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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Locally convex hypersurfaces of negatively curved spaces
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by S. Alexander PDF
Proc. Amer. Math. Soc. 64 (1977), 321-325 Request permission

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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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 64 (1977), 321-325
  • MSC: Primary 53C40
  • DOI: https://doi.org/10.1090/S0002-9939-1977-0448262-6
  • MathSciNet review: 0448262