Open Science: AltSU mathematicians developed a unique program for the study of geometric spaces

The Open Science scientific news portal published about  Pavel Klepikov, a post-graduate student at the Institute of Mathematics and Information Technologies of Altai State University, who is completing a project called “Locally Homogeneous Lorentzian Manifolds with an Isotropic Weyl Tensor”.

Space, or manifolds, studied by mathematicians at AltSU, is a mathematical object that does not physically exist, which differs from three-dimensional space in its multidimensionality. Realizing the project, the scientific team of the flagship university of Altai Krai solved the problems of researching the data of mathematical objects, collecting all the information received into a single database.

“In general, over a two-year work on the project, we developed a set of programs for studying locally homogeneous Lorentzian manifolds. We registered one of these programs and have already received a certificate of the Federal Service for Intellectual Property and its state registration. This is a software package for studying the Einstein equation of locally homogeneous (pseudo) Riemannian manifolds with vector torsion. In simple terms, this program allows to study the varieties stated in the project. It combines into one complex both the previously known and used for the first time formulas for the manifolds. Before, such programs did not exist”, emphasizes Pavel Klepikov.

Another result of the work of AltSU scientists in the framework of the project was the classification of research objects. This topic is covered in the article “On Classification of Four-dimensional Locally Homogeneous Pseudo-Riemannian Manifolds with the Isotropic Weyl Tensor”, which was published in the authoritative journal Russian Mathematics.

“The isotropic tensor was previously studied in the case of three-dimensional Lie groups with a left-invariant Lorentzian metric. In the case of locally homogeneous pseudo-Riemannian spaces with a nontrivial isotropy subgroup, we classified manifolds with the isotropic Weyl tensor. So, for example, in a work published in Russian Mathematics, we present one of the results of our work - the classification of four-dimensional locally homogeneous pseudo-Riemannian manifolds with an isotropic Weyl tensor. In addition, during the implementation of the project, we obtained some results about the curvature tensors of such manifolds”, noted the mathematician of AltSU.

Summing up the results of the project on studying locally homogeneous Lorentzian manifolds, the research team led by Pavel Klepikov continues research work. In particular, AltSU mathematicians are currently preparing for registration another software package for studying locally homogeneous pseudo-Riemannian manifolds with the isotropic Schouten – Weil tensor.

We want to note that the project of scientists from the AltSU Institute of Mathematics and Information Technology received financial support from the Russian Foundation for Basic Research in 2018 as the winner of “My First Grant” program.