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.

我们计算了具有形式 u (x^ty) 的成本函数的 R^n 上最优输运问题的 MTW 张量（或交叉曲率），其中 u 是具有逆 s 的标量函数，x^ty 是属于 R^n 的开子集上的向量 x、y 的非退化双线性配对。MTW 张量在 Kim-McCann 度量下在零向量上消失的条件是一个四阶非线性 ODE，可以简化为具有常系数 P 和 S 的线性 ODE 形式 s''−Ss'+Ps=0。所得到的逆函数包括 Lambert 函数和广义的逆双曲 / 三角函数。平方欧几里得度量和对数型成本是这些解的实例。该族的最优映射也是明确的。对于超球面和双曲空间的类似形式的成本函数，我们还使用 Gauss-Codazzi 方程将该张量表达为关于 s 的导数的代数表达式，得到这些流形上严格正则成本的新族，包括新的幂函数成本族。我们分析了 sinh 型双曲成本，提供了 c - 凸函数和离散度的示例。