Levi-Civita connections and vector fields for noncommutative differential calculi

We study covariant derivatives on a class of centered bimodules E over an algebra A. We begin by identifying a Z(A)-submodule X(A) which can be viewed as the analogue of vector fields in this context; X (A) is proven to be a Lie algebra. Connections on E are in one-to-one correspondence with covariant derivatives on X(A). 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.