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Cohomology of harmonic forms on Riemannian manifolds with boundary by Sylvain Cappell; Dennis DeTurck; Herman Gluck; Edward Y Miller is a Mathematics article available to read on EtoBox.

What is Cohomology of harmonic forms on Riemannian manifolds with boundary about?

On a smooth compact manifold M, the cohomology of the complex of di¤erential 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 di¤erentiation. When ðM; gÞ 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 ðM; gÞ 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 À 1-forms, represent an ''echo'' of the ordinary p À 1-dimensional cohomology within the p-dimensional harmonic cohomology.

Who reads Cohomology of harmonic forms on Riemannian manifolds with boundary?

Author

Sylvain Cappell; Dennis DeTurck; Herman Gluck; Edward Y Miller

Publisher

Walter de Gruyter GmbH

Published

2006

Language

EN

Field

Mathematics (Physical Sciences)