Totally complex submanifolds of a quaternionic Kaehlerian manifold

Abstract

BY SHOICHI FUNABASHI § 1. Introduction.We consider some kinds of submanifolds in a quaternionic Kaehlerian manifold M i. e., those invariant, totally real and totally complex submanifolds.These submanifolds are curvature invariant (see § 6).In M, invariant or totally real submanifolds are defined as quaternionic analogues of corresponding submanifolds in a Kaehlerian manifold ([6], [9]).It is known that each invariant submanifold of M is totally geodesic.But in general an n-dimensional totally real submanifold of M is not necessarily totally geodesic (see [3]).We shall define totally complex submanifold in § 2 and prove in § 2 that a totally complex submanifold is minimal.In the present paper, totally complex submanifolds of a quaternionic Kaehlerian manifold will be studied and the so-called pinching problem of the length of the second fundamental form will be discussed as quaternionic analogues of the arguments developed in [6] and [9] for submanifolds of a Kajehlerian manifold.Our main result is stated in the following main theorem which will be given in § 5.In the following theorem and hereafter, we mean by a complex space form (resp. a quaternionic space form) a Kaehlerian manifold with constant holomorphic sectional curvature (resp.a quaternionic Kaehlerian manifold with constant curvature).MAIN THEOREM.Let M 2n (n^2) be a 2n-dimensional compact, connected and complete submanifold of a An-dimensιonal quaternionic projective space HP 71 with constant Q-sectional curvature 4. Assume that M 2n is totally complex in HP n and the second fundamental form H of M 2n satisfies the inequality ||//|| being the length of H. Then M 2n is congruent to the complex projective space CP n of complex dimension n, which is naturally imbedded in HP n as a totally geodesic submanifold.