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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
DOI: https://doi.org/10.1090/S0002-9939-1976-0397624-3
MathSciNet review: 0397624
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Abstract | References | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] S. I. Goldberg and N. C. Petridis, The curvature groups of a pseudo-Riemannian manifold, J. Differential Geometry 9 (1974), 547-555. MR 0350670 (50:3162)
  • [2] S. I. Goldberg, Curvature and homology, Pure and Appl. Math., vol. 11, Academic Press, New York and London, 1962. MR 25 #2537. MR 0139098 (25:2537)
  • [3] I. Vaisman, The curvature groups of a space form, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 331-341. MR 37 #6955. MR 0231400 (37:6955)
  • [4] -, The curvature groups of a hypersurface in the Euclidean space, Acta Math. Acad. Sci. Hungar. 23 (1972), 21-31. MR 47 #2521. MR 0313969 (47:2521)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0397624-3
Keywords: Jet forms, curvature groups, closed vector fields, sheaf cohomology
Article copyright: © Copyright 1976 American Mathematical Society

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