  We define and study the theory of derivation-based connections on a recently
introduced class of bimodules over an algebra which reduces to the category of
modules whenever the algebra is commutative. This theory contains, in
particular, a noncommutative generalization of linear connections. We also
discuss the different noncommutative versions of differential forms based on
derivations. Then we investigate reality conditions and a noncommutative
generalization of pseudo-riemannian structures.
