Hofer’s metric is a metric on the group of Hamiltonian diffeomorphisms of a symplectic manifold. This group is infinite-dimensional, so that the existence of this metric is a very nontrivial due noncompactness. The subject of Hofer’s geometry is now a classic subject in symplectic geometry.
Helmut Hofer, The topology of symplectic maps, Proceedings of the Royal Society of Edinburgh, 115 (1990), 25-38.
Lev Buhovsky, Yaron Ostrover, On the uniqueness of Hofer’s geometry, arxiv:1008.0661