Totally umbilical CR-submanifolds of a Kaehler manifold

Abstract

A classification Theorem for totally umbilical CR-submanifolds of a Kaehler manifold is proved. Introduction.Cft-submanifolds of a Kaehler manifold [1] being generalization of holomorphic and totally real submanifolds of a Kaehler manifold, has recently become subject of sufficient interest. Totally umbilical Ci?-submanifolds of a Kaehler manifold have been studied by A. Bejancu [3], Blair and Chen [4]. The purpose of this paper is to classify all totally umbilical C7?-submanifolds of a Kaehler manifold. In fact we prove the following theorem.THEOREM. Let M, (dimM^δ) be a complete simply connected totally umbilical CR-submanifold of a Raheler manifold M. Then M is one of the following * (i) Locally the Riemanman product of a holomorphic submamfold and a totally real submanifold of M (ii) totally real submanifold (iii) isometric to an ordinary sphere (iv) homothetic to a Sasakian manifold. The cases (iii) and (iv) occur when dim M is odd. Preliminaries.Let M be an m-dimensional Kaehler manifold with almost complex structure /. Then the curvature tensor R of M satisfies [11].

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