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

 

Locally convex hypersurfaces of negatively curved spaces


Author: S. Alexander
Journal: Proc. Amer. Math. Soc. 64 (1977), 321-325
MSC: Primary 53C40
MathSciNet review: 0448262
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Abstract: A well-known theorem due to Hadamard states that if the second fundamental form of a compact immersed hypersurface M of Euclidean space $ {E^n}(n \geqslant 3)$ is positive definite, then M is imbedded as the boundary of a convex body. There have been important generalizations of this theorem concerning hypersurfaces of $ {E^n},{H^n}$, and $ {S^n}$, but there seem to be no versions for hypersurfaces of spaces of variable curvature, and no proofs which generalize to these spaces. Our main result is that Hadamard's theorem holds in any complete, simply connected Riemannian manifold of nonpositive sectional curvature.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0448262-6
PII: S 0002-9939(1977)0448262-6
Article copyright: © Copyright 1977 American Mathematical Society