Journal of Lie Theory

no. 4
Left Invariant Semi Riemannian Metrics on Quadratic Lie Groups
Shirley Bromberg1; Alberto Medina2; Andrés Villabón3
1Departamento de Matemáticas, Universidad Autónoma Metropolitana, Iztapalapa, México D.F.
2Institut A. Grothendieck, Université de Montpellier, France
3Escuela de Ciencias Básicas, Universidad Nacional Abierta y a Distancia, Medellín, Colombia
Journal of Lie Theory, Volume 34 (2024) no. 4, pp. 975-996
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.
DOI: 10.5802/jolt.1368
Classification: 53B05, 70G45, 22E30
Keywords: Left invariant semi-Riemannian metrics, flat semi Riemannian metrics, geodesically complete manifolds, quadratic Lie groups, Jacobi fields

Shirley Bromberg  1 ; Alberto Medina  2 ; Andrés Villabón  3

1 Departamento de Matemáticas, Universidad Autónoma Metropolitana, Iztapalapa, México D.F.
2 Institut A. Grothendieck, Université de Montpellier, France
3 Escuela de Ciencias Básicas, Universidad Nacional Abierta y a Distancia, Medellín, Colombia 
Shirley Bromberg; Alberto Medina; Andrés Villabón. Left Invariant Semi Riemannian Metrics on Quadratic Lie Groups. Journal of Lie Theory, Volume 34 (2024) no. 4, pp. 975-996. doi: 10.5802/jolt.1368
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     journal = {Journal of Lie Theory},
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