  Optimal transport and information geometry both study geometric structures on
spaces of probability distributions. Optimal transport characterizes the
cost-minimizing movement from one distribution to another, while information
geometry originates from coordinate-invariant properties of statistical
inference. Their connections and applications in statistics and machine
learning have started to gain more attention. In this paper we give a new
differential geometric connection between the two fields. Namely, the
pseudo-Riemannian framework of Kim and McCann, a geometric perspective on the
fundamental Ma-Trudinger-Wang (MTW) condition in the regularity theory of
optimal transport maps, encodes the dualistic structure of statistical
manifold. This general relation is described using the natural framework of
$c$-divergence, a divergence defined by an optimal transport map. As a
by-product, we obtain a new information-geometric interpretation of the MTW
tensor. This connection sheds light on old and new aspects of information
geometry. The dually flat geometry of Bregman divergence corresponds to the
quadratic cost and the pseudo-Euclidean space, and the
$L^{(\alpha)}$-divergence introduced by Pal and the first author has constant
sectional curvature in a sense to be made precise. In these cases we give a
geometric interpretation of the information-geometric curvature in terms of the
divergence between a primal-dual pair of geodesics.
