1. The Hodge theory of a smooth, oriented, compact Riemannian manifold

2. Adapted from lectures by Mark Andrea A. Cataldo
by William M. Faucette
Adapted from lectures by
Mark Andrea A. Cataldo

3. 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 F

4. 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*).

5. 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.

6. 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.

7. The adjoint of d: d
Define an inner product on the space of compactly supported p-forms on M by setting

8. 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)

9. 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.

10. The adjoint of d: d
Definition: The Laplace-Beltrami operator, or Laplacian, is defined as :Ep(M) Ep(M) by

11. 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.

12. The adjoint of d: d
Remark: Note that F= F. In particular, a form u is harmonic if and only if Fu is harmonic.

13. Harmonic forms and the Hodge Isomorphism Theorem

14. 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

15. 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

16. Harmonic forms and the Hodge Isomorphism Theorem
Theorem: (The Hodge Orthogonal Decomposition Theorem) Let (M, g) be a compact oriented Riemannian manifold.
Then
and . . .

17. Harmonic forms and the Hodge Isomorphism Theorem
we have a direct sum decomposition into
 , -orthogonal subspaces

18. 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)<.

19. Harmonic forms and the Hodge Isomorphism Theorem
Theorem: (Poincaré Duality) Let M be a compact oriented smooth manifold. The pairing
is non-degenerate.

20. 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.