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.

翻译：我们显式计算了$\mathbb{R}^n$上具有形如$\mathsf{c}(x, y) = \mathsf{u}(x^{\mathfrak{t}}y)$成本函数的最优输运问题的MTW张量（或交叉曲率），其中$\mathsf{u}$是标量函数且其逆函数为$\mathsf{s}$，$x^{\ft}y$是向量$x, y$（属于$\mathbb{R}^n$的开子集）的非退化双线性配对。在Kim-McCann度量下，MTW张量在零向量上消失的条件归结为一个四阶非线性常微分方程，该方程可化为形如$\mathsf{s}^{(2)} - S\mathsf{s}^{(1)} + P\mathsf{s} = 0$的线性常微分方程（$P$和$S$为常系数）。得到的逆函数包括{\it Lambert}函数和{\it广义反双曲/三角}函数。平方欧几里得度量和$\log$型成本是这些解的特例，且该族的显式最优映射也可给出。对于双曲空间（双曲模型）和单位球面上具有类似形式的成本函数，我们利用Gauss-Codazzi方程将该张量表示为$\mathsf{s}$导数的代数表达式，从而获得这些流形上新的严格正则成本函数族，包括新的{\it幂函数成本}族。我们进一步分析$\sinh$型双曲成本，并给出$\mathsf{c}$-凸函数和散度的示例。