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Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces

  • Gabriel Larotonda ORCID logo EMAIL logo
Published/Copyright: September 5, 2019
Forum Mathematicum
From the journal Forum Mathematicum Volume 31 Issue 6

Abstract

We study the geometry of Lie groups G with a continuous Finsler metric, in presence of a subgroup K such that the metric is right-invariant for the action of K. We present a systematic study of the metric and geodesic structure of homogeneous spaces M obtained by the quotient M≃G/K. Of particular interest are left-invariant metrics of G which are then bi-invariant for the action of K. We then focus on the geodesic structure of groups K that admit bi-invariant metrics, proving that one-parameter groups are short paths for those metrics, and characterizing all other short paths. We provide applications of the results obtained, in two settings: manifolds of Banach space linear operators, and groups of maps from compact manifolds.

Keywords: Lie group; Banach space; Finsler metric; homogeneous space; quotient metric; bi-invariant metric; geodesic; one-parameter group; diffeomorphism group; loop group; operator algebra; operator ideal; unitary group
MSC 2010: 22E65; 58B20; 53C22; 47B10; 58D05

Communicated by Karl-Hermann Neeb

Funding source: Consejo Nacional de Investigaciones Científicas y Técnicas

Award Identifier / Grant number: PIP 2014 11220130100525

Funding source: Fondo para la Investigació́n Científica y Tecnológica

Award Identifier / Grant number: PICT 2015 1505

Funding statement: This research was supported by CONICET (PIP 2014 11220130100525) and ANPCyT (PICT 2015 1505).

Acknowledgements

The author would like to thank the anonymous referee for her/his valuable suggestions, which helped to improve the quality of the final version of this paper.

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Received: 2019-05-13
Revised: 2019-08-10
Published Online: 2019-09-05
Published in Print: 2019-11-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Censored symmetric Lévy-type processes
  3. 𝐾-theory and immersions of spatial polygon spaces
  4. 𝐻𝑝 spaces for generalized Schrödinger operators and applications
  5. Commutative cocycles and stable bundles over surfaces
  6. Normal elements in the mod-𝑝 Iwasawa algebra over SL𝑛(ℤ𝑝): A computational approach
  7. Non-spectrality of self-affine measures on the three-dimensional Sierpinski gasket
  8. Plurisubharmonic functions on a neighborhood of a torus leaf of a certain class of foliations
  9. Discrete Littlewood–Paley–Stein characterization of multi-parameter local Hardy spaces
  10. Exceptional sets for sums of almost equal prime cubes
  11. Higher differentiability of solutions to a class of obstacle problems under non-standard growth conditions
  12. Beilinson–Flach elements, Stark units and 𝑝-adic iterated integrals
  13. On the bounded approximation property on subspaces of ℓp when 0 < p < 1 and related issues
  14. Refinement of the Chowla–Erdős method and linear independence of certain Lambert series
  15. Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces
  16. On commutator Krylov transitive and commutator weakly transitive Abelian p-groups
https://doi.org/10.1515/forum-2019-0127
Keywords for this article
Lie group; Banach space; Finsler metric; homogeneous space; quotient metric; bi-invariant metric; geodesic; one-parameter group; diffeomorphism group; loop group; operator algebra; operator ideal; unitary group

Articles in the same Issue

  1. Frontmatter
  2. Censored symmetric Lévy-type processes
  3. 𝐾-theory and immersions of spatial polygon spaces
  4. 𝐻𝑝 spaces for generalized Schrödinger operators and applications
  5. Commutative cocycles and stable bundles over surfaces
  6. Normal elements in the mod-𝑝 Iwasawa algebra over SL𝑛(ℤ𝑝): A computational approach
  7. Non-spectrality of self-affine measures on the three-dimensional Sierpinski gasket
  8. Plurisubharmonic functions on a neighborhood of a torus leaf of a certain class of foliations
  9. Discrete Littlewood–Paley–Stein characterization of multi-parameter local Hardy spaces
  10. Exceptional sets for sums of almost equal prime cubes
  11. Higher differentiability of solutions to a class of obstacle problems under non-standard growth conditions
  12. Beilinson–Flach elements, Stark units and 𝑝-adic iterated integrals
  13. On the bounded approximation property on subspaces of ℓp when 0 < p < 1 and related issues
  14. Refinement of the Chowla–Erdős method and linear independence of certain Lambert series
  15. Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces
  16. On commutator Krylov transitive and commutator weakly transitive Abelian p-groups
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