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2003 Groupes de Lie pseudo-riemanniens plats
Anne Aubert, Alberto Medina
Tohoku Math. J. (2) 55(4): 487-506 (2003). DOI: 10.2748/tmj/1113247126


The determination of affine Lie groups (i.e., which carry a left-invariant affine structure) is an open problem. In this work we begin the study of Lie groups with a left-invariant, flat pseudo-Riemannian metric (flat pseudo-Riemannian groups). We show that in such groups the left-invariant affine structure defined by the Levi-Civita connection is geodesically complete if and only if the group is unimodular. We also show that the cotangent manifold of an affine Lie group is endowed with an affine Lie group structure and a left-invariant, flat hyperbolic metric. We describe a double extension process which allows us to construct all nilpotent, flat Lorentzian groups. We give examples and prove that the only Heisenberg group which carries a left invariant, flat pseudo-Riemannian metric is the three dimensional one.


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Anne Aubert. Alberto Medina. "Groupes de Lie pseudo-riemanniens plats." Tohoku Math. J. (2) 55 (4) 487 - 506, 2003.


Published: 2003
First available in Project Euclid: 11 April 2005

zbMATH: 1058.53055
MathSciNet: MR2017221
Digital Object Identifier: 10.2748/tmj/1113247126

Primary: 53C50
Secondary: 22E60

Rights: Copyright © 2003 Tohoku University


Vol.55 • No. 4 • 2003
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