### 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 language | English (US) |
---|---|

Pages (from-to) | 923-931 |

Number of pages | 9 |

Journal | Forum Mathematicum |

Volume | 18 |

Issue number | 6 |

DOIs | |

State | Published - Nov 20 2006 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Forum Mathematicum*,

*18*(6), 923-931. https://doi.org/10.1515/FORUM.2006.046

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

Research output: Contribution to journal › Article

*Forum Mathematicum*, vol. 18, no. 6, pp. 923-931. https://doi.org/10.1515/FORUM.2006.046

}

TY - JOUR

T1 - Cohomology of harmonic forms on Riemannian manifolds with boundary

AU - Cappell, Sylvain

AU - DeTurck, Dennis

AU - Gluck, Herman

AU - Miller, Edward Y.

PY - 2006/11/20

Y1 - 2006/11/20

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33847412336&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33847412336&partnerID=8YFLogxK

U2 - 10.1515/FORUM.2006.046

DO - 10.1515/FORUM.2006.046

M3 - Article

AN - SCOPUS:33847412336

VL - 18

SP - 923

EP - 931

JO - Forum Mathematicum

JF - Forum Mathematicum

SN - 0933-7741

IS - 6

ER -