We study the information geometry of \({\textsf {c}}\) -divergences from families of costs of the form \({\textsf {c}}(x, {\bar{x}}) ={\textsf {u}}(x^{{\mathfrak {t}}}{\bar{x}})\) through the optimal transport point of view. Here, \({\textsf {u}}\) is a scalar function with inverse \({\textsf {s}}\) , \(x^{{\mathfrak {t}}}{\bar{x}}\) is a nondegenerate bilinear pairing of vectors \(x, {\bar{x}}\) belonging to an open subset of \({\mathbb {R}}^n\) . We compute explicitly the MTW tensor (or cross curvature) for the optimal transport problem on \({\mathbb {R}}^n\) with this cost. The condition that the MTW-tensor vanishes on null vectors under the Kim-McCann metric is a fourth-order nonlinear ODE, which could be reduced to a linear ODE of the form \({\textsf {s}}^{(2)} - S{\textsf {s}}^{(1)} + P{\textsf {s}} = 0\) with constant coefficients P and S. The resulting inverse functions include Lambert and generalized inverse hyperbolic/trigonometric functions. The square Euclidean metric and \(\log \) -type costs are equivalent to instances of these solutions. The optimal map may be written explicitly in terms of the potential function. For cost functions of a similar form on a hyperboloid model of the hyperbolic space and unit sphere, we also express this tensor in terms of algebraic expressions in derivatives of \({\textsf {s}}\) using the Gauss-Codazzi equation, obtaining new families of strictly regular costs for these manifolds, including new families of power function costs. We express the divergence geometry of the \({\textsf {c}}\) -divergence in terms of the Kim-McCann metric, including a \({\textsf {c}}\) -Crouzeix identity and a formula for the primal connection. We analyze the \(\sinh \) -type hyperbolic cost, providing examples of \({\textsf {c}}\) -convex functions, which are used to construct a new local form of the \(\alpha \) -divergences on probability simplices. We apply the optimal maps to the problem of sampling fat-tailed distributions, in particular, to sample the multivariate t-distribution.
