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Homogeneous pseudo-Riemannian structures of metrics of Kaluza-Klein type on the three-dimensional anti-de Sitter spacetime

Fumihiro Ueno

Abstract

We classify homogeneous pseudo-Riemannian structures of a three-parameter family of metrics called Kaluza-Klein type on the three-dimensional anti-de Sitter spacetime with their induced groups of isometries and reductive decompositions. We also obtain the classification of homogeneous almost contact and paracontact metric structures of metrics of Kaluza-Klein type on the three-dimensional anti-de Sitter spacetime with their isometry groups and reductive decompositions.

Homogeneous pseudo-Riemannian structures of metrics of Kaluza-Klein type on the three-dimensional anti-de Sitter spacetime

Abstract

We classify homogeneous pseudo-Riemannian structures of a three-parameter family of metrics called Kaluza-Klein type on the three-dimensional anti-de Sitter spacetime with their induced groups of isometries and reductive decompositions. We also obtain the classification of homogeneous almost contact and paracontact metric structures of metrics of Kaluza-Klein type on the three-dimensional anti-de Sitter spacetime with their isometry groups and reductive decompositions.
Paper Structure (21 sections, 24 theorems, 112 equations, 6 tables)

This paper contains 21 sections, 24 theorems, 112 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homogeneous pseudo-Riemannian structures
  4. Pseudo-Riemannian g-natural metrics on unit sphere bundles
  5. Metrics of Kaluza-Klein type on $\operatorname{AdS}_3$
  6. Homogeneous pseudo-Riemannian structures on $(\mathbb{H}^3_1, g_{\lambda\mu\nu})$
  7. The case of $-\lambda\neq\mu,\mu\neq\nu, \nu \neq -\lambda$
  8. The case of $- \lambda \neq \mu = \nu$
  9. The case of $- \lambda = \nu \neq \mu$
  10. The case of $-\lambda=\mu\neq\nu$
  11. The case of $-\lambda=\mu=\nu$
  12. Homogeneous almost contact metric structures on $(\mathbb{H}^3_1, g_{\lambda\mu\nu})$
  13. The case of $- \lambda \neq \mu = \nu$
  14. The case of $-\lambda = \nu \neq \mu$
  15. The case of $-\lambda=\mu=\nu$
  16. ...and 6 more sections

Key Result

Theorem 2.5

tricerri1983homogeneous Let $(M,g)$ be a reductive homogeneous pseudo-Riemannian manifold, and let $G$ and $G'$ be connected Lie subgroups of its isometry group that act transitively on $M.$ Assume that the Lie algebra $\mathfrak{g}$(resp. $\mathfrak{g}'$) of $G$(resp. $G'$) has a reductive decompos

Theorems & Definitions (48)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.9
  • Theorem 2.10
  • ...and 38 more