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Holonomy in pseudo-Hermitian geometry

Anton S. Galaev, Thomas Leistner, Felipe Leitner

TL;DR

The paper provides a comprehensive holonomy classification for pseudo-Hermitian sub-Riemannian spaces, establishing a dichotomy between the horizontal Schouten holonomy and the adapted/Wagner holonomies under torsion constraints and linking these to Riemannian holonomy. It analyzes the Tanaka–Webster connection, obtains explicit holonomy lists for the torsionful case, and shows that local symmetry emerges precisely in the $ rak{so}(m)$-holonomy scenario; it also characterizes locally sub-symmetric and pseudo-Einstein structures and connects to CR geometry via deformations. The work yields a complete taxonomy of holonomy algebras, derives consequences for sub-Riemannian symmetric spaces, and provides concrete geometric models and spinor-related consequences, including parallel horizontal spinors. Overall, it advances the understanding of CR/pseudo-Hermitian holonomy, offering explicit classifications, geometric constructions, and links to locally homogeneous and symmetric sub-Riemannian spaces.

Abstract

We study the holonomy that is associated to a sub-Riemannian structure defined on the kernel of a global contact form. This includes the holonomy of Schouten's horizontal connection as well as of the adapted connection, both canonical invariants of the structure. Under a condition on the torsion of the structure, we show that they are either equal or that the former is a codimension one normal subgroup of the latter. Furthermore, we establish a close relation to Riemannian holonomy, which yields a complete holonomy classification in the torsion-free case. For the main result we focus on the special case of pseudo-Hermitian structures and give a classification of holonomy algebras for both the Schouten and the adapted connection. Based on this, we derive a classification of symmetric sub-Riemannian structures and of of those holonomy groups that admit parallel spinors. Finally we exhibit a relation between locally symmetric sub-Riemannian contact structures and locally homogeneous Riemannian structures.

Holonomy in pseudo-Hermitian geometry

TL;DR

The paper provides a comprehensive holonomy classification for pseudo-Hermitian sub-Riemannian spaces, establishing a dichotomy between the horizontal Schouten holonomy and the adapted/Wagner holonomies under torsion constraints and linking these to Riemannian holonomy. It analyzes the Tanaka–Webster connection, obtains explicit holonomy lists for the torsionful case, and shows that local symmetry emerges precisely in the -holonomy scenario; it also characterizes locally sub-symmetric and pseudo-Einstein structures and connects to CR geometry via deformations. The work yields a complete taxonomy of holonomy algebras, derives consequences for sub-Riemannian symmetric spaces, and provides concrete geometric models and spinor-related consequences, including parallel horizontal spinors. Overall, it advances the understanding of CR/pseudo-Hermitian holonomy, offering explicit classifications, geometric constructions, and links to locally homogeneous and symmetric sub-Riemannian spaces.

Abstract

We study the holonomy that is associated to a sub-Riemannian structure defined on the kernel of a global contact form. This includes the holonomy of Schouten's horizontal connection as well as of the adapted connection, both canonical invariants of the structure. Under a condition on the torsion of the structure, we show that they are either equal or that the former is a codimension one normal subgroup of the latter. Furthermore, we establish a close relation to Riemannian holonomy, which yields a complete holonomy classification in the torsion-free case. For the main result we focus on the special case of pseudo-Hermitian structures and give a classification of holonomy algebras for both the Schouten and the adapted connection. Based on this, we derive a classification of symmetric sub-Riemannian structures and of of those holonomy groups that admit parallel spinors. Finally we exhibit a relation between locally symmetric sub-Riemannian contact structures and locally homogeneous Riemannian structures.

Paper Structure

This paper contains 30 sections, 36 theorems, 310 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries on sub-Riemannian contact structures
  3. The Schouten connection and its curvature
  4. Extended connections and the adapted connection
  5. The Wagner connection
  6. Holonomy
  7. Holonomy of the Schouten connection
  8. The connection $\nabla^\theta$
  9. The group $\widetilde{\operatorname{Hol}}_x(\nabla^N)$
  10. The group $\widehat{\operatorname{Hol}}_x(\nabla^N)$.
  11. A Codazzi condition
  12. A further analysis of the Codazzi case
  13. Holonomy of K-contact sub-Riemannian manifolds
  14. Pseudo-Hermitian structures
  15. Sub-Riemannian symmetric spaces
  16. ...and 15 more sections

Key Result

Lemma 2.1

The curvature tensor of the adapted connection $\nabla^{\boldsymbol{\tau}}$ satisfies In particular,

Theorems & Definitions (62)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • proof
  • ...and 52 more