Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53C42, 53C65

Retrieve articles in all journals with MSC: 53C42, 53C65

Additional Information

Keywords: Minkowski integral formulas, totally umbilic submanifold, Codazzi tensor, space form
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society