Compact 2-transnormal hypersurface in a Kaehler manifold of constant holomorphic sectional curvature

Abstract

The theory of transnormality has arisen from attempts to generalize a concept of curves with constant width in a Euclidian plane to an analogue in a Riemannian manifold. S.A. Robertson [4, 5] has accomplished this in the case where the ambient space is Euclidian. Subsequently, J. Bolton [1] has extended this to the case of a hypersurface in a Riemannian manifold. Let M be an w-dimensional connected complete Riemannian manifold isometricallyimbedded into an (w+l)-dimensional connected complete Riemannian manifold M. For each xgM there exists,up to parametrization, a unique geodesic rx of M which intersects M orthogonally at x. M is called a transnormal hypersurface of M if, for each pair x,y£M, the relation rx3y implies that rx ―xv. We difine an equivalent relation ~ for points on a transnormal hypersurface M by writing x~y to mean ysrx, with respect to which the quotient space M=M/~ with the quotient topology is considered. M is called an r-transnormal hypersurface if the natural projection <p of M onto M is an r-fold covering map. Recently, S. Nishikawa [2] has studied some global properties of transnormal hypersurfaces of a complete Riemannian manifold. He gave also in [3] differential geometric structures of a compact 2-transnormal hypersurface of a simply connected complete Riemannian manifold of constant curvature. In this paper, we shall investigate 2-transnormal hypersurfaces in a Kaehler manifold of constant holomorphic sectional curvature and prove that these hypersurfaces are geodesic hyperspheres if principal curvatures are bounded from below (or above) by certain constants which depend only upon the diameters of the hypersurfaces and the holomorphic sectional curvatures of the ambient Kaehler manifolds. (Theorem 3.3 and Theorem 4.1) The auther would like to express her hearty gratitude to Prof. Shun-ichi Tachibana for his kind advice and suggestion.