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Harmonicity of normal almost contact metric structures

M. Benyounes, T. Levasseur, E. Loubeau, E. Vergara-Diaz

Abstract

We consider normal almost contact structures on a Riemannian manifold and, through their associated sections of an ad-hoc twistor bundle, study their harmonicity, as sections or as maps. We rewrite these harmonicity equations in terms of the curvature tensor and find conditions relating the harmonicity of the almost contact metric and almost complex structures of the total and base spaces of the Morimoto fibration. We apply these results to homogeneous principal circle bundles over generalised flag manifolds, in particular Aloff-Wallach spaces, to mass-produce harmonic almost contact metric structures.

Harmonicity of normal almost contact metric structures

Abstract

We consider normal almost contact structures on a Riemannian manifold and, through their associated sections of an ad-hoc twistor bundle, study their harmonicity, as sections or as maps. We rewrite these harmonicity equations in terms of the curvature tensor and find conditions relating the harmonicity of the almost contact metric and almost complex structures of the total and base spaces of the Morimoto fibration. We apply these results to homogeneous principal circle bundles over generalised flag manifolds, in particular Aloff-Wallach spaces, to mass-produce harmonic almost contact metric structures.
Paper Structure (6 sections, 37 theorems, 184 equations)

This paper contains 6 sections, 37 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Normal almost contact metric structures
  3. Curvature equations of harmonicity
  4. The Morimoto submersion
  5. Generalised flag manifolds
  6. Homogeneous principal bundles over a generalised flag manifold

Key Result

Theorem 1

Let $G$ be a compact Lie group, $G/K$ be a generalised flag manifold and denote by $C$ the centre of $K$. Let $\mathsf{Q}^+ \subset \mathsf{R}$ be an invariant ordering of a root system $\mathsf{R}$ and let $R_T^+$ be the set of restrictions of the elements of $\mathsf{Q}^+$ to ${\mathfrak{z}} = \op where ${\mathfrak{g}}=\operatorname {Lie}(G)$, ${\mathfrak{k}} = \operatorname {Lie}(K)$ and the ${

Theorems & Definitions (64)

  • Theorem 1
  • Theorem 2
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Corollary 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 54 more