Normal curvature and real submanifold of the complex projective space

Abstract

It is a well-known fact that an odd-dimensional sphere is a principal circle bundle over the complex projective space and that the Riemannian structure of the complex projective space is the one induced from that of the sphere by the Hopf-fibration. For any real submanifold of the complex projective space we construct a circle bundle over it in such a way that the fibration is compatible with the Hopf-fibration. Then the circle bundle must be a submanifold of the sphere. Thus if we can say something about the circle bundle as a submanifold of a sphere we shall be able to say something about the base space which is a submanifold of the complex projective space. From this standpoint Lawson [I] and the present author [2], [3], [4] studied real hypersurface and some real submanifolds of the complex projective space. In the present paper the author, keeping this standpoint, studies relations between normal connection of the submanifold of the complex projective space and that of the circle bundle and then applies the results to theory of submanifolds of codimension 2 of the complex projective space. In §I, we state some formulas for real submanifold of the complex projective space and in §2 the relations between the normal connection of the circle bundle of the submanifold and that of the submanifold itself. Here we define L-flatness of the normal connection. In §3 we study, particularly, the submanifold of codimension 2 of the complex projective space and completely determine the submanifold of codimension 2 with L-fiat normal connection.