Levi-Civita connections and vector fields for noncommutative differential calculi

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