Proyecciones Journal of Mathematics Vol. 31, N^o 1, pp. 51-63, March 2012. Universidad Católica del Norte Antofagasta - Chile

The signature in actions of semisimple Lie groups on pseudo-Riemannian manifolds

José Rosales-Ortega

Universidad de Costa Rica, Costa Rica

 

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ABSTRACT

We study the relationship between the signature of a semisimple Lie group and a pseudoRiemannian manifold on wich the group acts topologically transitively and isometrically. We also provide a description of the bi-invariant pseudo-Riemannian metrics on a semisimple Lie Group over R in terms of the complexification of the Lie algebra associated to the group, and then we utilize it to prove a remark of Gromov.

Keywords : semisimple Lie groups, bi-invariant metric, local freeness.

Subjclass : [2000] Primary: 53C05; Secondary: 53C10.

 

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REFERENCES

[1] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, (1978).         [ Links ]

[2] M. Gromov, Rigid transformations groups, Geometrie diffórentielle (Paris 1986), Travaux en Cours 33, Hermann, Paris, pp. 65—139,(1988).         [ Links ]

[3] B. O'neill, SEMI-RIEMANNIAN GEOMETRY, Academic Press, New York, (1983).         [ Links ]

[4] J. Rosales-Ortega, The Gromov's Centralizer theorem for semisimple Lie group actions. Ph.D. Thesis, CINVESTAV-IPN, (2005).         [ Links ]

[5] J. Szaro, Isotropy of semisimple group actions on manifolds with geometric structures, Amer. J. Math.120, pp. 129—158, (1998).         [ Links ]

[6] R. J. Zimmer, Ergodic Theory and Semisimple Lie Groups, Birkhauser, Boston, (1984).

Jose Rosales-Ortega

Department of Mathematics

ITCR

Universidad de Costa Rica e Instituto Tecnológico de Costa Rica San Jose Cartago, Costa Rica

e-mail : jrosales@itcr.ac.cr jose.rosales@ucr.ac.cr

Received : September 2011. Accepted : November 2011