E) → Γ(E ⊗ T ∗M) compatible with the metric h·, ·i on E. The

5 adjoint D∗ : Γ(T ∗M ⊗ E) → Γ(E) of the covariant derivative D is given by the composition of DT ∗M⊗E defined earlier with the metric g−1 on cotangent bundles.

∗ T M⊗E g−1 D∗ : Γ(T ∗M ⊗ E) −−−−−−→D Γ(T ∗M ⊗ T ∗M ⊗ E) −−→ Γ(E).

In local coordinates, this can be written as n X ∗ D∗ = − gij∂ DT M⊗E , i y ∂j i,j=1 where y is the . Remark 1.21 (Interior product). Recall that the interior product is the contraction of a dif- k k−1 ferential form with a chosen vector field X i.e. X y :Ω (M) → Ω (M). In other words, it is like an “inverse” of the . Explicitly, if w ∈ Ωk(M) and X ∈ Γ(TM), we have:

(X y w)(Y1,...,Yk−1) = w(X,Y1,...,Yk−1), for any Y1,...,Yk−1 ∈ Γ(TM).

Remark 1.22. The adjoint D∗ of the connection D on a vector bundle E can also be written as: ∗ ] D (X ⊗ σ) = −(DX] σ + div(X ) · σ), for X ∈ Γ(T ∗M), σ ∈ Γ(E), div is the divergence operator as in Definition 2.3 in the next ∗ i section and ] : T M → TM the defined on cotangent vectors X = Xi dx ] ij as X = g Xi∂j.

Thus, we have the following lemma which gives the relation between the trace Laplacian and the composition of the connection and its adjoint.

Lemma 1.23. On Γ(E), we have the following equality of operators:

D∗D = −trD2 = ∆˜ .

Proof. Pick an arbitrary σ ∈ Γ(E). Then,

n n X ∗ X  ∗  D∗Dσ = − gij∂ DT M⊗EDσ = − gij DT M⊗EDσ (∂ ) i y ∂j ∂j i i,j=1 i,j=1 n X ij 2 = − g D σ (∂j, ∂i) = ∆˜ σ, i,j=1 which is what we wanted to prove.

2 Harmonic Functions

Next, we are going to define another notion of Laplacian on functions (or 0-forms), called the Laplace-Beltrami operator. In order to do so, we need to first define the gradient and divergence operators.

6 2.1 Gradient and Divergence Operators

For a differentiable function f : M → R on an n-dimensional Riemannian manifold (M, g), we define the following:

Definition 2.1 (Gradient). The gradient of a function f is defined as the vector field ∇f or grad(f) such that for any vector field X on M, we have that g(∇f, X) = df(X) = X(f). In coordinates, this can be written as

n X ij ∇f = grad(f) = g ∂if∂j. i=1

2 ij kl ij 2 Remark 2.2. Note that for norms, ||∇f|| = gjlg ∂ifg ∂kf = g ∂if∂jf = ||df|| where d is the usual differential operator on functions.

i Definition 2.3 (Divergence). For a vector field X = X ∂i, the divergence of X is:

n n X i X ji div(X) = tr(∇X) = (∇∂i X) = g h∇∂i X, ∂ji , i=1 i,j=1 where ∇ is the Levi-Civita connection. Using the formula in Remark 1.10, we have:

1 p j 1 p ij  div(X) = ∂j( |g|X ) = ∂j |g|g hX, ∂ii . p|g| p|g|

Another way of defining the divergence operator is via the exterior derivative, which we are going to define in the next section. The definition of the divergence operator using exterior derivatives is given by:

Definition 2.4 (Divergence - via exterior derivatives). Let dV be the volume form of (M, g) (in local coordinates, this is dV = p|g|dx1 ∧ ... ∧ dxn). Then, for any vector field X ∈ Γ(TM), we have: d(X y dV ) = div(X)dV.

2.2 Laplace-Beltrami Operator

Now, we can define the Laplace-Beltrami operator on functions which is defined as the negative composition of gradient and divergence operator, which is akin to the Laplacian operator in the Euclidean setting.

Definition 2.5 (Laplace-Beltrami Operator). For a differentiable function f : M → R, the Laplace-Beltrami operator is given by:

1 p ij  ∆f = −div(grad(f)) = − ∂j |g|g ∂if . (2) p|g|

Remark 2.6. Some literature defines the Laplace-Beltrami operator without the negative sign. We define it this way to make sure that it is consistent with the definition of Hodge Laplacian later. See Remark 3.16.

7 2.3 Harmonic Functions

2 2 Definition 2.7 (L inner product). For functions f, g : M → R, the L inner product (·, ·) is defined as: (f, g) = fg dV, ˆM where dV = p|g|dx1 ∧ ... ∧ dxn is the volume form.

Remark 2.8. For smooth functions f, g : M → R on a compact closed manifold M, the following holds: (∆f, g) = (df, dg) = (f, ∆g).

Definition 2.9 (Energy). The energy of a differentiable function f : M → R is given by:

1 2 1 ij E(f) = ||df|| dV = g ∂if∂jf dV. 2 ˆM 2 ˆM Lemma 2.10 (Harmonic functions). A smooth critical point f of the energy functional in the sense that: d E(f + tη)| = 0, dt t=0 for all smooth η : M → R with compact support in M is harmonic i.e. ∆f ≡ 0. Proof. We compute:

d 1 ij 0 = g (∂if + t∂iη)(∂jf + t∂jη) dV |t=0 dt 2 ˆM ij p 1 n = g ∂if∂jη |g| dx . . . dx ˆM p ij  1 n = − ∂j |g|g ∂if η dx . . . dx ˆM = (∆f)η dV, ˆM for all smooth and compactly supported η : M → R. For the equality to hold, we must have that ∆f ≡ 0.

2.4 Harmonic Maps

This is an aside from the main topic of Laplacians. The definition of harmonic functions can be extended to maps between . Suppose that f : M → N is a map between Riemannian m n manifolds (M, γ) and (N, g) of m and n respectively. Let {xα}α=1 and {fi}i=1 be local coordinates of M and N respectively. We define the energy density of the map f as: 1 1 e(f)(x) = ||df||2 = γαβ(x)g (f(x))∂ f i∂ f j, 2 2 ij α β where the Greek indices indicate the domain coordinates, Latin indices indicate the target coordinates and the norm || · || is taken over the tensor T ∗M ⊗ f −1TN. The energy of the map f is therefore defined as:

E(f) = e(f)dVM , (3) ˆM

8 p 1 m where dVM = |γ|dx ∧ ... ∧ dx is the volume form on the manifold M. Similar as before, harmonic maps are the critical point f of the energy functional (3) in the sense that

d E(f + tη)| = 0, dt t=0 for any smooth and compactly supported η : M → N. However, instead of the formula we had in (2), for harmonic maps f, they satisfy the following Euler-Lagrange equation:

1 p αβ i αβ (g) i j k ∂α |γ|γ ∂βf + γ Γ (f(x))∂αf ∂βf = 0, p|γ| jk

(g) i for i = 1, . . . , n where Γjk is the Christoffel symbol for the target metric g. Note that if the target manifold is just (R, δ), the Christoffel symbol vanishes and we would get the usual equation for Laplace-Beltrami operator in (2).

3 Hodge Laplacian

The aim of this section is to construct an analogous Laplacian operator on k-forms which extends from functions (or 0-forms) in the previous section. We begin by defining the exterior derivative d and construct its dual (called codifferential d∗). We require a tool called . In the next section, we shall look at an application of this Laplacian operator due to Hodge which decomposes the space of k-forms into three pairwise orthogonal parts.

3.1 Exterior Derivatives

Definition 3.1 (Exterior derivative). The exterior derivative d :Ωk(M) → Ωk+1(M) is defined for k = 0 as the usual differential. For any other k, in local coordinates, the exterior derivative is defined as: ! X I X I d aI dx = daI ∧ dx , I I where aI : M → R are smooth functions for an indexing set I of length k i.e. I = (i1, . . . , ik) where i1 < ··· < ik.

n Remark 3.2 (A basis formula for the exterior derivative). If {vi}i=1 is a local frame for (M, g) i n with Levi-Civita connection ∇, and {v }i=1 is its dual, then for every smooth k-form w, we have: n X i dw = v ∧ ∇vi w, i=1 for which we define the Levi-Civita connections on k-forms by using the Cartan connection forms. In particular, if the local frame is chosen to be the coordinate frame, then we get the usual definition of exterior derivative as in Definition 3.1: n X i X i I dw = dx ∧ (∂iw) = ∂iwI dx ∧ dx . i=1 i,I

9 Lemma 3.3. Properties of the exterior derivative:

1. d is a linear operator.

2. d obeys the product rule d(v ∧ w) = dv ∧ w + (−1)deg(v)v ∧ dw.

3. d commutes with pullback of a smooth map F : N → M by F ∗ ◦ d = d ◦ F ∗.

4. d ◦ d ≡ 0.

Definition 3.4 (Closed and exact forms). We call a k-form v closed if dv = 0 and exact if there exists a (k − 1)-form w such that v = dw.

One of the main reasons to consider the exterior derivatives on differential forms is the essential theorem by Stokes which is proven in any introductory courses on smooth manifolds.

Theorem 3.5 (Stokes’ Theorem). Suppose that M is an orientable n-dimensional manifold with boundary ∂M. Let ω ∈ Ωn−1(M) be compactly supported. Then, we have

ω = dω. ˆ∂M ˆM

3.2 Hodge Duals

In local coordinates, we let vi and wi be 1-forms and define an inner product on k-forms v1 ∧ ... ∧ vk and w1 ∧ ... ∧ wk on an n-dimensional manifold M by:

hv1 ∧ ... ∧ vk, w1 ∧ ... ∧ wki = det(hvi, wji), where the determinant is taken over the k × k and h·, ·i is the induced inner product on 1-forms. This definition can be extended linearly on all of Ωk(M). Next, we define the dual for a k-form which is given by an (n − k)-form obtained via the Hodge star operator.

Definition 3.6 (Hodge star). Let (M, g) be an n-dimensional Riemannian manifold and dV = p 1 n n |g|dx ∧...∧dx be its volume element in local coordinates {xi}i=1. The Hodge star operator ? :Ωk(M) → Ωn−k(M) maps any k-form v ∈ Ωk(M) to its dual (n − k)-form ?v ∈ Ωn−k(M) which is defined by the relation w ∧ ?v = hw, vi dV for any k-form w.

Lemma 3.7. Here are some immediate facts:

1. For a 0-form 1, its dual ?(1) = dV .

2. Conversely, the dual for the volume form is ?dV = 1.

3. The Hodge star operator is linear i.e. for v, w ∈ Ωk(M) and f, g ∈ C∞(M), we have ?(fv + gw) = f(?v) + g(?w).

4. By symmetry of h·, ·i, we have w ∧ ?v = v ∧ ?w.

5. If v ∧ ?v = 0, then v = 0.

10 n n In local coordinates {xi}i=1, the space of k-forms is k -dimensional and the basis is given i1 i by elements dx ∧ ... ∧ dx k where 1 ≤ i1 < ··· < ik ≤ n. Thus, by linearity, we can calculate the Hodge star on the basis explicitly and extend it linearly to the whole of Ωk(M).

n Lemma 3.8. In local coordinates {xi}i=1, we have p ?(dxi1 ∧ ... ∧ dxik ) = |g| det(hdxip , dxiq i) sgn(σ) dxj1 ∧ ... ∧ dxjn−k ,

ip iq ipiq where |g| = det(gij), det(hdx , dx i) is the determinant of the k × k matrix with entries (g ) for p, q = 1, . . . k, {j1, . . . jn−k} is the set of indices in {1, 2, . . . , n}\{i1, . . . , ik} and sgn(σ) is the signature of the permutation σ : (1 2 ··· n) 7→ (i1 i2 ··· ik j1 ··· jn−k).

n k Proof. Pick a local coordinate system {xi}i=1 on the manifold and v a basis element of Ω (M), say v = dxi1 ∧ ... ∧ dxik . By definition of the Hodge star operator, for any w ∈ Ωk(M), we have w ∧ ?v = hw, vi dV . In particular, if we choose w = v, we must have that ?v is of the form

j1 j Cdx ∧ ... ∧ dx n−k where {j1, . . . jn−k} is the set of indices in {1, 2, . . . , n}\{i1, . . . , ik} and C is some constant which is going to be determined. Thus:

v ∧ ?v = hv, vi dV p Cdxi1 ∧ ... ∧ dxik ∧ dxj1 ∧ ... ∧ dxjn−k = det(hdxip , dxiq i) |g|dx1 ∧ ... ∧ dxn p C sgn(σ−1) dx1 ∧ ... ∧ dxn = |g| det(hdxip , dxiq i) dx1 ∧ ... ∧ dxn,

−1 −1 ip iq where σ is the permutation σ :(i1 ··· ik j1 ··· jn−k) 7→ (1 2 ··· n) and det(hdx , dx i) is taken from the definition of inner products on k-forms. Equating coefficients and noting that sgn(σ) = sgn(σ−1) completes the proof.

Corollary 3.9 (Double Hodge star). For any v ∈ Ωk(M) where M is an n-dimensional Rie- mannian manifold, we have ? ? v = (−1)k(n−k)v.

Now, we can define a global L2 inner product (called Hodge L2 inner product) on compactly supported k-forms as follows:

(v, w) = hv, wi dV = v ∧ ?w. (4) ˆM ˆ By the properties of the Hodge star, we get that the inner product is indeed symmetric, bilinear and (v, v) 0 iff v 6= 0. Note also that (?v, ?w) = (v, w).

Remark 3.10. Note that the definition of the Hodge L2 inner product agrees with the usual definition of the L2 inner product on functions (i.e. 0-forms) as in Definition 2.7.

Definition 3.11 (Codifferential). The codifferential operator d∗ :Ωk+1(M) → Ωk(M) is the formal adjoint operator for the exterior derivative d in the Hodge L2 inner product in (4) i.e. for every v ∈ Ωk(M) and w ∈ Ωk+1(M), we have

(dv, w) = (v, d∗w).

11 The following properties for the codifferential operator d∗ are straightforward to check. The final property is akin to the exterior derivative property d ◦ d ≡ 0.

Lemma 3.12. Let d∗ :Ωk+1(M) → Ωk(M) be the codifferential of the exterior derivative d :Ωk(M) → Ωk+1(M). Then we have:

1. d∗ = (−1)n(k+1)+1 ? d ?.

2. d = (−1)nk ? d∗ ?.

3. d∗ ◦ d∗ ≡ 0.

Just as exterior derivatives, we can write the codifferential operator in terms of the Levi- Civita connection. In fact, this form is useful for explicit calculations and will be used later in the final section.

n Remark 3.13 (Another expression for codifferential). Let {xi}i=1 be local coordinates on (M, g). Then, the codifferential operator on w ∈ Ωk(M) can be written as:

n ∗ X ij d w = − g (∇∂i w). i,j=1

n If the local frame {ei}i=1 is chosen to be orthonormal, then we can write

n ∗ X d = − ei y ∇ei . i=1

3.3 Hodge Laplacian

In Section 1, we had the Laplace-Beltrami operator on 0-forms. We are now going to define the Hodge Laplacian which generalises the Laplace-Beltrami to k-forms.

Definition 3.14 (Hodge Laplacian). The Hodge Laplacian on Ωk(M) is defined as

∆ = (d + d∗)2 = dd∗ + d∗d :Ωk(M) → Ωk(M).

A k-form v ∈ Ωk(M) is called harmonic if ∆v = 0.

Definition 3.15 (Harmonic forms). The space of harmonic k-forms on M is denoted as

Hk(M) = {v ∈ Ωk(M) : ∆v = 0}.

Also, h ∈ Hk(M) iff h is both closed and coclosed, that is dh ≡ 0 and d∗h ≡ 0.

Remark 3.16. Note that this extension agrees with the Laplace-Beltrami operator on functions in Definition 2.5. Indeed, if k = 0, then d∗ ≡ 0, thus ∆ = d∗d and by local computation, for a

12 0-form f, we have

∗ ∗ j ∆f = d df = d ∂jfdx j = − ? d ? ∂jfdx

 p ij 1 j n = − ? d sgn(σj) |g|g ∂if dx ∧ ... ∧ dxd ∧ ... ∧ dx

−1 1 p ij  = −sgn(σj) sgn(σ ) ∂j |g|g ∂if ? dV j p|g|

1 p ij  = − ∂j |g|g ∂if , p|g| where σj is the permutation σj : (1 2 ··· n) 7→ (j 1 2 ··· bj ··· n). Thus, on 0-forms, the Hodge Laplacian agrees with the Laplace-Beltrami operator in (2).

The following properties of the Hodge Laplacian are easy to verify:

Lemma 3.17. Properties of the Hodge Laplacian:

1. ∆ is formally self-adjoint in the Hodge L2 inner product i.e. (∆v, w) = (v, ∆w).

2. (∆v, v) ≥ 0 and equality occurs iff dv = d∗v = 0.

3. ∆ commutes with the Hodge star i.e. ?∆ = ∆?.

The last property of the above gives us a relation between harmonic k-forms and harmonic (n − k)-forms, which is called the Poincar´eduality:

Corollary 3.18 (Poincar´eduality). Since (?v, ?w) = (v, w) and ?∆ = ∆?, we have that ? : Hk(M) → Hn−k(M) is an for 0 ≤ k ≤ n. Thus, dim(Hk(M)) = dim(Hn−k(M)) (which is finite by Fredholm alternative).

4 Hodge Decomposition

4.1 De Rham Cohomology

We recall from Remark 3.3 that the external derivative of a k-form d :Ωk(M) → Ωk+1(M) has the property d ◦ d ≡ 0. Recall also the definition of closed and exact differential forms from Definition 3.4. All exact forms are closed as d ◦ d ≡ 0. Thus, the set Bp(M) := Im(d :Ωp−1(M) → Ωp(M)) is contained in the set Zp(M) := Ker(d :Ωp(M) → Ωp+1(M)). Two k-forms are called cohomologous if their difference is exact. This defines an equivalence relation on the space of closed k forms and we define the set of equivalence classes, the de Rham cohomology group, as follows:

Definition 4.1 (de Rham cohomology). The p-th de Rham cohomology group is a topological invariant of a manifold M, given by the quotient Zp(M) Hp (M) = Hp(M) = . dR Bp(M)

13 Definition 4.2 (Betti number). The of the p-th de Rham cohomology group bp = dim(Hp(M)) is called the p-th Betti number.

The Betti number is used to calculate the Euler-Poincar´echaracteristic χ(M) of the manifold M of dimension n, a topological invariant, which is given by:

n X p χ(M) = (−1) bp. p=0

4.2 Hodge Decomposition Theorem

Now we are going to state an application of the Hodge Laplacian and elliptic partial differential equations.

Theorem 4.3 (Hodge decomposition theorem). Let M be an orientable compact smooth man- ifold of dimension n. Then, any k-form in Ωk(M) for 0 ≤ k ≤ n can be written as a unique sum of an exact form, a coexact form and a harmonic form. In other words,

Ωk(M) = dΩk−1(M) ⊕ d∗Ωk+1(M) ⊕ Hk(M).

Remark 4.4. The proof of this theorem uses the Fredholm alternative on the elliptic operator ∆ = dd∗ + d∗d. Note that also by Fredholm alternative, we get that dim(Hk(M)) is finite.

Applications of the Hodge decomposition theorem is the following result, which is more difficult to show using different methods:

Theorem 4.5. Let M be an orientable compact smooth manifold of dimension n. Then, every cohomology class in Hk(M) contains a unique harmonic form.

Proof. Uniqueness is straightforward. The proof for existence uses Dirichlet’s principle. Choose k a closed differential form w0 which represents the class [w0] in H (M). Then, any element k−1 w in the equivalence class [w0] can be expressed as w = w0 + dv for some v ∈ Ω (M). If we minimise the Hodge L2 inner product over all such w, the minimiserw ˜ satisfies the Euler- Lagrange equation for the variational problem for the Hodge L2 inner product, that is for any v ∈ Ωk−1(M), we have

d 0 = (w ˜ + tdv, w˜ + tdv)| dt t=0 = 2(w, ˜ dv) = 2(d∗w,˜ v).

Since this holds for any v ∈ Ωk−1(M), we have that d∗w˜ ≡ 0. Furthermore, we can show that this minimiser is attained in [w], implying that dw˜ ≡ 0. This shows that the minimiser is indeed a harmonic form.

From the above theorem, the following corollaries can be concluded easily.

14 Corollary 4.6. The space of harmonic k-forms Hk(M) is isomorphic to the k-th de Rham cohomology group. In fact, by definition of Betti numbers and Poincar´eduality, for 0 ≤ k ≤ n, we have

k k n−k n−k bk = dim(H (M)) = dim(H (M)) = dim(H (M)) = dim(H (M)) = bn−k.

Corollary 4.7. Closed orientable smooth manifold M of odd dimension has Euler-Poincar´e characteristic equal to 0.

5 Weitzenb¨ock and B¨ochner Formulas

From the previous sections, we have two notions of Laplacian, namely the Hodge Laplacian and the trace Laplacian. Now, we are going to relate the two. The relation of the two Laplacians are given by the Weitzenb¨ock formula.

5.1 Weitzenb¨ock Formula

5.1.1 0-forms

If f ∈ C∞(M) is just a function (i.e. 0-form), then the trace Laplacian is given by the negative of the trace of the Hessian where the Hess(f) ∈ Γ(T ∗M ⊗ T ∗M) is given by:

Hess(f)(X,Y ) = h∇X grad(f),Y i = X(Y (f)) − df(∇X Y ), for any X,Y ∈ Γ(TM). So,

n ˜ ij X ij ∆f = −g Hess(f)ij = − g h∇∂i grad(f), ∂ji . i,j=1

The Hodge Laplacian of a 0-form is just the Laplacian-Beltrami operator by the formula for divergence in Definition 2.3: ∆f = −div(grad(f)) = ∆˜ f. (5)

5.1.2 k-forms

Despite the equality of the two Laplacians on 0-forms, we would get different expressions for the Laplacians on 1-forms. This is because the curvature term would appear in higher forms.

Theorem 5.1. If σ ∈ Γ(T ∗M) is a 1-form on the manifold M, the Hodge and the trace Laplacians ∆σ, ∆˜ σ ∈ Γ(T ∗M) differ by a term involving the Ricci curvature of (M, g). More explicitly: ∆σ = ∆˜ σ + RicT (σ), (6) where RicT : T ∗M → T ∗M is the Ricci transform defined as RicT (σ)(Y ) = Ric(σ],Y ) for any X,Y ∈ Γ(TM) and ] is the musical isomorphism defined in Remark 1.22.

In fact, this can be generalised using the Weitzenb¨ock formula:

15 n Theorem 5.2 (Weitzenb¨ock formula). Let {ei}i=1 be a local orthonormal frame with the dual i n frame {e }i=1. Then, we have the following relation of the Hodge Laplacian and the trace ∗ Laplacian on k-forms for any k ∈ Z : n ˜ X i ∆ = ∆ + e ∧ ej y R(ei, ej), (7) i,j=1 where R is the curvature tensor on Ωk(M) with connection D as in Definition 1.12. Proof. We are going to check this identity at every point on M. Pick an arbitrary point x ∈ M ∂ and choose centred at x. Putting ∂xi = ei at the point x and extend the frame at x to an orthonormal frame in a local patch U by parallel transport, we have that the Christoffel symbols and the Lie brackets all vanish at x. Therefore, at the point x, we have:

2 ∇eiei = ∇ei ∇ei

R(ei, ej) = ∇ej ∇ei − ∇ei ∇ej . Now, we calculate the expression for ∆ = dd∗ +d∗d at the point x. Here, we use the expressions in Remarks 3.2 and 3.13.

 n  n ∗ X X i dd = d − ej y ∇ej  = − e ∧ ∇ei (ej y ∇ej ) j=1 i,j=1 n X i = − e ∧ ej y ∇ei ∇ej , i,j=1 and n ! n ∗ ∗ X i X i d d = d e ∧ ∇ei = − ej y ∇ej (e ∧ ∇ei ) i=1 i,j=1 n X i = − ej y (e ∧ ∇ej ∇ei ) i,j=1 n n X X i = − ∇ei ∇ei + e ∧ ej y ∇ej ∇ei . i=1 i,j=1 Thus, adding the two together, we get: n n n X X i ˜ X i ∆ = − ∇ei ∇ei + e ∧ ej y (∇ej ∇ei − ∇ei ∇ej ) = ∆ + e ∧ ej y R(ei, ej), i=1 i,j=1 i,j=1 which is what we wanted to show.

Remark 5.3. Note that if (7) acts on a 0-form f, since R(ei, ej)f = 0, we have (5). Similarly, if (7) acts on a 1-form σ, for any X ∈ Γ(TM), we have:

 n  X i ]  e ∧ ej y R(ei, ej)σ (X) = Ric(σ ,X), i,j=1 which gives us (6).

16 5.2 B¨ochner Formula

Apart from the Weitzenb¨ock formula, there is another formula that relates the two different types of Laplacian called the B¨ochner formula.

Theorem 5.4 (B¨ochner formula). Let D be a connection and h·, ·i be the metric on k-forms Ωk(M). Then, for any smooth k-form w, we have:

n 2 X ˜ −∆hw, wi = −∆|w| = 2 hDei w, Dei wi − 2h∆w, wi. i=1 Putting B¨ochner and Weitzenb¨ock formulas together, we get an expression for the Hodge Laplacian of the norm of a k-form:

n i n Theorem 5.5. Let {ei}i=1 be a local orthonormal frame with the dual frame {e }i=1. Then, for any w ∈ Ωk(M), we have:

n n 1 X X − ∆|w|2 = −h∆w, wi + hD w, D wi + hw, ei ∧ e R(e , e )wi. (8) 2 ei ei j y i j i=1 i,j=1

In particular, for the 1-form case, we have a nice expression:

n i n Corollary 5.6. Let {ei}i=1 be a local orthonormal frame with the dual frame {e }i=1. For any σ ∈ Γ(T ∗M), we have:

n 1 X − ∆|σ|2 = −h∆σ, σi + hD σ, D σi + Ric(σ], σ]). 2 ei ei i=1 The result above gives us a bit of information on the harmonic forms on the manifold by its curvature property.

Corollary 5.7. Suppose that (M, g) is a compact manifold of dimension n. Then:

1. If (M, g) has non-negative Ricci curvature, every harmonic 1-form σ are parallel (i.e. ∇σ = 0). In particular, the first de Rham cohomology group satisfies:

dim(H1(M)) ≤ n.

2. If (M, g) has positive curvature, them M has no nontrivial harmonic forms. Thus:

H1(M) = {0}.

17