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Proceedings of the American Mathematical Society

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



The Gauss map in spaces of constant curvature

Author: Joel L. Weiner
Journal: Proc. Amer. Math. Soc. 38 (1973), 157-161
MSC: Primary 53C40
MathSciNet review: 0310813
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Abstract: Let $ N$ be a complete simply connected Riemannian manifold of constant sectional curvature $ \ne 0$. Let $ M$ be an immersed Riemannian hypersurface of $ N$. The Gauss map on $ M$ based at a point $ p$ in $ N$ is defined. Suppose a Gauss map on $ M$ has constant rank less than the dimension of $ M$; then $ M$ is generated by Riemannian submanifolds with constant sectional curvature. The sectional curvature of each of these generating submanifolds of $ M$ has the same sign as the sectional curvature of $ N$.

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Keywords: Parallel displacement, Gauss map, submanifold of a sphere, submanifold of hyperbolic space, conformally equivalent, constant rank, foliated, constant sectional curvature
Article copyright: © Copyright 1973 American Mathematical Society