Normally flat Ric-semisymmetric submanifolds in Euclidean spaces
Consider a Riemannian manifold \(M\) with Riemann curvature tensor \(R\) and Ricci tensor \(R_1\). The manifold \(M\) is called Ric-semisymmetric if \(R(X,Y) \cdot R_1 = 0\). This paper's topic is the characterization of normally flat Ric-semisymmetric submanifolds in Euclidean spaces in terms of principal curvature vectors. Such a classification is know for normally flat Ric-semisymmetric hypersurfaces [\textit{V. A. Mirzoyan}, Sb. Math. 191, No. 9, 1323--1338 (2000); translation from Mat. Sb. 191, No. 9, 65--80 (2000; Zbl 1001.53006)]. In the non-hypersurface case, the situation is more complicated. The authors manage to characterize Einstein and semi-Einstein submanifolds and, as their central contribution, give a local geometric description for a certain class of submanifolds with co-dimension greater than one.
scientific article; zbMATH DE number 6148294
scientific article