Curvature groups of a hypersurface
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- by Samuel I. Goldberg PDF
- Proc. Amer. Math. Soc. 54 (1976), 271-275 Request permission
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
- Samuel I. Goldberg and Nicholas C. Petridis, The curvature groups of a pseudo-Riemannian manifold, J. Differential Geometry 9 (1974), 547–555. MR 350670
- Samuel I. Goldberg, Curvature and homology, Pure and Applied Mathematics, Vol. XI, Academic Press, New York-London, 1962. MR 0139098
- Izu Vaisman, The curvature groups of a space form, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 331–341. MR 231400
- I. Vaisman, The curvature groups of a hypersurface in the Euclidean space, Acta Math. Acad. Sci. Hungar. 23 (1972), 21–31. MR 313969, DOI 10.1007/BF01889900
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 54 (1976), 271-275
- DOI: https://doi.org/10.1090/S0002-9939-1976-0397624-3
- MathSciNet review: 0397624