AI 中文总结
在无边界光滑n维黎曼流形上，证明任意向量场可表示为至多max{6,6n−3}个梯度向量场对的深度1李括号的线性组合，结合强Trotter性质，得到紧致连通流形上微分同胚型最优传输映射生成的群在微分同胚群恒等分支中稠密的结论。
AI 中文摘要
我们证明，在无边界的光滑n维黎曼流形上，任意向量场都可表示为至多max{6,6n−3}个梯度向量场对的深度1李括号的线性组合。结合强Trotter性质，我们进一步证明，若该流形还是紧致连通的，则由微分同胚型最优传输映射生成的群在微分同胚群的恒等分支中稠密。
英文摘要
We prove that, on a smooth manifold $M$ equipped with a constant rank horizontal distribution $H$ and a smooth inner product $g_H$ on $H$, any horizontal vector field can be written as a linear combination of, at most, $3N$ Lie brackets of horizontal gradient fields of depth-$(1)$, where $N$ is the minimal immersion dimension of $M$, recovering the Riemannian case trivially when $H$ is the entire tangent bundle. When $H$ is also bracket-generating with a uniformly bounded step, we show that any vector field can be written as a linear combination of a uniformly bounded number of iterated Lie brackets of horizontal gradient fields with uniformly bounded depth, both bounds depending solely on the immersion dimension $N$ and the uniform upper bound on the step of $H$. Utilizing this, in conjunction with the Trotter property, we show that, if the manifold is compact, connected, and without boundary, the group generated by flows of horizontal gradient fields is dense in the identity component of the diffeomorphism group. When the manifold is also Riemannian, we show that the same density statement holds for the group generated by diffeomorphic optimal mass transport maps.
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