On naturally reductive left-invariant metrics of SL ( 2 , ℝ )

Stefan Halverscheid; Andrea Iannuzzi

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2006)

  • Volume: 5, Issue: 2, page 171-187
  • ISSN: 0391-173X

Abstract

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On any real semisimple Lie group we consider a one-parameter family of left-invariant naturally reductive metrics. Their geodesic flow in terms of Killing curves, the Levi Civita connection and the main curvature properties are explicitly computed. Furthermore we present a group theoretical revisitation of a classical realization of all simply connected 3-dimensional manifolds with a transitive group of isometries due to L. Bianchi and É. Cartan. As a consequence one obtains a characterization of all naturally reductive left-invariant riemannian metrics of SL ( 2 , ℝ ) .

How to cite

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Halverscheid, Stefan, and Iannuzzi, Andrea. "On naturally reductive left-invariant metrics of ${\rm SL}(2,\mathbb {R})$." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 5.2 (2006): 171-187. <http://eudml.org/doc/241055>.

@article{Halverscheid2006,
abstract = {On any real semisimple Lie group we consider a one-parameter family of left-invariant naturally reductive metrics. Their geodesic flow in terms of Killing curves, the Levi Civita connection and the main curvature properties are explicitly computed. Furthermore we present a group theoretical revisitation of a classical realization of all simply connected 3-dimensional manifolds with a transitive group of isometries due to L. Bianchi and É. Cartan. As a consequence one obtains a characterization of all naturally reductive left-invariant riemannian metrics of $\operatorname\{SL\}\nolimits (2,\mathbb \{R\})$.},
author = {Halverscheid, Stefan, Iannuzzi, Andrea},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {2},
pages = {171-187},
publisher = {Scuola Normale Superiore, Pisa},
title = {On naturally reductive left-invariant metrics of $\{\rm SL\}(2,\mathbb \{R\})$},
url = {http://eudml.org/doc/241055},
volume = {5},
year = {2006},
}

TY - JOUR
AU - Halverscheid, Stefan
AU - Iannuzzi, Andrea
TI - On naturally reductive left-invariant metrics of ${\rm SL}(2,\mathbb {R})$
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2006
PB - Scuola Normale Superiore, Pisa
VL - 5
IS - 2
SP - 171
EP - 187
AB - On any real semisimple Lie group we consider a one-parameter family of left-invariant naturally reductive metrics. Their geodesic flow in terms of Killing curves, the Levi Civita connection and the main curvature properties are explicitly computed. Furthermore we present a group theoretical revisitation of a classical realization of all simply connected 3-dimensional manifolds with a transitive group of isometries due to L. Bianchi and É. Cartan. As a consequence one obtains a characterization of all naturally reductive left-invariant riemannian metrics of $\operatorname{SL}\nolimits (2,\mathbb {R})$.
LA - eng
UR - http://eudml.org/doc/241055
ER -

References

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