Complete 2-transnormal hypersurfaces in a Kaehler manifold of negative constant holomorphic sectional curvature

Abstract

The idea of constant width has been developed in a somewhat differentspirit, as a topic in differentialgeometry, and the concept of transnormality has been introduced as the generalized one of constant width in a Riemannian manifold. Let M be a connected complete hypersurface of a connected complete Riemannian manifold M. For each igM, there exists,up to parametrization, a unique geodesic zx of M which intersectsM orthogonally at x. M is called a transnormal hypersurface of M it, for each pair x,y<E.M, the relationyr for xjgM, x~y means that yE.rx. Then we can consider the quotient space M―Mj~ with the quotient topology with respect to thisrelation. We call M an r-transnormal hypersurface if the natural projection of M onto M is an r-fold covering map. Topological structures of transnormal submanifolds are full of interest and have been investigated from various angles (for example, see [3]). On the other hand, differentialgeometric structures of 2-transnormal hypersurfaces in a space form have been given in [2] and [4]. Recentry, the author has studied in [5] differentialgeometric structures of compact 2-transnormal hypersurfaces in a complex space form. The purpose of this paper is to generalize the result in [5] to the case where 2-transnormal hypersurfaces are complete. Namely we shall prove that 2-transnormal hypersurfaces in a Kaehler manifold of negative constant holomorphic sectionalcurvature are tubes over some submanifolds or geodesic hyperspheres if any principal curva-