NDLI: Sub-Riemannian Geometry on Infinite-Dimensional Manifolds

Content Provider	Springer Nature Link
Author	Grong, Erlend Markina, Irina Vasil’ev, Alexander
Copyright Year	2014
Abstract	We generalize the concept of sub-Riemannian geometry to infinite-dimensional manifolds modeled on convenient vector spaces. On a sub-Riemannian manifold $$M$$ , the metric is defined only on a sub-bundle $${{\mathrm{\mathcal {H}}}}$$ of the tangent bundle $$TM$$ , called the horizontal distribution. Similarly to the finite-dimensional case, we are able to split possible candidates for minimizing curves into two categories: semi-rigid curves that depend only on $${{\mathrm{\mathcal {H}}}}$$ , and normal geodesics that depend both on $${{\mathrm{\mathcal {H}}}}$$ itself and on the metric on $${{\mathrm{\mathcal {H}}}}$$ . In this sense, semi-rigid curves in the infinite-dimensional case generalize the notion of singular curves for finite dimensions. In particular, we study the case of regular Lie groups with invariant sub-Riemannian structure. As examples, we consider the group of sense-preserving diffeomorphisms $${{\mathrm{Diff}}}\, S^1$$ of the unit circle and the Virasoro–Bott group with their respective horizontal distributions chosen to be the Ehresmann connections with respect to a projection to the space of normalized univalent functions. In these cases we prove controllability and find formulas for the normal geodesics with respect to the pullback of the invariant Kählerian metric on the class of normalized univalent functions. The geodesic equations are analogues to the Camassa–Holm, Hunter–Saxton, KdV, and other known non-linear PDE.
Starting Page	2474
Ending Page	2515
Page Count	42
File Format	PDF
ISSN	10506926
Journal	Journal of Geometric Analysis
Volume Number	25
Issue Number	4
e-ISSN	1559002X
Language	English
Publisher	Springer US
Publisher Date	2014-09-04
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Sub-Riemannian geometry Infinite-dimensional manifold Semi-rigid curves Group of diffeomorphisms of the circle Hamiltonian structures, symmetries, variational principles, conservation laws Symplectic structures of moduli spaces Group structures and generalizations on infinite-dimensional manifolds General theory of conformal mappings Completely integrable systems and methods of integration Differential Geometry Convex and Discrete Geometry Fourier Analysis Abstract Harmonic Analysis Dynamical Systems and Ergodic Theory Global Analysis and Analysis on Manifolds
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Sub-Riemannian geometry on infinite-dimensional manifolds

Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms

Quasiregular Mappings on Sub-Riemannian Manifolds

Classification, Casimir Invariants, and Stability of Lie-Poisson Systems

Geometric hydrodynamics and infinite-dimensional Newton's equations

Poisson geometry and first integrals of geostrophic equations

Induced Riemannian Structure on a Reduced Symplectic Manifold

Integrable Background Geometries

Poisson surfaces and algebraically completely integrable systems

Sub-Riemannian Geometry on Infinite-Dimensional Manifolds

Similar Documents

Sub-Riemannian geometry on infinite-dimensional manifolds

Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms

Quasiregular Mappings on Sub-Riemannian Manifolds

Classification, Casimir Invariants, and Stability of Lie-Poisson Systems

Geometric hydrodynamics and infinite-dimensional Newton's equations

Poisson geometry and first integrals of geostrophic equations

Induced Riemannian Structure on a Reduced Symplectic Manifold

Integrable Background Geometries

Poisson surfaces and algebraically completely integrable systems

Sub-Riemannian Geometry on Infinite-Dimensional Manifolds