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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  2. [B] L. Bianchi, Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti, Mem. Soc. It. delle Sc. (dei XL), (3) 11 (1898), 267–252 (also in Luigi Bianchi, Opere, Vol. IX, Edizioni Cremonese, Roma, 1958). JFM29.0415.01
  3. [BTV] J. Berndt, F. Tricerri, L. Vanhecke, “Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces”, LNM 1598, Springer-Verlag, 1995. Zbl0818.53067MR1340192
  4. [C] E. Cartan, “Leçons sur la Géométrie des Espaces de Riemann”, Gauthier-Villars, Paris, 1951. Zbl0044.18401MR44878
  5. [DZ] J. E. D’Atri, W. Ziller, Naturally reductive metrics and Einstein metrics on compact Lie groups. Mem. Amer. Math. Soc. 215 (1979). Zbl0404.53044MR519928
  6. [G] C. S. Gordon, Naturally reductive homogeneous Riemannian manifolds, Canad. J. Math. 37 (1985), 467–487. Zbl0554.53035MR787113
  7. [H] S. Helgason, “Differential Geometry, Lie Groups and Symmetric Spaces”, GSM 34, AMS, Providence, 2001. Zbl0993.53002MR1834454
  8. [HI] S. Halverscheid, A. Iannuzzi, A family of adapted complexifications for noncompact semisimple Lie groups, ArXiv: math.CV/0503377. Zbl1180.53053
  9. [KN] S. Kobayashi, K. Nomizu, “Foundations of Differential Geometry”, Vol. II., Interscience, New York, 1969. Zbl0119.37502MR238225
  10. [M1] J. Milnor“Morse Theory”, Annals of Math. Studies 51, Princeton University Press, Princeton, N.J. 1963. Zbl0108.10401MR163331
  11. [M2] J. Milnor, Curvature of left invariant metrics on Lie groups, Adv. Math. 21 (1976), 293–329. Zbl0341.53030MR425012
  12. [O’N] B. O’Neill, “Semi-Riemannian Geometry”, Academic Press, 1983. Zbl0531.53051MR719023
  13. [V] G. Vranceanu, “Leçons de Géométrie Différentielle”, Vol. I, Ed. Acad. Rp. Pop. Roumaine, 1957. Zbl0119.17102MR124823
  14. [W] J. A. Wolf, “Spaces of Constant Curvature”, New York: McGraw-Hill, 1967. Zbl0162.53304MR217740

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