Application of Computer Mathematics Systems to the Study of Homogeneous (pseudo)Riemannian Manifolds with the Trivial Schouten - Weyl Tensor
Papers of many mathematicians are devoted to the investigation of (pseudo)Riemannian Einstein manifolds, locally symmetric, Ricci parallel and conformally flat manifolds. All these manifolds (as special cases) are contained in the class of (pseudo)Riemannian manifolds with trivial Schouten - Weyl tensor. In the case of low dimension, it is possible to apply computer mathematics systems for studying the homogeneous (pseudo)Riemannian manifolds with trivial Schouten - Weyl tensor, since all invariant tensor fields are expressed in terms of Lie algebra structure constants of isometry group and the components of the metric tensor. A key step to solving the problem of classifying homogeneous (pseudo)Riemannian manifolds with the zero Schouten - Weyl tensor is a sequential consideration of all possible Segre types of the Ricci operator. This study is aimed at developing a mathematical model, as well as a computer program for studying and classifying the homogeneous (pseudo)Riemannian manifolds with zero Schouten - Weyl tensors of finite dimensions. Also, this paper contains an example that shows the basic steps of the developed algorithm.