Compact Minimal $CR$ Submanifolds of a Complex Projective Space with Positive Ricci Curvature

Abstract

We give a reduction theorem for the codimension of a compact $n$-dimensional minimal proper $CR$ submanifold $M$ immersed in a complex projective space $CP^m$ with complex structure $J$, under the assumption that the Ricci curvature of $M$ is equal to or greater than $n-1$. Moreover, we classify compact $n$-dimensional minimal $CR$ submanifolds whose Ricci tensor $S$ satisfies $S(X,X)\geq (n-1)g(X,X)+kg(PX,PX)$, $k=0,1,2$, for any vector field $X$ tangent to $M$, where $PX$ is the tangential part of $JX$.