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Submanifolds of almost quaternionic skew-Hermitian manifolds

Ioannis Chrysikos, Jan Gregorovič

TL;DR

The paper addresses the geometry of submanifolds in almost quaternionic skew-Hermitian manifolds, formulating a unified framework via $\mathsf{SO}^*(2n)\mathsf{Sp}(1)$-structures and analyzing both intrinsic torsion and extrinsic geometry. It develops a complete extrinsic setup for almost symplectic and almost complex submanifolds, introduces compatibility and integrability criteria expressed through induced connections and torsion projections, and provides a torsion-free theory paralleling submanifold results in quaternionic Kähler settings. The results yield explicit homogeneous submanifold realizations in semisimple qs-H symmetric spaces, including Lagrangian and pseudo-Kähler-type examples, thereby connecting intrinsic torsion classifications with concrete geometric models. Overall, the work advances the understanding of submanifold geometry in the skew-Hermitian quaternionic framework and offers tools for constructing and recognizing torsion-free phenomena in ambient symmetric spaces.

Abstract

We investigate several classes of submanifolds of almost quaternionic skew-Hermitian manifolds $(M^{4n}, Q, ω)$, including almost symplectic, almost complex, almost pseudo-Hermitian and almost quaternionic submanifolds. In the torsion-free case, we realize each type of submanifold considered in the theoretical part by constructing explicit examples of submanifolds of semisimple quaternionic skew-Hermitian symmetric spaces.

Submanifolds of almost quaternionic skew-Hermitian manifolds

TL;DR

The paper addresses the geometry of submanifolds in almost quaternionic skew-Hermitian manifolds, formulating a unified framework via -structures and analyzing both intrinsic torsion and extrinsic geometry. It develops a complete extrinsic setup for almost symplectic and almost complex submanifolds, introduces compatibility and integrability criteria expressed through induced connections and torsion projections, and provides a torsion-free theory paralleling submanifold results in quaternionic Kähler settings. The results yield explicit homogeneous submanifold realizations in semisimple qs-H symmetric spaces, including Lagrangian and pseudo-Kähler-type examples, thereby connecting intrinsic torsion classifications with concrete geometric models. Overall, the work advances the understanding of submanifold geometry in the skew-Hermitian quaternionic framework and offers tools for constructing and recognizing torsion-free phenomena in ambient symmetric spaces.

Abstract

We investigate several classes of submanifolds of almost quaternionic skew-Hermitian manifolds , including almost symplectic, almost complex, almost pseudo-Hermitian and almost quaternionic submanifolds. In the torsion-free case, we realize each type of submanifold considered in the theoretical part by constructing explicit examples of submanifolds of semisimple quaternionic skew-Hermitian symmetric spaces.
Paper Structure (21 sections, 30 theorems, 103 equations, 1 table)

This paper contains 21 sections, 30 theorems, 103 equations, 1 table.

Table of Contents

  1. Introduction and preliminary facts
  2. Motivation and summary
  3. A review of $\mathop{\mathrm{\mathsf{SO}}}\nolimits^*(2n)$-structures and $\mathop{\mathrm{\mathsf{SO}}}\nolimits^*(2n)\mathop{\mathrm{\mathsf{Sp}}}\nolimits(1)$-structures
  4. Adapted connections and integrability conditions
  5. On the integrability of an element in an admissible basis
  6. Submanifolds of almost quaternionic skew-Hermitian manifolds
  7. Almost symplectic submanifolds
  8. Extrinsic geometry of almost symplectic submanifolds
  9. Almost symplectic submanifolds of almost qs-H manifolds
  10. Almost complex submanifolds
  11. Compatible almost complex submanifolds
  12. Remarks regarding the torsion-free case and the $\mathcal{X}_{345}$-class
  13. Almost pseudo-Hermitian submanifolds
  14. Compatible almost pseudo-Hermitian submanifolds
  15. Integrability conditions in terms of the induced connection
  16. ...and 6 more sections

Key Result

Theorem 1.6

(CGWPartICGWPartII) (1) On an almost hypercomplex skew-Hermitian manifold $(M, H, \omega)$ the adapted connection '$\nabla^{H, \omega}$ is given by $\nabla^{H, \omega}=\nabla^{H}+A$, where $\nabla^{H}$ is the Obata connection associated to $H$, and $A$ is the $(1, 2)$-tensor field on $M$ defined by The connection $\nabla^{H, \omega}$ is torsion-free if and only if In other words, $T^{H, \omega}=

Theorems & Definitions (75)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Lemma 1.8
  • proof
  • Theorem 1.9
  • ...and 65 more