Properties of metrics and pairs consisting of left and right connectio
ns are studied on the bimodules of differential 1-forms. Those bimodul
es are obtained from the derivation based calculus of an algebra of ma
trix valued functions, and an SL(q)(2,C)-covariant calculus of the qua
ntum plane at a generic q and the cubic root of unity. It is shown tha
t, in the aforementioned examples, giving up the middle-linearity of m
etrics significantly enlarges the space of metrics. A metric compatibi
lity condition for the pairs of left and right connections is defined.
Also, a compatibility condition between a left and right connection i
s discussed. Consequences entailed by reducing to the center of a bimo
dule the domain of those conditions are investigated in detail. Altern
ative ways of relating left and right connections are considered. (C)
1996 American Institute of Physics.