Certain application of an integral formula to CR submanifold of complex projective space

Abstract

Let M be an n-dimensional CR submanifold of CR dimension n-1/2 of complex projective space. In this case M is necessarily odd-dimensional and there exists a unit vector field ξ1 normal to M such that JT(M) ⊂ T(M) ⊕ ξ1. Under the assumption that ξ1 is parallel with respect to the normal connection, we bring into use an integral formula which leads to an inequality between the Ricci tensor, the scalar curvature and the mean curvature of M. Using this inequality, we provide a sufficient condition for the submanifold M to be a tube over a totally geodesic complex subspace of Pn+k/2 (C).