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Ricci curvature; shape operator; real hypersurface; algebraic lemma; tubular hypersurface; horosphere; complex hyperbolic space
First we prove a general algebraic lemma. By applying the algebraic lemma we establish a general inequality involving the Ricci curvature of an arbitrary real hypersurface in a complex hyperbolic space. We also classify real hypersurfaces with constant principal curvatures which satisfy the equality case of the inequality.
[1] Berndt, J.: Real hypersurfaces with constant principal curvatures in complex hyperbolic space. J. Reine Angew. Math. 395 (1989), 132-141. MR 0983062 | Zbl 0655.53046
[2] Chen, B. Y.: Geometry of Submanifolds. M. Dekker, New York, 1973. MR 0353212
[3] Chen, B. Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math. (Basel) 60 (1993), 568–578. MR 1216703 | Zbl 0811.53060
[4] Chen, B. Y.: A general inequality for submanifolds in complex-space-forms and its applications. Arch. Math. (Basel) 67 (1996), 519–528. MR 1418914 | Zbl 0871.53043
[5] Chen, B. Y.: Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension. Glasgow Math. J. 41 (1999), 33-41. MR 1689730
[6] De Smet, P. J., Dillen, F., Verstraelen, L. and Vrancken, L.: The normal curvature of totally real submanifolds of $S^6(1)$. Glasgow Math. J. 40 (1998), 199–204. MR 1630238
[7] De Smet, P. J., Dillen, F., Verstraelen, L. and Vrancken, L.: A pointwise inequality in submanifold theory. Arch. Math. (Brno) 35 (1999), 115–128. MR 1711669
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