Advanced

Riemannian geometry on the diffeomorphism group of the circle

Lenells, Jonatan LU (2007) In Arkiv för matematik 45(2). p.297-325
Abstract
The topological group D-k(S) of diffeomorphisms of the unit circle 5 of Sobolev class H-k, for k large enough, is a Banach manifold modeled on the Hilbert space H-k(S). In this paper we show that the H-1 right-invariant metric obtained by right-translation of the H-1 inner product on TidDk(S)similar or equal to H-k(S) defines a smooth Riemannian metric on D-k(S), and we explicitly construct a compatible smooth affine connection. Once this framework has been established results from the general theory of affine connections on Banach manifolds can be applied to study the exponential map, geodesic flow, parallel translation, curvature etc. The diffeomorphism group of the circle provides the natural geometric setting for the Camassa-Holm... (More)
The topological group D-k(S) of diffeomorphisms of the unit circle 5 of Sobolev class H-k, for k large enough, is a Banach manifold modeled on the Hilbert space H-k(S). In this paper we show that the H-1 right-invariant metric obtained by right-translation of the H-1 inner product on TidDk(S)similar or equal to H-k(S) defines a smooth Riemannian metric on D-k(S), and we explicitly construct a compatible smooth affine connection. Once this framework has been established results from the general theory of affine connections on Banach manifolds can be applied to study the exponential map, geodesic flow, parallel translation, curvature etc. The diffeomorphism group of the circle provides the natural geometric setting for the Camassa-Holm equation - a nonlinear wave equation that has attracted much attention in recent years - and in this context it has been remarked in various papers how to construct a smooth Riemannian structure compatible with the H-1 right-invariant metric. We give a self-contained presentation that can serve as a detailed mathematical foundation for the future study of geometric aspects of the Camassa-Holm equation. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Arkiv för matematik
volume
45
issue
2
pages
297 - 325
publisher
Springer
external identifiers
  • wos:000250460200008
  • scopus:38149115470
ISSN
0004-2080
DOI
10.1007/s11512-007-0047-8
language
English
LU publication?
yes
id
ab79fd4a-6838-4d74-ad9f-3e77a6796eb9 (old id 653281)
date added to LUP
2007-12-12 13:50:14
date last changed
2017-03-05 04:06:43
@article{ab79fd4a-6838-4d74-ad9f-3e77a6796eb9,
  abstract     = {The topological group D-k(S) of diffeomorphisms of the unit circle 5 of Sobolev class H-k, for k large enough, is a Banach manifold modeled on the Hilbert space H-k(S). In this paper we show that the H-1 right-invariant metric obtained by right-translation of the H-1 inner product on TidDk(S)similar or equal to H-k(S) defines a smooth Riemannian metric on D-k(S), and we explicitly construct a compatible smooth affine connection. Once this framework has been established results from the general theory of affine connections on Banach manifolds can be applied to study the exponential map, geodesic flow, parallel translation, curvature etc. The diffeomorphism group of the circle provides the natural geometric setting for the Camassa-Holm equation - a nonlinear wave equation that has attracted much attention in recent years - and in this context it has been remarked in various papers how to construct a smooth Riemannian structure compatible with the H-1 right-invariant metric. We give a self-contained presentation that can serve as a detailed mathematical foundation for the future study of geometric aspects of the Camassa-Holm equation.},
  author       = {Lenells, Jonatan},
  issn         = {0004-2080},
  language     = {eng},
  number       = {2},
  pages        = {297--325},
  publisher    = {Springer},
  series       = {Arkiv för matematik},
  title        = {Riemannian geometry on the diffeomorphism group of the circle},
  url          = {http://dx.doi.org/10.1007/s11512-007-0047-8},
  volume       = {45},
  year         = {2007},
}