Infinitesimal variations of invariant submanifolds of a Kaehlerian manifold

Abstract

Recently infinitesimal variations of submanifolds have been studied by Chen [1], Goldstein [2], Ryan [2], Tachibana [3, 4] and one of the present authors [1, 4]. The purpose of the present paper is to study infinitesimal variations of invariant submanifolds of a Kaehlerian manifold and to generalize some of recent results of Tachibana and one of the present authors. In the preliminary § 1, we state some properties of invariant submanifolds of a Kaehlerian manifold. In § 2 we prove fundamental formulas in the theory of infinitesimal variations and study complex variations, that is, infinitesimal variations which carry an invariant submanifold into an invariant submanifolds. In § 3, we study holomorphic variations, that is, complex variations which preserve complex structures induced on invariant submanifolds. In § 4, we study complex conformal variations and prove that a complex conformal variation of a compact invariant submanifold of a Kaehlerian manifold is necessarily isometric and hence holomorphic, (Theorem 4.1). In the last § 5 we prove an integral formula and show some of its applications.