Cohomology of Harmonic Forms on Riemannian Manifolds With Boundary

TL;DR

This paper characterizes the cohomology of harmonic forms on compact Riemannian manifolds with boundary, revealing a direct sum structure involving the manifold's ordinary cohomology and boundary effects.

Contribution

It provides a new explicit description of harmonic form cohomology on manifolds with boundary, extending classical results for closed manifolds.

Findings

Harmonic cohomology decomposes into ordinary cohomology and boundary-related terms.

Harmonic forms on manifolds with boundary can be exact yet not co-closed.

The structure differs significantly from the classical Hodge theorem for closed manifolds.

Abstract

Theorem. Let M be a compact, connected, oriented smooth Riemannian n-manifold with non-empty boundary. Then the cohomology of the complex (Harm*(M),d) of harmonic forms on M is given by the direct sum H^p(Harm*(M),d) = H^p(M;R) + H^(p-1)(M;R) for p=0,1,...,n. When M is a closed manifold, a form is harmonic if and only if it is both closed and co-closed. In this case, all the maps in the complex (Harm*(M),d) are zero, and so H^p(Harm*(M),d) = Harm^p(M) = H^p(M;R) according to the classical theorem of Hodge. By contrast, when M is connected and has non-empty boundary, it is possible for a p-form to be harmonic without being both closed and co-closed. Some of these, which are exact, although not exterior derivatives of harmonic p-1-forms, represent the "echo" of the ordinary p-1-dimensional cohomology within the p-dimensional harmonic cohomology that appears in the above theorem.