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Some relations between differential geometric invariants and topological invariants of submanifolds1)

Published online by Cambridge University Press:  22 January 2016

Bang-Yen Chen
Bang-Yen Chen*
Affiliation:
Michigan State University
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Let M be an n-dimensional manifold immersed in an m-dimensional euclidean space Em and let ∇ and ∇̃ be the covariant differentiations of M and Em, respectively. Let X and Y be two tangent vector fields on M. Then the second fundamental form h is given by

(1.1) ∇̃XY = ∇XY + h(X,Y).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 60 , February 1976 , pp. 1 - 6
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1976

Footnotes

2)

This work was partially supported by NSF Grant GP-36684.

1)

A partial result of this paper was announced in the following article “Some integral inequalities of two geometric invariants” appeared in Bull. Amer. Math. Soc. 81 (1975), 177-178.

References

[1] Beckenbach, E. F. and Bellman, R., Inequalities, Springer-Verlag, Berlin, 1961.Google Scholar
[2] Chen, B.-Y., Geometry of Submanifolds, M. Dekker, New York, 1973.Google Scholar
[3] Chen, B.-Y., On the total curvature of immersed manifolds, I, Amer. J. Math. 93 (1971), 148–162; On the total curvature of immersed manifolds, II, Amer. J. Math. 94 (1972), 799–809; On the total curvature of immersed manifolds, III, Amer. J. Math. 95 (1973), 636–642.Google Scholar
[4] Chern, S. S. and Lashof, R. K., On the total curvature of immersed manifolds, I, Amer. J. Math. 79 (1957), 306–318; On the total curvature of immersed manifolds, II, Michigan Math. J. 5 (1958), 5–12.Google Scholar