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

DOI:
https://doi.org/10.1090/S0002-9939-1983-0691289-2

MathSciNet review:
691289

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Abstract: Let denote a connected compact hypersurface without boundary contained in Euclidean or hyperbolic space or in an open hemisphere of . We show that if two consecutive mean curvatures of are constant then 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 possesses a generalized position vector field; that is a vector field whose covariant derivative is at each point a multiple of the identity. In addition we prove that if is complete and connected with the covariant derivative of exactly the identity at each point then is isometric to Euclidean space.

**[1]**I. Bivens,*Codazzi tensors and reducible submanifolds*, Trans. Amer. Math. Soc.**268**(1981), 231-245. MR**628456 (82k:53072)****[2]**R. Gardner,*The Dirichlet integral in differential geometry*, Global Analysis, S. S. Chern and S. Smale, editors, Proc. Sympos. Pure Math, vol. 15, Amer. Math. Soc., Providence, R. I., 1970, pp. 231-237. MR**0262986 (41:7591)****[3]**G. H. Hardy, J. E. Littlewood and G. Pólya,*Inequalities*, Cambridge Univ. Press, Cambridge, 1934.**[4]**C. C. Hsiung,*Some integral formulas for closed hypersurfaces*, Math. Scand.**2**( 1954), 286-294. MR**0068236 (16:849j)****[5]**Y. Tashiro,*Complete Riemannian manifolds and some vector fields*, Trans. Amer. Math. Soc.**117**(1965), 251-275. MR**0174022 (30:4229)**

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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