Certain almost contact hypersurfaces in Kaehlerian manifolds of constant holomorphic sectional curvatures

Abstract

Introduction. An odd-dimensional differentiable manifold is said to have an almost contact structure or to be an almost contact manifold if the structural group of its tangent bundle is reducible to the product of a unitary group with the 1-dimensional identity group. The study of almost contact manifolds, at the first time, has been developed by W. M. Boothby and H. C. Wang [ l ] υ and J. W. Gray [2] using a topological method. Recently, S. Sasaki [7] found a differential geometric method of investigation into the almost contact manifold and using this method Y. Tashiro [12] proved that in any orientable differentiable hypersurface in an almost complex manifold we can naturally define an almost contact structure. Hereafter, the almost contact structure of the hypersurface is studied by M. Kurita [4], Y. Tashiro and S. Tachibana [13] and the present author [5]. The purpose of the paper is to discuss normal almost contact hypersurfaces in a Kaehlerian manifold of constant holomorphic sectional curvature and to prove some fundamental properties of the hypersurfaces. In §1, we give first of all some preliminaries of almost contact manifold and prove a certain condition for a Riemannian manifold to be a normal contact manifold for the later use. In §2, we consider hypersurfaces in a Kaehlerian manifold and give a condition for the induced almost contact structure of a hypersurface in a Kaehlerian manifold to be normal. After proving a lemma in §3, we show in §4 that, in a normal almost contact hypersurface of a Kaehlerian manifold of constant holomorphic sectional curvature, the second fundamental tensor can admit at most three distinct characteristic roots and that they are all constants. The distributions corresponding these characteristic roots are studied in §5 and integrability of these distributions is discussed. In §6, the integral submanifolds of certain distributions are considered and using the theorem in §1, we prove that the integral submanifolds admit normal contact metric structures.