  We develop a "metrically selfdual" variational calculus for $c$-monotone
vector fields between general manifolds $X$ and $Y$, where $c$ is a coupling on
$X\times Y$. Remarkably, many of the key properties of classical monotone
operators known to hold in a linear context, extend to this non-linear setting.
This includes an integral representation of $c$-monotone vector fields in terms
of $c$-convex selfdual Lagrangians, their characterization as a partial
$c$-gradients of antisymmetric Hamiltonians, as well as the property that these
vector fields are generically single-valued. We also use a symmetric
Monge-Kantorovich transport to associate to any measurable map its closest
possible $c$-monotone "rearrangement". We also explore how this metrically
selfdual representation can lead to a global variational approach to the
problem of inverting $c$-monotone maps, an approach that has proved efficient
for resolving non-linear equations and evolutions driven by monotone vector
fields in a Hilbertian setting.
