Submanifolds whose quadric representations satisfy Δ˜{x}=B˜{x}+C

Abstract

Let x : Mⁿ→E^m be an isometric immersion of an n-dimensional Riemannian manifold into the m-dimensional Euclidean space. Then the map ˜{x}=xx^t (where t denotes transpose) is called the quadric representation of Mⁿ. In this paper, we give some results on submanifolds in the Euclidean space E^m which satisfy Δ˜{x}=B˜{x}+C, where B and C are two constant matrices.