Cohomology of Harmonic Forms on Riemannian Manifolds With Boundary
| dc.creator | Cappell, Sylvain | |
| dc.creator | DeTurck, Dennis | |
| dc.creator | Gluck, Herman | |
| dc.creator | Miller, Edward Y. | |
| dc.date | 2005-08-19 | |
| dc.date.accessioned | 2026-07-07T05:22:31Z | |
| dc.date.available | 2026-07-07T05:22:31Z | |
| dc.description | 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. | |
| dc.identifier | https://arxiv.org/abs/math/0508372 | |
| dc.identifier | http://arxiv.org/abs/math/0508372 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/76088 | |
| dc.subject | Differential Geometry | |
| dc.subject | Algebraic Topology | |
| dc.subject | 58A12; 58A14; 14F40; 53C20; 53C21 | |
| dc.title | Cohomology of Harmonic Forms on Riemannian Manifolds With Boundary | |
| dc.type | text |