In this paper we characterize all totally $\eta$-umbilic real hypersurfaces $M$'s in complex projective or complex hyperbolic spaces by using an inequality related to the shape operator $A$ of $M$.
References
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Adachi, T., Kimura, M., and Maeda, S.: Real hypersurfaces some of whose geodesics are plane curves in nonflat complex space forms. (To appear in Tohoku Math. J.). MR2137467 10.2748/tmj/1119888336 euclid.tmj/1119888336
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Niebergall, R., and Ryan, P. J.: Real hypersurfaces in complex space forms. Tight and Taut Submanifolds (eds. Cecil, T. E. and Chern, S. S.). Cambridge Univ. Press, Cambridge, pp. 233–305 (1997). MR1486875 Niebergall, R., and Ryan, P. J.: Real hypersurfaces in complex space forms. Tight and Taut Submanifolds (eds. Cecil, T. E. and Chern, S. S.). Cambridge Univ. Press, Cambridge, pp. 233–305 (1997). MR1486875
