A remark on foliations on a complex projective space with complex leaves

Abstract

Let EF be a foliation on a Riemannian manifold M. The distribution on M which is defined to be orthogonal to SF is said to be normal to £Fand denoted by 3L. Nakagawa and Takagi [8] showed that any harmonic foliation on a compact Riemannian manifold of non-negative constant sectional curvature is totally geodesic if the normal distribution is minnimal. And succesively the present author [2] proved a complex version of their result, that is, the above result holds also on a complex projective space with a Fubini-Study metric. However, recently, Li [4] pointed out a serious mistake in the proof of the result of Nakagawa and Takagi, and so of the author's. Therefore those results are now open yet. On the other hand, Li [4] have studied a harmonic foliationon the sphere along the method of Nakagawa and Takagi, and obtained some interesting results. The purpose of this paper is to give a complex analogue of the Li's results. Let Pn+p(C) be the complex projective space with the Fubini-Study metric of constant holomorphic sectional curvature c. Let £Fbe a complex foliation on Pn+p(C) with ^-complex codimention and h the second fundamental tensor of <7. Then we shall Drove the following: