Hodge Theory

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Hodge Theory

• The Hodge theory of a smooth, oriented, compact
• Riemannian manifold

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by William M. Faucette

• Adapted from lectures by
• Mark Andrea A. Cataldo

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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

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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).

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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.

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The adjoint of d d?

• Note that in the example MR with the standard
  orientation and the Euclidean metric shows that ?
  and d do not commute. In particular, ?does not
  preserve closed forms.

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The adjoint of d d?

• Define an inner product on the space of compactly
  supported p-forms on M by setting

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The adjoint of d d?

• Definition Let TEp(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)

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The adjoint of d d?

• Definition Define dFEp(M)? Ep-1(M) by
• This operator, so defined, is the formal adjoint
  of exterior differentiation on the algebra of
  differential forms.

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The adjoint of d d?

• Definition The Laplace-Beltrami operator, or
  Laplacian, is defined as ?Ep(M)? Ep(M) by

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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.

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The adjoint of d d?

• Remark Note that F? ?F. In particular, a form
  u is harmonic if and only if Fu is harmonic.

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Harmonic forms and the Hodge Isomorphism Theorem
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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

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Harmonic forms and the Hodge Isomorphism Theorem

• Lemma A p-form u2Ep(M) is harmonic if and only
  if du0 and dFu0.
• This follows immediately from the fact that

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Harmonic forms and the Hodge Isomorphism Theorem

• Theorem (The Hodge Orthogonal Decomposition
  Theorem) Let (M, g) be a compact oriented
  Riemannian manifold.
• Then
• and . . .

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Harmonic forms and the Hodge Isomorphism Theorem

• we have a direct sum decomposition into
• ?? , ??-orthogonal subspaces

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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)lt?.

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Harmonic forms and the Hodge Isomorphism Theorem

• Theorem (Poincaré Duality) Let M be a compact
  oriented smooth manifold. The pairing
• is non-degenerate.

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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.