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Homogeneous 8-manifolds admitting invariant Spin(7)-structures

Dmitri Alekseevsky, Ioannis Chrysikos, Anna Fino, Alberto Raffero

Abstract

We study compact, simply connected, homogeneous 8-manifolds admitting invariant Spin(7)-structures, classifying all canonical presentations G/H of such spaces, with G simply connected. For each presentation, we exhibit explicit examples of invariant Spin(7)-structures and we describe their type, according to Fernández classification. Finally, we analyse the associated Spin(7)-connection with torsion.

Homogeneous 8-manifolds admitting invariant Spin(7)-structures

Abstract

We study compact, simply connected, homogeneous 8-manifolds admitting invariant Spin(7)-structures, classifying all canonical presentations G/H of such spaces, with G simply connected. For each presentation, we exhibit explicit examples of invariant Spin(7)-structures and we describe their type, according to Fernández classification. Finally, we analyse the associated Spin(7)-connection with torsion.

Paper Structure

This paper contains 14 sections, 24 theorems, 112 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Invariant $\mathop{\mathrm{\mathsf{Spin}}}\nolimits(7)$-structures on homogeneous spaces
  4. Simply connected 8-dimensional homogeneous spaces
  5. Proof of Theorem \ref{['MainTheoremA']}
  6. Proof of Theorem \ref{['MainTheoremB']}
  7. The infinite family $C_{k, \ell, m}$
  8. The Calabi-Eckmann manifold ${(\mathop{\mathrm{\mathsf{SU}}}\nolimits(3)/\mathop{\mathrm{\mathsf{SU}}}\nolimits(2))\times\mathop{\mathrm{\mathsf{SU}}}\nolimits(2)}$
  9. The space $(\mathop{\mathrm{\mathsf{SU}}}\nolimits(2) \times \mathop{\mathrm{\mathsf{SU}}}\nolimits(2) \times \mathop{\mathrm{\mathsf{SU}}}\nolimits(2))/\Delta(\mathop{\mathrm{\mathsf{SU}}}\nolimits(2)))\times(\mathop{\mathrm{\mathsf{SU}}}\nolimits(2)/\mathop{\mathrm{\mathsf{U}}}\nolimits(1))$
  10. Explicit examples of invariant $\mathop{\mathrm{\mathsf{Spin}}}\nolimits(7)$-structures
  11. The infinite family $C_{k, \ell, m}$
  12. The Calabi-Eckmann manifold $(\mathop{\mathrm{\mathsf{SU}}}\nolimits(3)/\mathop{\mathrm{\mathsf{SU}}}\nolimits(2))\times\mathop{\mathrm{\mathsf{SU}}}\nolimits(2)$
  13. Presentations for $\mathop{\mathrm{\mathbb{S}}}\nolimits^3\times\mathop{\mathrm{\mathbb{S}}}\nolimits^3\times\mathop{\mathrm{\mathbb{S}}}\nolimits^2$ not included in the family $C_{k, \ell, m}$.
  14. Details on the infinite family $C_{k, \ell, m}$

Key Result

Theorem A

Theorems & Definitions (50)

  • Theorem A
  • Theorem B
  • Remark 2.1
  • Proposition 2.2: Law
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 4.1
  • Proposition 4.2
  • Example 4.3
  • ...and 40 more