Cohomology of harmonic forms on Riemannian manifolds with boundary

Sylvain Cappell, Dennis DeTurck, Herman Gluck, Edward Y. Miller

Research output: Contribution to journalArticle

Abstract

On a smooth compact manifold M, the cohomology of the complex of differential forms is isomorphic to the ordinary cohomology by the classical theorem of de Rham. When M has a Riemannian metric g, the harmonic forms constitute a subcomplex of the de Rham complex because the Laplacian commutes with exterior differentiation. When (M, g) has no boundary, all of its harmonic forms are closed, and hence the cohomology of this subcomplex is isomorphic to the ordinary cohomology by the classical theorem of Hodge. But when the boundary of (M, g) is non-empty, it is possible for a p-form to be harmonic without being closed, and some of these, which are exact, although not the exterior derivatives of harmonic p - 1-forms, represent an "echo" of the ordinary p - 1-dimensional cohomology within the p-dimensional harmonic cohomology.

Original languageEnglish (US)
Pages (from-to)923-931
Number of pages9
JournalForum Mathematicum
Volume18
Issue number6
DOIs
StatePublished - Nov 20 2006

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Harmonic Forms
Manifolds with Boundary
Riemannian Manifold
Cohomology
Derivatives
Isomorphic
Harmonic
P-harmonic
Closed
Smooth Manifold
Differential Forms
Riemannian Metric
Commute
Theorem
Compact Manifold
Derivative

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Cohomology of harmonic forms on Riemannian manifolds with boundary. / Cappell, Sylvain; DeTurck, Dennis; Gluck, Herman; Miller, Edward Y.

In: Forum Mathematicum, Vol. 18, No. 6, 20.11.2006, p. 923-931.

Research output: Contribution to journalArticle

Cappell, Sylvain ; DeTurck, Dennis ; Gluck, Herman ; Miller, Edward Y. / Cohomology of harmonic forms on Riemannian manifolds with boundary. In: Forum Mathematicum. 2006 ; Vol. 18, No. 6. pp. 923-931.
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