Journal of Lie Theory 34 (2024), No. 4, 975--996
Copyright Heldermann Verlag 2024
Left Invariant Semi Riemannian Metrics on Quadratic Lie Groups
Shirley Bromberg
Departamento de Matemáticas, Universidad Autónoma Metropolitana, Iztapalapa, México D.F.
stbs@xanum.uam.mx
Alberto Medina
Institut A. Grothendieck, Université de Montpellier, France
alberto.medina@umontpellier.fr
Andrés Villabón
Escuela de Ciencias Básicas, Universidad Nacional Abierta y a Distancia, Medellín, Colombia
edgar.villabon@unad.edu.co
To determine the Lie groups that admit a flat (eventually geodesically complete) left invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of left invariant semi-Riemannian metrics on quadratic Lie groups i.e. Lie groups endowed with a bi-invariant semi-Riemannian metric. We give a useful necessary and sufficient condition that guarantees the flatness of a left invariant semi-Riemannian metric defined on a quadratic Lie group. All these semi-Riemannian metrics are complete. We show that there are no Riemannian flat left invariant metrics on non Abelian quadratic Lie groups. We study the Jacobi fields of any left invariant semi-Riemannian metric on a Lie group, using the notion of reflections. The case of Oscillator groups is addressed. This paper is a modification of a 2011 previous version due to the first two authors.
Keywords: Left invariant semi-Riemannian metrics, flat semi Riemannian metrics, geodesically complete manifolds, quadratic Lie groups, Jacobi fields.
MSC: 53B05, 70G45, 22E30.
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