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Harmonic Differential Forms and the Hodge Decomposition Theorem

Matthew Romney

Abstract This is an expository paper for Rui Fernandes’s Spring 2014 Differentiable 2 (Math 519) class.

Introduction

This paper will explain Hodge’s theorem giving an orthogonal decomposition of a dif- ferential form on a compact Riemannian . This is one of the central ideas of , developed in the 1930’s by the Scottish William V.D. Hodge and laid out in his 1941 book [1]. It had origins in , par- ticularly the work of Solomon Lefschetz, but it is immediately relevant to differential geometry. This is due to the corollary that every de Rham class for a com- pact has a unique harmonic representative, where “harmonic” refers to the appropriate generalization of the usual . In the intro- duction to the 1989 edition of Hodge’s book, Michael Atiyah describes Hodge’s work as one of the great mathematical landmarks of the century. He adds that Hodge was “the victim of his own success,” in the sense that Hodge’s original work would be over- shadowed as other refined and expanded on his ideas and found new applications. A presentation of the Hodge decomposition theorem has two main obstacles. First is the large amount of notation and definitions needed to simply state the result. Second is the difficulty of the proof, which draws heavily on harmonic and functional analysis. This paper will communicate as much of the ideas as is reasonable possible, but this will fall well short of a complete rigorous proof. The interested reader will find a self- contained proof in Warner’s 1971 textbook on manifold theory [5]. A recent expository paper of Nikolai Nowaczyk explains Warner’s approach in a more accesible form [4]. The paper is bases on these two sources, along with the standard textbook on smooth manifolds by John Lee [3]. Another recent work for further study is the 2011 book by J. Jost [2], although this book has a more analytic point of view. In the first we will define the on an oriented . Following that, we review the properties of Riemmanian manifolds and extend the definition of the Hodge star operator to this setting. This allows us to define the Laplace-Beltrami operator, which is a suitable generalization of the usual Laplace operator. After a few more definitions, we will finally be able to state the Hodge decomposition theorem and sketch the proof. The last section is devoted to applications to .

1 Before proceeding further, we will review the main definitions and notation for ten- sor fields, following that used by Lee. Let V be a and k be a positive integer. A covariant k- on V is a multilinear α : V k = V × · · · × V → R. Such a tensor α is said to be alternating if α(v1, . . . , vi, . . . , vj, . . . , vk) = −α(v1, . . . , vj, . . . , vi, . . . , vk) for any pair of indices i, j. Similarly, α is symmetric if α(v1, . . . , vi, . . . , vj, . . . , vk) = α(v1, . . . , vj, . . . , vi, . . . , vk). The space of all alternating covariant k- on V is denoted by Λk(V ∗). From here, we can extend these defini- tions to objects defined on manifolds. We let ΛkT ∗M denote the bundle of alternating k ∗ G k ∗ covariant k-tensors on M; more precisely, Λ T M = Λ (Tp M). A (differential) p∈M k-form is a smooth section of ΛkT ∗M. The space of all k-forms is denoted by Ωk(M). ∞ M We also use the notation Ω∗(M) := Ωk(M). i=1 1 Hodge Star Operator

In this section we will start with an oriented inner product space V of finite n and build up to the definition of the Hodge star operator. The existence of an inner product on V provides a large amount of structure to work with. The most basic consequence is the existence of a positive orthonormal (ei), which follows from the Gram-Schmidt process. Next is the existence of the element dV . This the n ∗ 0 0 unique element of Λ (V ) satisfying dV (e1, . . . , en) = 1 for any positive orthonormal 0 basis (ei). Another feature of an inner product is that it provides a canonical ∗ between V and V by associating v ∈ V with the linear functional Tv given by Tv(w) = hv, wi. Note that there is no such canonical isomorphism between an arbitrary finite dimensional vector space and its dual. We will typically use (ei) to denote the dual ∗ basis of (ei). In particular, we now have a natural inner product on V defined for basis i j vectors in the obvious way by he , e i := hei, eji. An important fact is that this inner product on V ∗ can be extended to an inner product on Λk(V ∗) for any 0 ≤ k ≤ n. Proposition 1. Let 0 ≤ k ≤ n. There exists a unique inner product on Λk(V ∗), also denoted by h·, ·i, such that for any B = (ei) of V ∗, the basis ΛkB is orthonormal with respect to this inner product. In fact, this larger inner product is given explicitly by hv1 ∧...∧vk, w1 ∧...∧wki = det(hvi, wji) for any basis elements v1 ∧ ... ∧ vk, w1 ∧ ... ∧ wk ∈ ΛkB. Next, the definition of the Hodge star operator is based on the following result. Proposition 2. For all 0 ≤ k ≤ n, there exists a unique isomorphism ∗ :Λk(V ∗) → Λn−k(V ∗) such that ω ∧ ∗η = hω, ηidV for all ω, η ∈ Λk(V ∗). We define the Hodge star operator to be the isomorphism ∗ in Proposition 2. To un- derstand it better, in the proof of this proposition it is shown that the Hodge star oper- i1 ik j1 jk ator is defined locally by ∗(e ∧· · ·∧e ) = ±e ∧· · ·∧e , where (i1, . . . , ik, j1, . . . , jn−k) is a permutation of (1, . . . , n). That is, (i1, . . . , ik, j1, . . . , jn−k) = (σ(1), . . . , σ(n)) for some permutation σ. The choice of sign in this definition is given by sgn(σ). A number of interesting elementary properties can be derived immediately from the definition. One that will be useful for us is the composition rule ∗ ∗ ω = (−1)k(n−k)ω.

2 The reader can also easily check that hω, ηi = h∗ω, ∗ηi and that ∗(ω ∧ ∗η) = ∗(η ∧ ∗ω) for all ω, η ∈ Λk(V ∗).

2 Riemannian Manifolds

Our next goal is to transfer these definitions to the manifold level. To do this, we need a brief overview of Riemannian manifolds. A Riemannian on a manifold M is a smooth symmetric covariant 2-tensor field that is positive definite at each point. Such a tensor field gives an inner product at each point p ∈ M, which we normally denote by gp(u, v) or hu, vip. Positive definiteness means that gp(u, u) > 0 for all u 6= 0. The requirement is equivalent to the property that, for any smooth vector fields X,Y ∈ X(M), the function p 7→ hX(p),Y (p)ip is smooth (see [3], Proposition 12.19, p. 317). A Riemannian manifold is a pair (M, g), where M is a manifold and g is a Riemannian metric defined on M. We will state some of the essential facts for a Riemannian manifold (M, g). These are analogous to the properties of inner product spaces discussed in the last section. First is the existence of a smooth orthonormal frame (E1,...,En) in a neighborhood of each point, as well as the corresponding coframe (E1,...,En). From this, we can prove the existence of the Riemannian dVg, which is the unique smooth orientation form satisfying dVg(E1,...,En) = 1 for every local oriented orthonormal frame (Ei). Note that dVg is a notational convenience and does not imply that the Riemannian volume form is an exact differential form. Also, we have a natural isomor- ∗ phism between the and cotangent bundles. This is given by gb : TM → T M ∗ by v ∈ TpM 7→ gb(v) ∈ Tp M, where gb(v)(w) = gp(v, w). This is much like the case of [ inner product spaces, although one new feature is the common notation X for gb(X) ] −1 and ω for gb (ω). For this reason gb and its inverse are sometimes called the “musical .” As expected, for all 0 ≤ k ≤ n the Hodge star operator extends to a mapping ∗ :Ωk(M) → Ωn−k(M). It is not immediately clear that ∗ is smooth, but in fact this follows from the local representation ∗(Ei1 ∧ · · · ∧ Eik ) = ±Ej1 ∧ · · · ∧ Ejk , as this implies that ∗ in local coordinates is either ±1 or 0. An interesting immediate application is in providing a convenient notation for the classical vector field operations of multivariable . The operator can be written as div X = ∗d ∗ X[. In the case that M = 3, the operator is given by [ ] curl X = (∗dX ) . In addition, interior multiplication into dVg can be written as X y [ dVg = ∗X .

3 Laplace-Beltrami Operator

Our goal is to define the Laplace-Beltrami operator ∆ : Ωk(M) → Ωk(M). This is a ∞ n ∞ n generalization of the usual Laplace operator ∆ : C (R ) → C (R ) defined by

n X ∂2f ∆f = −div(grad f) = − . (1) (∂xi)2 i=1

First, we must define the codifferential (or Beltrami operator) δ :Ωk(M) → Ωk−1(M). This is by the formulas δω = (−1)n(k+1)+1 ∗ d ∗ ω for 1 ≤ k ≤ n and

3 δω = 0 for k = 0. In terms of the codifferential, the Laplace-Beltrami operator is defined as ∆ : Ωk(M) → Ωk(M) by ∆ω = dδω + δdω. Naturally, a form ω is said to be harmonic if ∆ω = 0. We use the notation Hk(M) to denote the set of harmonic n M k-forms on M, as well as the notation H∗(M) := Hk(M). A fairly straightforward k=0 unraveling of the definitions shows that this is consistent with Equation (1) in the case n 0 ∞ that M = R and f ∈ Ω (M) = C (M). 2 k k Our last major definition is the L product h·, ·iL2 :Ω (M) × Ω (M) → R, Z where 0 ≤ k ≤ n. This is given by hα, βiL2 := α ∧ ∗β, noting that α ∧ ∗β is an M ∗ j n-form. We extend h·, ·iL2 to an inner product on Ω (M) by declaring Ω (M) and Ωk(M) to be orthogonal for all j 6= k. We want to consider now how the various operators we have defined relate to each other. One important fact is that the Laplace and Hodge star operators commute: ∗∆ = ∆∗. Even more important is the following lemma.

Lemma 3. The codifferential δ is the adjoint of the exterior with respect ∗ to h·, ·, iL2 . That is, for all α, β ∈ Ω (M), hdα, βiL2 = hα, δβiL2 . In proving this lemma, we need only consider the case that α ∈ Ωk−1(M) and β ∈ Ωk(M), since forms of different degree are orthogonal. Applying Stoke’s Theorem gives Z Z 0 = α∧∗β = d(α∧∗β). A computation shows that d(α∧∗β) = dα∧∗β−α∧∗δβ, ∂M Z M Z implying that dα ∧ ∗β = α ∧ ∗δβ. The lemma follows. M M This has some important corollaries. First is that the Laplace-Beltrami operator k is self-adjoint: h∆α, βiL2 = hα, ∆βiL2 . Next is the result that, for any α ∈ Ω (M), ∆α = 0 if and only if dα = 0 and δα = 0. This fact places significant restrictions on which forms can be harmonic. For example, we can now immediately see that every harmonic function f ∈ C∞(M) defined on a connected compact manifold is constant.

4 The Hodge Decomposition Theorem

We are finally in a position to state the main result.

Theorem 4 (Hodge Decomposition Theorem). For all 0 ≤ k ≤ n, Hk(M) is finite di- mensional, and we have the following orthogonal direct sum decompositions of Ωk(M):

Ωk(M) = ∆(Ωk(M)) ⊕ Hk(M) (2) = dδ(Ωk(M)) ⊕ δd(Ωk(M)) ⊕ Hk(M) (3) = d(Ωk−1(M)) ⊕ δ(Ωk+1(M)) ⊕ Hk(M) (4)

The proof as presented by Warner is dependent on two technical lemmas outside the scope of this paper. Here we will simply state them. Given α ∈ Ωk(M), consider the equation ∆ω = α. If ω is a solution, then we have h∆ω, ϕi = hα, ϕi for all ϕ ∈ Ωk(M). Since ∆ is self-adjoint, it follows that hω, ∆ϕi = hα, ϕi. Hence we can consider ω as k determining a continuous linear functional l :Ω (M) → R given by l(ϕ) = hα, ϕi and satisfying l(∆ϕ) = hα, ϕi for all ϕ ∈ Ωk(M). Any linear functional l satisfying this

4 condition is said to be a weak solution of ∆ω = α. The first lemma says that every weak solution l determines a ordinary solution of ∆ω = α. Lemma 5. Let l be a weak solution of the equation ∆ω = α, where α ∈ Ωk(M). There exists ω ∈ Ωk(M) such that l(ϕ) = hω, ϕi for all ϕ ∈ Ωk(M). Thus ∆ω = α. The next lemma places a restriction on the size of Ωk(M).

k Lemma 6. Suppose {ωn} ⊂ Ω (M) satisfy kωnk ≤ C and k∆ωnk ≤ C for all n ∈ N k and some C > 0. Then {ωn} has a subsequence which is Cauchy in Ω (M). Armed with these two lemmas, we sketch a proof of Theorem 4. First off, the second lemma immediately implies that Hk(M) is finite dimensional, since otherwise there would be an infinite orthonormal sequence. Next, for an arbitrary ω ∈ Ωk(M), we can l X k ⊥ l express it as ω = η+ hω, ωjiωj, where η ∈ (H (M)) and {ωj}j=1 is an orthonormal j=1 basis for Hk(M). The task then becomes to show that (Hk(M))⊥ = ∆(Ωk(M)). The inclusion ∆(Ωk(M)) ⊂ (Hk(M))⊥ is immediate since h∆ω, αi = hω, ∆αi for all ω ∈ Ωk(M) and α ∈ Hk(M). For the reverse inclusion, we consider an arbitrary k ⊥ k α ∈ (H ) and define a linear functional l : ∆(Ω (M)) → R by l(∆ϕ) = hα, ϕi for all ϕ ∈ Ωk(M). One shows that l is a bounded functional and then uses the Hahn-Banach Theorem to extend the definition of l to the whole space Ωk(M). This l is then a weak solution of ∆ω = α. By the first lemma, there exists an ordinary solution ω ∈ Ωk(M), showing that (Hk(M))⊥ ⊂ ∆(Ωk(M)). This establishes the theorem.

5 Applications to Differential Geometry

The major immediate application of the Hodge decomposition theorem is in studying the de Rham cohomology of a compact oriented Riemannian manifold. To review the basic concepts, recall that a k-form ω is closed if dω = 0 and exact if there exists a (k−1)-form η such that dη = ω. The set of exact forms is a of the space of closed forms. This gives an equivalence relation ∼ on the set of closed k-forms, where ω1 ∼ ω2 if ω1 −ω2 is exact. We define the k-th de Rham cohomology group, denoted by k k k k k HdR(M), to be the quotient vector space HdR(M) := Z (M)/B (M), where Z (M) denotes the set of closed k-forms and Bk(M) denotes the set of exact k-forms. The cohomology class of ω, denoted by [ω], is the equivalence class of ω under ∼. The main theorem is the following. Theorem 7. Every de Rham cohomology class on a compact oriented Riemannian manifold M has a unique harmonic representative.

k k In particular, Theorem 7 proves that HdR(M) is isomorphic to H (M). We have already seen that Hk(M) is finite dimensional. Since any smooth manifold can be endowed with a Riemannian metric, the k-th de Rham cohomology group of a com- pact oriented smooth manifold is finite dimensional as well. The proof of Theorem 7 is relatively short and involves defining Green’s operator G :Ωk(M) → (Hk)⊥ as the unique form G(ω) satisfying ∆G(ω) = ω − h(ω). Here h denotes the orthogo- nal projection Ωk(M) = ∆(Ωk) ⊕ Hk(M) → Hk(M). One then proves that G com- mutes with the operators d, δ, and ∆. An arbitrary α ∈ Ωk(M) can be written as

5 α = dδGα + δdGα + h(α) = dδGα + δGdα + h(α). If α is closed, this simplifies to α = dδGα + h(α). This implies that α − h(α) is exact, so that α and h(α) are in the same cohomology class. Uniqueness is straightforward to prove, and we arrive at the conclusion. As another interesting consequence, Theorem 7 yields a quick proof of Poincar´e in the case of a compact manifold. To introduce this, observe that the mapping Z k n−k ([ω], [η]) 7→ ω ∧ η gives a well-defined bilinear mapping HdR(M) × HdR (M) → R. M The result is the following.

Theorem 8 (Poincar´eDuality). The previous bilinear map is nondegenerate and there- n−k k ∗ fore determines an isomorphism of HdR (M) with the (HdR(M)) . k The proof is quite simple. For a non-zero cohomology class in HdR(M) with har- monic representative ϕ, ∗ϕ is also harmonic by the identity ∆∗ = ∗∆. Thus ∗ϕ is a Z n−k representative of a cohomology class in HdR (M). Since ([ϕ], [∗ϕ]) = ϕ ∧ ∗ϕ = M kϕk2 6= 0, this is a nondegenerate pairing, and we obtain the desired isomorphism. n ∼ An interesting special case of Poincar´eduality is the result that HdR(M) = R for any compact, connected, orientable smooth manifold. This provides another example of the usefulness of Hodge theory.

References

[1] Hodge, William V. D., The Theory and Applications of Harmonic , Cambridge University Press, Cambridge, 1941. [2] Jost, J¨urgen, Riemannian Geometry and , Universitext, Springer-Verlag, Berlin, 2011. [3] Lee, John M., Introduction to Smooth Manifolds, 2nd ed., Graduate Texts in Mathematics 218, Springer, New York, 2013. [4] Nowaczyk, Nikolai, The Hodge Decomposition, http://math.nikno.de/, 2010. [5] Warner, Frank W., Foundations of Differentiable Manifolds and Lie Groups, Scott, Foresman and Company, Glenview, 1971.

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