The study of diffeomorphism groups and their applications to problems in analysis and geometry has a long history. In geometric hydrodynamics, pioneered by V.~Arnold in the 1960s, one considers an ideal fluid flow as the geodesic motion on the infinite-dimensional group of volume-preserving diffeomorphisms of the fluid domain with respect to the metric defined by the kinetic energy. Similar considerations on the space of densities lead to a geometric description of optimal mass transport and the Kantorovich-Wasserstein metric. Likewise, information geometry associated with the Fisher-Rao metric and the Hellinger distance has an equally beautiful infinite-dimensional geometric description and can be regarded as a higher-order Sobolev analogue of optimal transportation. In this work we review various metrics on diffeomorphism groups relevant to this approach and introduce appropriate topology, smooth structures and dynamics on the corresponding infinite-dimensional manifolds. Our main goal is to demonstrate how, alongside topological hydrodynamics, Hamiltonian dynamics and optimal mass transport, information geometry with its elaborate toolbox has become yet another exciting field for applications of geometric analysis on diffeomorphism groups.

翻译：微分同胚群及其在分析和几何问题中的应用研究具有悠久历史。在由V.~Arnold于20世纪60年代开创的几何流体动力学中，理想流体流动可视为流体域上保体积微分同胚群（关于动能定义的度量）的测地运动。在密度空间上的类似考量引出了最优质量传输的几何描述与Kantorovich-Wasserstein度量。同样地，与Fisher-Rao度量和Hellinger距离相关联的信息几何也具有同样优美的无穷维几何描述，可视为最优传输的高阶Sobolev类比。本文回顾了与此方法相关的微分同胚群上的各类度量，并在相应的无穷维流形上引入了恰当的拓扑结构、光滑结构及动力学。我们的主要目标是阐明：除了拓扑流体动力学、哈密顿动力学和最优质量传输之外，拥有精密工具箱的信息几何如何成为微分同胚群上几何分析应用的另一个激动人心的领域。