Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions

First we define the notion of k -Ricci curvature of a Riemannian n -manifold. Then we establish sharp relations between the k -Ricci curvature and the shape operator and also between the k -Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Several applications of such relationships are also presented.