  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 ). $ 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.
