Riemannian metrics induced by two immersions
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- by M. do Carmo and M. Dajczer PDF
- Proc. Amer. Math. Soc. 86 (1982), 115-119 Request permission
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.References
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M. do Carmo and M. Dajczer, General conformally flat hypersurfaces (preprint).
- John Douglas Moore, Submanifolds of constant positive curvature. I, Duke Math. J. 44 (1977), no. 2, 449–484. MR 438256
- Patrick J. Ryan, Homogeneity and some curvature conditions for hypersurfaces, Tohoku Math. J. (2) 21 (1969), 363–388. MR 253243, DOI 10.2748/tmj/1178242949
Additional Information
- © Copyright 1982 American Mathematical Society
- 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