Hofer’s metric is a Riemannian metric on the group of Hamiltonian diffeomorphisms of a symplectic manifold. This group is infinite-dimensional, so that the existence of this metric is nontrivial, due noncompactness.
Hofer’s geometry is now a classical subject in symplectic geometry.
Helmut Hofer, On the topological properties of symplectic maps, Proceedings of the Royal Society of Edinburgh. Section A: Mathematics, Volume 115 Issue 1-2, 1990 (doi:10.1017/S0308210500024549)
Lev Buhovsky, Yaron Ostrover, On the uniqueness of Hofer’s geometry, arxiv:1008.0661
Michael Usher, Hofer geometry and cotangent fibers, arxiv/1301.4484
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