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Proceedings of the American Mathematical Society

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

 

 

Integral formulas and hyperspheres in a simply connected space form


Author: Irl Bivens
Journal: Proc. Amer. Math. Soc. 88 (1983), 113-118
MSC: Primary 53C42; Secondary 53C65
MathSciNet review: 691289
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Abstract: Let $ {M^n}$ denote a connected compact hypersurface without boundary contained in Euclidean or hyperbolic $ n + 1$ space or in an open hemisphere of $ {S^{n + 1}}$. We show that if two consecutive mean curvatures of $ M$ are constant then $ M$ is in fact a geodesic sphere. The proof uses the generalized Minkowski integral formulas for a hypersurface of a complete simply connected space form. These Minkowski formulas are derived from an integral formula for submanifolds in which the ambient Riemannian manifold $ \overline M $ possesses a generalized position vector field; that is a vector field $ Y$ whose covariant derivative is at each point a multiple of the identity. In addition we prove that if $ \overline M $ is complete and connected with the covariant derivative of $ Y$ exactly the identity at each point then $ \overline M $ is isometric to Euclidean space.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0691289-2
Keywords: Minkowski integral formulas, totally umbilic submanifold, Codazzi tensor, space form
Article copyright: © Copyright 1983 American Mathematical Society