A characterization of classical Riemannian manifolds by curvature operators

Abstract

In the paper using the skew-symmetric curvature operator and the generalized Jacobi operator (both defined by G. Stanilov in 1990) we give a characterization for the classical conformally flat Riemannian manifolds and the Einstein manifolds of dimension 4. Namely these are the only manifolds for which the skew-symmetric curvature operator induced from any two dimensional tangent subspace and the generalized Jacobi operator induced from the same tangent subspace are orthogonal on the orthogonal complement of such tangent subspaces at any point of the manifold. For dimension 3 such a characterization is given for the spaces of constant sectional curvature and the foliated manifolds but in this case in place of the generalized Jacobi operator we take the Jacobi operator on the orthogonal complement of the two dimensional tangent subspace and the orthogonality is taken on the two dimensional tangent subspace.