Sectional curvatures of geodesic spheres in a complex hyperbolic space

Abstract

We characterize geodesic spheres with su‰ciently small radii in a complex hyperbolic space of constant holomorphic sectional curvature cð< 0Þ by using their geometric three properties. These properties are based on their contact forms, geodesics and shape operators. These geodesic spheres are the only examples of hypersurfaces of type (A) which are of nonnegative sectional curvature in this ambient space. Moreover, in particular, when À1 e c < 0, the class of these geodesic spheres has just one example of Sasakian space forms.

Extrinsic circular trajectories on totally η-umbilic real hypersurfaces in a complex hyperbolic space

“…In order to make clear our standing point of studies, we shall start by recalling a characterization of geodesic spheres of small radius given by Kajiwara and Maeda [12], and by giving its supplemental results. On a real hypersurface M of a Ka ¨hler manifold M M, we have a canonical contact metric structure which is induced by the complex structure J on M M. For a (local) unit normal vector field N on M in M M and the Riemannian metric h ; i on M M, we define a vector field x by x ¼ ÀJN, a 1-form h by hðvÞ ¼ hv; xi and a ð1; 1Þ-tensor field f by fðvÞ ¼ Jv À hðvÞN for each tangent vector v A TM.…”

“…Let i : M ! M M denote an isometric embedding of a real hypersurface M into the ambient space M M. For a smooth curve g on M, we can regard it as a curve in M M by considering i g and call it the extrinsic shape of g. For the sake of simplicity, we usually denote the extrinsic shape i g also by g. In their paper [12] Kajiwara and Maeda characterized geodesic spheres of nonnegative curvature by a property of extrinsic shapes of geodesics.…”

“…In their paper [12] Kajiwara and Maeda gave several characterizations of nonnegatively curved geodesic spheres in a complex hyperbolic space. Among them the authors are interested in the characterization of geodesic spheres by using extrinsic circular geodesics.…”