Tohoku Mathematical Journal

Groupes de Lie pseudo-riemanniens plats

Anne Aubert and Alberto Medina

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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.

Article information

Tohoku Math. J. (2), Volume 55, Number 4 (2003), 487-506.

First available in Project Euclid: 11 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C50: Lorentz manifolds, manifolds with indefinite metrics
Secondary: 22E60: Lie algebras of Lie groups {For the algebraic theory of Lie algebras, see 17Bxx}

Flat pseudo-Riemannian Lie groups affine Lie groups geodesic completeness


Aubert, Anne; Medina, Alberto. Groupes de Lie pseudo-riemanniens plats. Tohoku Math. J. (2) 55 (2003), no. 4, 487--506. doi:10.2748/tmj/1113247126.

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