Holonomy of the Obata connection on 2-step hypercomplex nilmanifolds
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918385342742528 |
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| author | Andrada, Adrián Barberis, María Laura Brienza, Beatrice |
| author_facet | Andrada, Adrián Barberis, María Laura Brienza, Beatrice |
| contents | We study the holonomy of the Obata connection on 2-step hypercomplex nilmanifolds. By explicitly computing the curvature tensor, we determine the conditions under which the Obata connection is flat, showing that this depends on the nilpotency step of each complex structure. In particular, we show that for 2-step hypercomplex nilmanifolds the holonomy algebra of the Obata connection is always an abelian subalgebra of $\mathfrak{sl}(n, \mathbb{H})$ and we prove that the $\mathbb{H}$-solvable conjecture holds in this case. Furthermore, we provide new examples of $k$-step nilpotent hypercomplex nilmanifolds, with arbitrary $k$, which are not Obata flat. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07889 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Holonomy of the Obata connection on 2-step hypercomplex nilmanifolds Andrada, Adrián Barberis, María Laura Brienza, Beatrice Differential Geometry Rings and Algebras 53C26, 22E25, 53C29, 53C55, 17B05 We study the holonomy of the Obata connection on 2-step hypercomplex nilmanifolds. By explicitly computing the curvature tensor, we determine the conditions under which the Obata connection is flat, showing that this depends on the nilpotency step of each complex structure. In particular, we show that for 2-step hypercomplex nilmanifolds the holonomy algebra of the Obata connection is always an abelian subalgebra of $\mathfrak{sl}(n, \mathbb{H})$ and we prove that the $\mathbb{H}$-solvable conjecture holds in this case. Furthermore, we provide new examples of $k$-step nilpotent hypercomplex nilmanifolds, with arbitrary $k$, which are not Obata flat. |
| title | Holonomy of the Obata connection on 2-step hypercomplex nilmanifolds |
| topic | Differential Geometry Rings and Algebras 53C26, 22E25, 53C29, 53C55, 17B05 |
| url | https://arxiv.org/abs/2508.07889 |