zbMATH — the first resource for mathematics

Einstein Kaehler submanifolds of a complex linear or hyperbolic space. (English) Zbl 0648.53031
The author proves that every Einstein Kaehler submanifold of \({\mathbb{C}}^ n\) or \({\mathbb{C}}H^ n\) (the complex hyperbolic space of negative constant holomorphic sectional curvature) is always totally geodesic. He uses in his proof some previous results (by himself) on the so-called “diastasis” associated to a metric, a notion introduced by E. Calabi in Ann. Math., II. Ser. 58, 1-23 (1953; Zbl 0051.131)].
Reviewer: J.Girbau

53C40 Global submanifolds
Full Text: DOI
[1] E. CALABI, Isometric imbedding of complex manifolds, Ann. of Math. 58 (1953), 1-23. · Zbl 0051.13103
[2] S. S. CHERN, On Einstein hypersurfaces in a Kaehlerian manifold of constant holomorphi sectional curvature, J. Differential Geom. 1 (1967), 21-31. · Zbl 0168.19505
[3] S. KOBAYASHI AND K. NOMIZU, Foundation of Differential Geometry, vol. 2, Wiley Interscience, New York, 1969. · Zbl 0526.53001
[4] B. SMYTH, Differential geometry of complex hypersurface, Ann. of Math. 85 (1967), 246-266. · Zbl 0168.19601
[5] T. TAKAHASHI, Hypersurface with parallel Ricci tensor in a space of constant holomorphi sectional curvature, J. Math. Soc. Japan 19 (1967), 199-204. · Zbl 0147.40603
[6] K. TSUKADA, Einstein Kaehler submanifolds with codimension 2 in a complex spac form, Math. Ann. 274 (1986), 503-516. · Zbl 0592.53046
[7] M. UMEHARA, Kaehler submanifolds of complex space forms, to appear in Tokyo J. Math · Zbl 0679.53016
[8] M. UMEHARA, Diastases and real analytic functions on complex manifolds, to appear i J. Math. Soc. Japan. · Zbl 0651.53046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.