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Proceedings of the American Mathematical Society

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

 

 

Riemannian metrics induced by two immersions


Authors: M. do Carmo and M. Dajczer
Journal: Proc. Amer. Math. Soc. 86 (1982), 115-119
MSC: Primary 53C42
DOI: https://doi.org/10.1090/S0002-9939-1982-0663878-1
MathSciNet review: 663878
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Abstract: We consider the situation where a Riemannian manifold $ {M^n}$ can be isometrically immersed into spaces $ {N^{n + 1}}(c)$ and $ {N^{n + q}}(\tilde c)$ with constant curvatures $ c < \tilde c$, $ q \leqslant n - 3$, and show that this implies the existence, at each point $ p \in M$, of an umbilic subspace $ {U_p} \subset {T_p}M$, for both immersions, with $ {U_p} \geqslant n - q$. In particular, if $ {M^n}$ can be isometrically immersed as a hypersurface into two spaces of distinct constant curvatures, $ {M^n}$ is conformally flat.


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DOI: https://doi.org/10.1090/S0002-9939-1982-0663878-1
Keywords: Isometric immersions, umbilic subspaces, conformally flat Riemannian manifolds
Article copyright: © Copyright 1982 American Mathematical Society