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Lorentzian connections with parallel twistor-free torsion

Igor Ernst, Anton S. Galaev

Abstract

We describe Lorentzian manifolds that admit metric connections with parallel torsion having zero twistorial component and non-zero vectorial component. We also describe Lorentzian manifolds admitting metric connections with closed parallel skew-symmetric torsion.

Lorentzian connections with parallel twistor-free torsion

Abstract

We describe Lorentzian manifolds that admit metric connections with parallel torsion having zero twistorial component and non-zero vectorial component. We also describe Lorentzian manifolds admitting metric connections with closed parallel skew-symmetric torsion.
Paper Structure (8 sections, 13 theorems, 135 equations)

This paper contains 8 sections, 13 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Proof of Theorem \ref{['Th_nonis']}
  5. Proof of Theorem \ref{['ThLordT=0']}
  6. Proof of Theorem \ref{['Th_isotropic_nondeg_dim=3']}
  7. Proof of Theorems \ref{['Th_isotropic_deg']} and \ref{['Th_isotropic_nondeg']}
  8. Proof of Theorem \ref{['ThComplg']}

Key Result

Lemma 1

Let $(M,g)$ be a pseudo-Riemannian manifold and $\nabla$ a metric connection with skew-symmetric torsion $\tau$ on it. Then the following conditions are equivalent

Theorems & Definitions (16)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Example 1
  • Theorem 5
  • Theorem 6
  • Remark 1
  • Example 2
  • ...and 6 more