  We study connections on hermitian modules, and show that metric connections
exist on regular hermitian modules; i.e finitely generated projective modules
together with a non-singular hermitian form. In addition, we develop an index
calculus for such modules, and provide a characterization in terms of the
existence of a pseudo-inverse of the matrix representing the hermitian form
with respect to a set of generators. As a first illustration of the above
concepts, we find metric connections on the fuzzy sphere. Finally, the
framework is applied to a class of noncommutative minimal surfaces, for which
there is a natural concept of torsion, and we prove that there exist metric and
torsion-free connections for every minimal surface in this class.
