Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-06T07:16:01.620Z Has data issue: false hasContentIssue false
  • English
  • Français

Integral Formulas for Submanifolds and their Applications

Published online by Cambridge University Press:  20 November 2018

Kentaro Yano
Kentaro Yano*
Affiliation:
University of Illinois, Urbana, Illinois
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Liebmann [12] proved that the only ovaloids with constant mean curvature in a 3-dimensional Euclidean space are spheres. This result has been generalized to the case of convex closed hypersurfaces in an m-dimensional Euclidean space by Alexandrov [1], Bonnesen and Fenchel [3], Hopf [4], Hsiung [5], and Süss [14].

The result has been further generalized to the case of closed hypersurfaces in an m-dimensional Riemannian manifold by Alexandrov [2], Hsiung [6], Katsurada [7; 8; 9], Ōtsuki [13], and by myself [15; 16].

The attempt to generalize the result to the case of closed submanifolds in an m-dimensional Riemannian manifold has been recently done by Katsurada [10; 11], Kôjyô [10], and Nagai [11].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 376 - 388
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Alexandrov, A. D., Uniqueness theorems for surfaces in the large. V, Vestnik Leningrad. Univ. 18 (1958), no. 19, 5–8.Google Scholar
2. Alexandrov, A. D., A characteristic property of spheres, Ann. Mat. Pura Appl. (4) 58 (1962), 303–315.Google Scholar
3. Bonnesen, T. and Fenchel, W., Théorie der konvexen Körper (Springer, Berlin, 1934).Google Scholar
4. Hopf, H., Uber Fldchen mit einer Relation zwischen den Hauptkriimmungen, Math. Nachr. 4 (1951), 232–249.Google Scholar
5. Hsiung, C. C., Some integral formulas for closed hyper surf aces, Math. Scand. 2 (1954), 286–294.Google Scholar
6. Hsiung, C. C., Some integral formulas for closed hypersurfaces in Riemannian space, Pacific J. Math. 6 (1956), 291–299.Google Scholar
7. Katsurada, Y., Generalized Minkowski formulas for closed hypersurfaces in Riemann space, Ann. Mat. Pura Appl. (4) 57 (1962), 283–293.Google Scholar
8. Katsurada, Y., Qn a certain property of closed hypersurfaces in an Einstein space, Comment. Math. Helv. 38 (1964), 165–171.Google Scholar
9. Katsurada, Y., Qn tfa isoperimetric problem in a Riemann space, Comment. Math. Helv. 41 (1966/ 67), 18–29.Google Scholar
10. Katsurada, Y. and H., Köjyö, Some integral formulas for closed submanifolds in a Riemann space, J. Fac. Sci. Hokkaido Univ. Ser. I 22 (1968), 90–100.Google Scholar
11. Katsurada, Y. and Nagai, T., On some properties of a submanifold with constant mean curvature in a Riemann space, J. Fac. Sci. Hokkaido Univ. Ser. I 22 (1968), 79–89.Google Scholar
12. Liebmann, H., Uber die Verbiegung der geschlossen Flachen positiver Krummung, Math. Ann. 53 (1900), 91–112.Google Scholar
13. Otsuki, T., Integral formulas for hypersurfaces in a Riemannian manifold and their applications, Tökoku Math. J. (2) 17 (1965), 335–348.Google Scholar
14. Suss, W., Zur relativen Differentialgeometrie. V, Töhoku Math. J. 30 (1929), 202–209.Google Scholar
15. Yano, K., Closed hypersurfaces with constant mean curvature in a Riemannian manifold, J. Math. Soc. Japan 17 (1965), 333–340.Google Scholar
16. Yano, K., Notes on hypersurfaces in a Riemannian manifold, Can. J. Math. 19 (1967), 439–446.Google Scholar