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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Submanifolds in a euclidean hypersphere


Author: Bang-yen Chen
Journal: Proc. Amer. Math. Soc. 27 (1971), 627-628
MSC: Primary 53.74
MathSciNet review: 0270309
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Abstract: Let $ M$ be an oriented closed $ n$-dimensional submanifold of a euclidean ($ (n + N)$-space $ {E^{n + N}}$. Let $ X$ and $ H$ be the position vector field and the mean curvature vector field of $ M$ in $ {E^{n + N}}$. Then $ M$ is contained in a hypersphere of $ {E^{n + N}}$ centered at $ c$ when and only when either $ (X - c) \cdot H \geqq - 1$ or $ (X - c)\cdot H \leqq - 1$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0270309-5
PII: S 0002-9939(1971)0270309-5
Keywords: Position vector, mean curvature normal, mean curvature, Laplacian, support function, submanifold in a hypersphere
Article copyright: © Copyright 1971 American Mathematical Society