Tohoku Mathematical Journal
2003
December
SECOND SERIES VOL. 55, NO. 4

Tohoku Math. J.
55 (2003), 487-506

Title GROUPES DE LIE PSEUDO-RIEMANNIENS PLATS

Author Anne Aubert and Alberto Medina

(Received October 11, 2001, revised September 19, 2002)
Abstract. The determination of affine Lie groups (i.e., which carry a left-invariant affine structure) is an open problem ([12]). 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.

2000 Mathematics Subject Classification. Primary 53C50; Secondary 22E60.
Key words and phrases. Flat pseudo-Riemannian Lie groups, affine Lie groups, geodesic completeness.

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