  We consider metrics which are preserved under a $p$-Wasserstein transport
map, up to a possible contraction. In the case $p=1$ this corresponds to a
metric which is uniformly curved in the sense of coarse Ricci curvature. We
investigate the existence of such metrics in the more general sense of
pseudo-metrics, where the distance between distinct points is allowed to be 0,
and show the existence for general Markov chains on compact Polish spaces.
Further we discuss a notion of algebraic reducibility and its relation to the
existence of multiple true pseudo-metrics with constant curvature. Conversely,
when the Markov chain is irreducible and the state space finite we obtain
effective uniqueness of a metric with uniform curvature up to scalar
multiplication and taking the $p$th root, making this a natural intrinsic
distance of the Markov chain. An application is given in the form of
concentration inequalities for the Markov chain.
