Levi-Civita connections and vector fields for noncommutative differential calculi
International Journal of Mathematics
We study covariant derivatives on a class of centered bimodules over an algebra . We begin by identifying a ()-submodule () which can be viewed as the analogue of vector fields in this context; () is proven to be a Lie algebra. Connections on are in one-to-one correspondence with covariant derivatives on (). We recover the classical formulas of torsion and metric compatibility of a connection in the covariant derivative form. As a result, a Koszul formula for the Levi-Civita connection is also derived.
Bhowmick, Jyotishman; Goswami, Debashish; and Landi, Giovanni, "Levi-Civita connections and vector fields for noncommutative differential calculi" (2020). Journal Articles. 205.