Metric and Levi-Civita Connections on Regular Hom-Lie-Rinehart Algebras

Abstract

In this paper, we study regular Hom-Lie-Rinehart algebras and extend several classical geometric notions to this framework. In particular, we develop the theory of pseudo-Riemannian metrics, connections, curvature tensors, and torsion tensors in the setting of regular Hom-Lie-Rinehart algebras. We also investigate representations, differential operators, and the associated cohomology theory, establishing several results in differential calculus. Finally, a number of examples are presented to illustrate the developed theory.