Submanifolds withL-flat normal connection of the complex projective space

Abstract

Introduction* As is well known an odd-dimensional sphere S is a principal circle bundle over a complex projective space P(C). The Riemannian structure on P{C) is given by the submersion π: S -> P(C) which is defined by the Hopf-fibration. If we construct a circle bundle over a real submanifold of P(C) in such a way that it is compatible with the Hopf-fibration, the circle bundle is a submanifold of the odd-dimensional sphere. Thus when we want to study submanifolds of the complex projective space it is useful to study the circle bundle over the submanifold. From this point of view, H. B. Lawson, Jr. [2] and the present author [3, 4, 5] have studied real submanifolds of the complex projective space. In the previous paper [5], the author studied relatious between the normal connection of a submanifold of P(C) and that of the circle bundle over the submanifold and established the notion of L-flatness for the normal connection of a real submanifold of P(C). The purpose of the present paper is to study submanifolds with L-flat normal connection of P(C). The main result is the following.