Submanifolds with parallel Ricci tensor in Euclidean spaces (Q1902759)

Let \(M\) be a submanifold in a Euclidean space \(E_n\) with second fundamental form \(h\), mean curvature vector \(H\), induced Riemannian connection \(\nabla\) and normal connection \(\nabla^\perp\), curvature tensor \(R\) and Ricci tensor \(R_1\) of \(\nabla\), and curvature tensor \(R^\perp\) of \(\nabla^\perp\). The pair \((R, R^\perp)\) is the curvature tensor \(\overline R\) of the van der Waerden-Bortolotti connection \(\overline\nabla= (\nabla, \nabla^\perp)\). If \(\overline\nabla h= 0\), then \(M\) is said to be parallel. If \(\overline R(X, Y)\cdot h= 0\), or, equivalently, \(\overline\nabla_X \overline\nabla_Y h= \overline\nabla_Y \overline\nabla_X h\) for any vectors \(X\) and \(Y\) tangent to \(M\), the submanifold is said to be semi-parallel. Theorem 1. Let a semiparallel submanifold \(M\) in \(E_n\) satisfy the conditions \(\nabla R_1= 0\) and \(\nabla^\perp H= 0\). Then \(M\) is parallel. Theorem 2. A submanifold \(M\) with flat \(\nabla^\perp\) (i.e. \(R^\perp= 0\)) in \(E_n\) satisfies \(\nabla R_1= 0\) if and only if \(M\) is either Einsteinian (i.e. \(R_1= \lambda I\)), or locally decomposable in a product \(M_1\times\cdots\times M_p\), every \(M_k\) being Einsteinian with \(R^\perp_k= 0\) in some \(E_{n_k}\), \(E= E_{n_1}\times\cdots\times E_{n_s}\), \(E_{n_k}\) and \(E_{n_l}\) completely orthogonal if \(k\neq l\).