Ellipsoids, Complete Integrability and Hyperbolic Geometry

Ellipsoids, Complete Integrability and Hyperbolic Geometry

We describe a new proof of the complete integrability of the two related dynamical systems: the billiard inside the ellipsoid and the geodesic flow on the ellipsoid (in Euclidean, spherical or hyperbolic space). The proof is based on the construction of a metric on the ellipsoid whose nonparameterized geodesics coincide with those of the standard metric. This new metric is induced by the hyperb...

Geometry of Diffeomorphism Groups, Complete Integrability and Geometric Statistics

We study the geometry of the space of densities Dens(M), which is the quotient space Diff(M)/Diffμ(M) of the diffeomorphism group of a compact manifold M by the subgroup of volume-preserving diffeomorphisms, endowed with a right-invariant homogeneous Sobolev Ḣ-metric. We construct an explicit isometry from this space to (a subset of) an infinite-dimensional sphere and show that the associated E...

Geometry of Diffeomorphism Groups, Complete Integrability and Optimal Transport

We study the geometry of the space of densities Dens(M), which is the quotient space Diff(M)/Diffμ(M) of the diffeomorphism group of a compact manifold M by the subgroup of volume-preserving diffemorphisms, endowed with a right-invariant homogeneous Sobolev Ḣ-metric. We construct an explicit isometry from this space to (a subset of) an infinite-dimensional sphere and show that the associated Eu...

Contact complete integrability

Complete integrability in a symplectic setting means the existence of a Lagrangian foliation leaf-wise preserved by the dynamics. In the paper we describe complete integrability in a contact set-up as a more subtle structure: a flag of two foliations, Legendrian and coLegendrian, and a holonomy-invariant transverse measure of the former in the latter. This turns out to be equivalent to the exis...

Hyperbolic Geometry