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A characterization of totally η-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form

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Abstract

We give a characterization of totally η-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator A of a real hypersurface M of a complex space form M n(c), c ≠ 0, n ⩾ 3, satisfies g(AX, Y) = ag(X, Y) for any X, Y ∈ T 0(x), a being a function, where T 0 is the holomorphic distribution on M, then M is a totally η-umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of η-umbilical real hypersurfaces.

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  1. Department of Mathematics, Hokkaido University, Kita 10 Nishi 8, Sapporo, 060-0810, Japan

    Mayuko Kon

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  1. Mayuko Kon
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Correspondence to Mayuko Kon.

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Kon, M. A characterization of totally η-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form. Czech Math J 58, 1279–1287 (2008). https://doi.org/10.1007/s10587-008-0086-8

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  • DOI: https://doi.org/10.1007/s10587-008-0086-8

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