Twists, Codazzi Tensors, and the $6$-sphere
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908868851793920 |
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| author | Pham, David N. |
| author_facet | Pham, David N. |
| contents | Let $(M,g,J,ω)$ be an almost Hermitian manifold. Given an automorphism $ψ\in \mathrm{Aut}(TM)$, the existing structure can be twisted to obtain a new almost Hermitian manifold $(M,g^ψ,J^ψ,ω^ψ)$. In the current paper, we study these $ψ$-twisted almost Hermitian structures with particular emphasis on questions regarding the integrability of $J^ψ$ and the Riemannian geometry of $g^ψ$. By studying the latter, we identity a certain class of $\mathrm{Aut}(TM)$ with nice transformation properties. We call these automorphisms $g$-\textit{Codazzi maps} because of their close relationship with Codazzi tensors. The aforementioned results are ultimately applied to the standard nearly Kähler structure on the $6$-sphere where we prove a nonintegrability result for the class of $g$-Codazzi maps. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_05790 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Twists, Codazzi Tensors, and the $6$-sphere Pham, David N. Differential Geometry 32Q60, 53C15, 53C55 Let $(M,g,J,ω)$ be an almost Hermitian manifold. Given an automorphism $ψ\in \mathrm{Aut}(TM)$, the existing structure can be twisted to obtain a new almost Hermitian manifold $(M,g^ψ,J^ψ,ω^ψ)$. In the current paper, we study these $ψ$-twisted almost Hermitian structures with particular emphasis on questions regarding the integrability of $J^ψ$ and the Riemannian geometry of $g^ψ$. By studying the latter, we identity a certain class of $\mathrm{Aut}(TM)$ with nice transformation properties. We call these automorphisms $g$-\textit{Codazzi maps} because of their close relationship with Codazzi tensors. The aforementioned results are ultimately applied to the standard nearly Kähler structure on the $6$-sphere where we prove a nonintegrability result for the class of $g$-Codazzi maps. |
| title | Twists, Codazzi Tensors, and the $6$-sphere |
| topic | Differential Geometry 32Q60, 53C15, 53C55 |
| url | https://arxiv.org/abs/2603.05790 |