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Harmonic Differential Forms and the Hodge Decomposition Theorem

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Harmonic Differential Forms and the Hodge Decomposition Theorem Harmonic Differential Forms and the Hodge Decomposition Theorem Matthew Romney Abstract This is an expository paper for Rui Fernandes's Spring 2014 Differentiable Manifolds 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 manifold. This is one of the central ideas of Hodge theory, developed in the 1930's by the Scottish mathematician William V.D. Hodge and laid out in his 1941 book [1]. It had origins in algebraic geometry, 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 cohomology class for a com- pact Riemannian manifold has a unique harmonic representative, where \harmonic" refers to the appropriate generalization of the usual Laplace operator. 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 mathematicians 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 current 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 section we will define the Hodge star operator on an oriented inner product space. 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 de Rham cohomology. 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 vector space and k be a positive integer. A covariant k-tensor on V is a multilinear function 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-tensors 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) p2M k-form is a smooth section of ΛkT ∗M. The space of all k-forms is denoted by Ωk(M). 1 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 dimension 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 basis (ei), which follows from the Gram-Schmidt process. Next is the existence of the volume 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 isomorphism ∗ between V and V by associating v 2 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 orthonormal basis 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 2 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 ! ^ ∗η = h!; ηidV for all !; η 2 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 ∗ ∗ ! = (−1)k(n−k)!. 2 The reader can also easily check that h!; ηi = h∗!; ∗ηi and that ∗(! ^ ∗η) = ∗(η ^ ∗!) for all !; η 2 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 metric 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 2 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 smoothness requirement is equivalent to the property that, for any smooth vector fields X; Y 2 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 volume form 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 tangent and cotangent bundles. This is given by gb : TM ! T M ∗ by v 2 TpM 7! gb(v) 2 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 ! for gb (!). For this reason gb and its inverse are sometimes called the \musical isomorphisms." 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 calculus. The divergence operator can be written as div X = ∗d ∗ X[. In the case that M = 3, the curl 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 1 n 1 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 δ! = (−1)n(k+1)+1 ∗ d ∗ ! for 1 ≤ k ≤ n and 3 δ! = 0 for k = 0.
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