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Proceedings of the American Mathematical Society

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A characterization of totally geodesic hypersurfaces of $ S\sp{n+1}$ and $ {\bf C}P\sp{n+1}$


Author: Kinetsu Abe
Journal: Proc. Amer. Math. Soc. 81 (1981), 603-606
MSC: Primary 53C42
DOI: https://doi.org/10.1090/S0002-9939-1981-0601739-3
MathSciNet review: 601739
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Abstract: Let $ M$ be a complete hypersurface in $ {S^{n + 1}}$ (or $ {\mathbf{C}}{P^{n + 1}}$). Assume that through each point $ x$ of $ M$ a (local) $ \mu (x)$-dimensional totally geodesic submanifold $ {S_x}$ of $ {S^{n + 1}}$ (or $ {\mathbf{C}}{P^{n + 1}}$) exists in $ M$. A sufficient condition for $ M$ itself to be totally geodesic is given in terms of $ \mu (x)$.


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DOI: https://doi.org/10.1090/S0002-9939-1981-0601739-3
Article copyright: © Copyright 1981 American Mathematical Society

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