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

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.