On an integral formula for closed hypersurfaces of the sphere

Author:
Chorng-shi Houh

Journal:
Proc. Amer. Math. Soc. **35** (1972), 234-237

MSC:
Primary 53C45

DOI:
https://doi.org/10.1090/S0002-9939-1972-0296867-3

MathSciNet review:
0296867

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Abstract | References | Similar Articles | Additional Information

Abstract: In a compact oriented hypersurface of the sphere the integral formula is proved where is the *r*th mean curvature, *e* is the unit normal of in . Some applications are considered.

**[1]**B. Y. Chen,*Some integral formulas for hypersurfaces in Euclidean space*, Nagoya Math. J.**43**(1971), 117-125. MR**0293549 (45:2626)****[2]**S. S. Chern,*Minimal submanifolds in a Riemannian manifold*, University of Kansas, Department of Mathematics Technical Report, 19, Univ. of Kansas, Lawrence, Kan., 1968. MR**40**#1899. MR**0248648 (40:1899)****[3]**C. C. Hsiung,*Some integral formulas for closed hypersurfaces*, Math. Scand.**2**(1954), 286-294. MR**16**, 849. MR**0068236 (16:849j)****[4]**C. C. Hsiung and J. K. Shahin,*Affine differential geometry of closed hypersurfaces*, Proc. London Math. Soc. (3)**17**(1967), 715-735. MR**36**#2069. MR**0218986 (36:2069)****[5]**R. C. Reilly,*Extrinsic rigidity theorems for compact submanifolds of the sphere*, J. Differential Geometry**4**(1970), 487-497. MR**0290296 (44:7480)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0296867-3

Keywords:
Combined operation of the vector product,
combined operation of the exterior product,
principal curvatures,
*r*th mean curvature,
Stokes theorem

Article copyright:
© Copyright 1972
American Mathematical Society