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On Some Complex Submanifolds in Kaehler Manifolds

Published online by Cambridge University Press:  20 November 2018

Masahiro Kon*
Affiliation:
Science University of Tokyo, Tokyo, Japan
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The purpose of this paper is to give some conditions for complex submanifolds in a Kaehler manifold of constant holomorphic sectional curvature to be Einstein.

For a complex hypersurface which is Einstein, Smyth [8] has obtained its classification and Chern [2] has proved the corresponding local result. Moreover, Takahashi [9] and Nomizu-Smyth [3] generalized this to a complex hypersurface with parallel Ricci tensor. We shall consider a condition weaker than the requirement that the Ricci tensor be parallel, that is we shall consider a complex submanifold with commuting curvature and Ricci operator, which condition was treated by Bishop-Goldberg [1].

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Bishop, R. L. and Goldberg, S. I., On conformally flat spaces with commuting curvature and Ricci transformations, Can. J. Math. 24 (1972), 799804.CrossRefGoogle Scholar
2. Chern, S. S., Einstein hypersurfaces in a Kaehler manifold of constant holomorphic curvature, J. Differential Geometry 1 (1967), 2131.CrossRefGoogle Scholar
3. Nomizu, K. and Smyth, B., Differential geometry of complex hypersurfaces, II, J. Math. Soc. Japan 20 (1968), 498521.CrossRefGoogle Scholar
4. Ogiue, K., Complex submanifolds of the complex projective space with second fundamental form of constant length, Ködai Math. Sem. Rep. 21 (1969), 252254.CrossRefGoogle Scholar
5. Ogiue, K., On Kaehler immersions, Can. J. Math. 24 (1972), 11781182.CrossRefGoogle Scholar
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7. Simons, J., Minimal varieties in Riemannian manifolds, Ann. of Math. 88 (1968), 62105.CrossRefGoogle Scholar
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9. Takahashi, T., Hyper surf ace with parallel Ricci tensor in a space of constant holomorphic sectional curvature, J. Math. Soc. Japan 19 (1967), 199204.CrossRefGoogle Scholar
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