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## Hodge Theory

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**Hodge Theory**The Hodge theory of a smooth, oriented, compact Riemannian manifold**by William M. Faucette**Adapted from lectures by Mark Andrea A. Cataldo**Structure of Lecture**• The Inner Product on compactly supported forms • The adjoint dF • The Laplacian • Harmonic Forms • Hodge Orthogonal Decomposition Theorem • Hodge Isomorphism Decomposition Theorem • Poincaré Duality**The adjoint of d: d**Let (M,g) be an oriented Riemannian manifold of dimension m. Then the Riemannian metric g on M defines a smoothly varying inner product on the exterior algebra bundle (TM*).**The adjoint of d: d**The orientation on M gives rise to the F operator on the differential forms on M: In fact, the star operator is defined point-wise, using the metric and the orientation, on the exterior algebras (TM,q*) and it extends to differential forms.**The adjoint of d: d**Note that in the example M=R with the standard orientation and the Euclidean metric shows that and d do not commute. In particular, does not preserve closed forms.**The adjoint of d: d**Define an inner product on the space of compactly supported p-forms on M by setting**The adjoint of d: d**Definition: Let T:Ep(M)Ep(M) be a linear map. We say that a linear map is the formal adjoint to T with respect to the metric if, for every compactly supported u2Ep(M) and v2Ep(M)**The adjoint of d: d**Definition: Define dF:Ep(M) Ep-1(M) by This operator, so defined, is the formal adjoint of exterior differentiation on the algebra of differential forms.**The adjoint of d: d**Definition: The Laplace-Beltrami operator, or Laplacian, is defined as :Ep(M) Ep(M) by**The adjoint of d: d**While F is defined point-wise using the metric, dF and are defined locally (using d) and depend on the metric.**The adjoint of d: d**Remark: Note that F= F. In particular, a form u is harmonic if and only if Fu is harmonic.**Harmonic forms and the Hodge Isomorphism Theorem**Let (M,g) be a compact oriented Riemannian manifold. Definition: Define the space of real harmonic p-forms as**Harmonic forms and the Hodge Isomorphism Theorem**Lemma: A p-form u2Ep(M) is harmonic if and only if du=0 and dFu=0. This follows immediately from the fact that**Harmonic forms and the Hodge Isomorphism Theorem**Theorem: (The Hodge Orthogonal Decomposition Theorem) Let (M, g) be a compact oriented Riemannian manifold. Then and . . .**Harmonic forms and the Hodge Isomorphism Theorem**we have a direct sum decomposition into , -orthogonal subspaces**Harmonic forms and the Hodge Isomorphism Theorem**Corollary: (The Hodge Isomorphism Theorem) Let (M, g) be a compact oriented Riemannian manifold. There is an isomorphism depending only on the metric g: In particular, dimRHp(M, R)<.**Harmonic forms and the Hodge Isomorphism Theorem**Theorem: (Poincaré Duality) Let M be a compact oriented smooth manifold. The pairing is non-degenerate.**Harmonic forms and the Hodge Isomorphism Theorem**In fact, the F operator induces isomorphisms for any compact, smooth, oriented Riemannian manifold M. The result follows by the Hodge Isomorphism Theorem.