Holonomy in pseudo-Hermitian geometry
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912676461936640 |
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| author | Galaev, Anton S. Leistner, Thomas Leitner, Felipe |
| author_facet | Galaev, Anton S. Leistner, Thomas Leitner, Felipe |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_25598 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Holonomy in pseudo-Hermitian geometry Galaev, Anton S. Leistner, Thomas Leitner, Felipe Differential Geometry 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. |
| title | Holonomy in pseudo-Hermitian geometry |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2510.25598 |