Nijenhuis geometry of parallel tensors
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866911409567170560 |
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| author | Derdzinski, Andrzej Piccione, Paolo Terek, Ivo |
| author_facet | Derdzinski, Andrzej Piccione, Paolo Terek, Ivo |
| contents | A tensor -- meaning here a tensor field $Θ$ of any type $(p,q)$ on a manifold -- may be called integrable if it is parallel relative to some torsion-free connection. We provide analytical and geometric characterizations of integrability for differential $q$-forms, $q=0,1,2,n-1,n$ (in dimension $n$), vectors, bivectors, symmetric $(2,0)$ and $(0,2)$ tensors, as well as complex-diagonalizable and nilpotent tensors of type $(1,1)$. In most cases, integrability is equivalent to algebraic constancy of $Θ$ coupled with the vanishing of one or more suitably defined Nijenhuis-type tensors, depending on $Θ$ via a quasilinear first-order differential operator. For $(p,q)=(1,1)$, they include the ordinary Nijenhuis tensor. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_04539 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nijenhuis geometry of parallel tensors Derdzinski, Andrzej Piccione, Paolo Terek, Ivo Differential Geometry Primary 53C15, Secondary 53D17 A tensor -- meaning here a tensor field $Θ$ of any type $(p,q)$ on a manifold -- may be called integrable if it is parallel relative to some torsion-free connection. We provide analytical and geometric characterizations of integrability for differential $q$-forms, $q=0,1,2,n-1,n$ (in dimension $n$), vectors, bivectors, symmetric $(2,0)$ and $(0,2)$ tensors, as well as complex-diagonalizable and nilpotent tensors of type $(1,1)$. In most cases, integrability is equivalent to algebraic constancy of $Θ$ coupled with the vanishing of one or more suitably defined Nijenhuis-type tensors, depending on $Θ$ via a quasilinear first-order differential operator. For $(p,q)=(1,1)$, they include the ordinary Nijenhuis tensor. |
| title | Nijenhuis geometry of parallel tensors |
| topic | Differential Geometry Primary 53C15, Secondary 53D17 |
| url | https://arxiv.org/abs/2407.04539 |