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Mean curvature and shape operator of isometric immersions in real-space-forms

Published online by Cambridge University Press:  18 May 2009

Bang-Yen Chen
Bang-Yen Chen
Affiliation:
Michigan State University, Department of Mathematics, East Lansing, Michigan 48824-1027U.S.A. E-mail: bychen@math.msu.edu
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According to the well-known Nash's theorem, every Riemannian n-manifold admits an isometric immersion into the Euclidean space En(n+1)(3n+11)/2. In general, there exist enormously many isometric immersions from a Riemannian manifold into Euclidean spaces if no restriction on the codimension is made. For a submanifold of a Riemannian manifold there are associated several extrinsic invariants beside its intrinsic invariants. Among the extrinsic invariants, the mean curvature function and shape operator are the most fundamental ones.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 38 , Issue 1 , January 1996 , pp. 87 - 97
Copyright
Copyright © Glasgow Mathematical Journal Trust 1996

References

REFERENCES

1.Chen, B. Y., Geometry of Submanifolds, (Marcel Dekker, New York, 1973).Google Scholar
2.Chen, B. Y., Some pinching and classification theorems for minimal submanifolds, Archiv der Math. 60 (1993), 568–578.CrossRefGoogle Scholar
3.Chen, B. Y., A Riemannian invariant for submanifolds in space forms and its applications, Geometry and Topology of Submanifolds (Dillen, F. and Verstraelen, L., eds.), vol. VI, 1994, pp. 58–81.Google Scholar
4.Chen, B. Y., A Riemannian invariant and its applications to submanifold theory, Results in Math. 27 (1995), 17–26.CrossRefGoogle Scholar
5.Erbacher, J., Reduction of the codimension of an isometric immersion, J. Differential Geometry 5 (1971), 333–340.CrossRefGoogle Scholar
6.Leung, P. F., On a relation between the topology and the intrinsic and extrinsic geometries of a compact submanifold, Proc. Edinburgh Math. Soc. 28 (1985), 305–311.CrossRefGoogle Scholar