Geometry of diffeomorphism groups, complete integrability and geometric statistics

DOI10.1007/s00039-013-0210-2zbMath1275.58006arXiv1105.0643OpenAlexW2033360875MaRDI QIDQ352112

Stephen C. Preston, Gerard Misiołek, Jonatan Lenells, Boris A. Khesin

Publication date: 4 July 2013

Published in: Geometric and Functional Analysis. GAFA (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1105.0643

zbMATH Keywords

Hellinger distancegeodesicsintegrable systemscurvatureRiemannian metricsdiffeomorphism groupsFisher-Rao metricEuler-Arnold equations

Mathematics Subject Classification ID

Groups of diffeomorphisms and homeomorphisms as manifolds (58D05) Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds (58B20) Methods of global Riemannian geometry, including PDE methods; curvature restrictions (53C21) Manifolds of metrics (especially Riemannian) (58D17)

Related Items (34)

Deformed Neumann model from spinning strings on \({\mathrm{AdS}}_5\times\mathrm{S}^5_{\eta}\) ⋮ About simple variational splines from the Hamiltonian viewpoint ⋮ Geometry of matrix decompositions seen through optimal transport and information geometry ⋮ When optimal transport meets information geometry ⋮ Diffeomorphic Density Matching by Optimal Information Transport ⋮ On the smoothness of the geodesic spray for the Camassa-Holm equation ⋮ On geodesic completeness for Riemannian metrics on smooth probability densities ⋮ Invariant variational problems on homogeneous spaces ⋮ Gaussian distributions, Jacobi group, and Siegel-Jacobi space ⋮ The measurement and analysis of shapes. An application of hydrodynamics and probability theory ⋮ On the relation between geometrical quantum mechanics and information geometry ⋮ Quantum information geometry of driven CFTs ⋮ A two-component geodesic equation on a space of constant positive curvature ⋮ The Wasserstein geometry of nonlinear σ models and the Hamilton–Perelman Ricci flow ⋮ Geometry of generalized fluid flows ⋮ The \(L^p\)-Fisher-Rao metric and Amari-Čencov \(\alpha\)-connections ⋮ On Optimal Transport of Matrix-Valued Measures ⋮ Riemannian Hilbert Manifolds ⋮ The Schrödinger problem on the non-commutative Fisher-Rao space ⋮ Global estimates and blow-up criteria for the generalized Hunter-Saxton system ⋮ Semi-invariant Riemannian metrics in hydrodynamics ⋮ The Camassa-Holm equation as an incompressible Euler equation: a geometric point of view ⋮ Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics ⋮ Stochastic wave-current interaction in thermal shallow water dynamics ⋮ Vortex sheets and diffeomorphism groupoids ⋮ Convex Sobolev inequalities related to unbalanced optimal transport ⋮ Overview of the geometries of shape spaces and diffeomorphism groups ⋮ Generalized compressible flows and solutions of the \(H(\text{div})\) geodesic problem ⋮ Uniqueness of the Fisher–Rao metric on the space of smooth densities ⋮ Amari-Chentsov connections and their geodesics on homogeneous spaces of diffeomorphism groups ⋮ Geometry of the Madelung transform ⋮ Geometric hydrodynamics and infinite-dimensional Newton’s equations ⋮ Generalized Hunter-Saxton equations, optimal information transport, and factorization of diffeomorphisms ⋮ Partial differential equations with quadratic nonlinearities viewed as matrix-valued optimal ballistic transport problems

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• The basic geometry of the manifold of Riemannian metrics and of its quotient by the diffeomorphism group
• A metric on shape space with explicit geodesics
• Well-posedness of the generalized Proudman-Johnson equation without viscosity
• Defining the curvature of a statistical problem (with applications to second order efficiency)
• Further comments on some comments on a paper by Bradley Efron
• The Euler-Poincaré equations and semidirect products with applications to continuum theories
• Topological methods in hydrodynamics
• Construction of a smooth mapping with prescribed Jacobian. I
• Euler equations on homogeneous spaces and Virasoro orbits
• A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem
• The Hunter-Saxton equation describes the geodesic flow on a sphere
• Weak geodesic flow and global solutions of the Hunter-Saxton equation
• 2D-shape analysis using conformal mapping
• Groups of diffeomorphisms and the motion of an incompressible fluid
• Curvatures of Sobolev metrics on diffeomorphism groups
• Fredholm properties of Riemannian exponential maps on diffeomorphism groups
• THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION
• Dynamics of Director Fields
• Ricci flow and the metric completion of the space of Kähler metrics
• On the Volume Elements on a Manifold
• Optimal Transport
• Shapes and diffeomorphisms

This page was built for publication: Geometry of diffeomorphism groups, complete integrability and geometric statistics