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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Curvature groups of a hypersurface

Author: Samuel I. Goldberg
Journal: Proc. Amer. Math. Soc. 54 (1976), 271-275
MathSciNet review: 0397624
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Abstract: A cochain complex associated with the vector $ 1$-form determined by the first and second fundamental tensors of a hypersurface $ M$ in $ {E^{n + 1}}$ is introduced. Its cohomology groups $ {H^p}(M)$, called curvature groups, are isomorphic with the cohomology groups of $ M$ with coefficients in a subsheaf $ {\mathcal{S}_R}$ of the sheaf $ \mathcal{S}$ of closed vector fields on $ M$. If $ M$ is a minimal variety, the same conclusion is valid with $ {\mathcal{S}_R}$ replaced by a sheaf of harmonic vector fields. If the Ricci tensor is nondegenerate the $ {H^p}(M)$ vanish. If $ {\mathcal{S}_R} \ne \emptyset $, and there are no parallel vector fields, locally, the $ {H^p}(M)$ are isomorphic with the corresponding de Rham groups.

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Keywords: Jet forms, curvature groups, closed vector fields, sheaf cohomology
Article copyright: © Copyright 1976 American Mathematical Society