Families of costs with zero and nonnegative MTW tensor in optimal
  transport
Du Nguyen
We compute explicitly the MTW tensor (or cross curvature) for the optimal
transport problem on $\mathbb{R}^n$ with a cost function of form $\mathsf{c}(x,
y) = \mathsf{u}(x^{\mathfrak{t}}y)$, where $\mathsf{u}$ is a scalar function
with inverse $\mathsf{s}$, $x^{\ft}y$ is a nondegenerate bilinear pairing of
vectors $x, y$ belonging to an open subset of $\mathbb{R}^n$. 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
$\mathsf{s}^{(2)} - S\mathsf{s}^{(1)} + P\mathsf{s} = 0$ with constant
coefficients $P$ and $S$. The resulting inverse functions include {\it Lambert}
and {\it generalized inverse hyperbolic\slash trigonometric} functions. The
square Euclidean metric and $\log$-type costs are equivalent to instances of
these solutions. The optimal map for the family is also explicit. 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 $\mathsf{s}$ using the Gauss-Codazzi equation, obtaining new
families of strictly regular costs for these manifolds, including new families
of {\it power function costs}. We analyze the $\sinh$-type hyperbolic cost,
providing examples of $\mathsf{c}$-convex functions and divergence.