  Arbitrary connections on a generic Hopf algebra $H$ are studied and shown to
extend to connections on tensor fields. On this ground a general definition of
metric compatible connection is proposed. This leads to a sufficient criterion
for the existence and uniqueness of the Levi-Civita connection, that of
invertibility of an $H$-valued matrix. Provided invertibility for one metric,
existence and uniqueness of the Levi-Civita connection for all metrics
conformal to the initial one is proven. This class consists of metrics which
are neither central (bimodule maps) nor equivariant, in general. For central
and bicoinvariant metrics the invertibility condition is further simplified to
a metric independent one. Examples include metrics on $SL_q(2)$.
