CERTAIN LIE ALGEBRAIC STRUCTURES ON RIEMANNIAN MANIFOLDS WITH SEMI-SYMMETRIC NON-METRIC CONNECTION

Abstract

As a natural consequence of the Levi-Civita connection on a Riemannian manifold, there is a Lie algebra structure on a Riemannian manifold. Lie Algebras and Lie Groups are the mathematical structure of continuous symmetries in physics. In this paper, semi-symmetric non-metric connection is considered instead of Levi-Civita connection of Riemann manifold, and accordingly  the existence of algebraic structures is investigated. First, it is shown that there is not always a Lie algebra structure on a Riemannian manifold with a semi-symmetric non-metric connection.
Then, necessary and sufficient conditions for Lie admissible algebra, pre-Lie algebra and post Lie algebra on a Riemann manifold with semi-symmetric non-metric connection are obtained depending on geometric terms. In addition, the cases of the Riemannian manifold with such algebraic structures according to the semi-symmetric non-metric connection being Einstein manifold and being flat manifold have been also investigated.