Submanifolds in pseudo-Euclidean spaces satisfying the condition ∆x = Ax+B

In this paper we study pseudo-Riemannian submanifolds in Rn+kt satisfying the condition ∆x = Ax+B, whereA is an endomorphism ofRn+kt andB is a constant vector inRn+kt. We give a characterization theorem when A is a self-adjoint endomorphism. As for hypersurfaces we are able to obtain a classification theorem for any endomorphism A. An old and celebrated result due to T. Takahashi, [13], gives a characterization of the minimal submanifolds in an ordinary sphere Sm(r) by means of a nonzero eigenvalue of the Laplacian, getting also an explicit expression of that eigenvalue in terms of the dimension n of the subman-ifold and the radius r of the sphere. This can be seen in some sense as a starting point of a