Canonical connections attached to generalized quaternionic and para-quaternionic structures
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909714424528896 |
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| author | Blaga, Adara M. Nannicini, Antonella |
| author_facet | Blaga, Adara M. Nannicini, Antonella |
| contents | We put into light some generalized almost quaternionic and almost para-quaternionic structures and characterize their integrability with respect to a $\nabla$-bracket on the generalized tangent bundle $TM\oplus T^*M$ of a smooth manifold $M$, defined by an affine connection $\nabla$ on $M$. Also, we provide necessary and sufficient conditions for these structures to be $\hat \nabla$-parallel and $\hat \nabla^*$-parallel, where $\hat \nabla$ is an affine connection on $TM\oplus T^*M$ induced by $\nabla$, and $\hat\nabla^*$ is its generalized dual connection with respect to a bilinear form $\check h$ on $TM\oplus T^*M$ induced by a non-degenerate symmetric or skew-symmetric $(0,2)$-tensor field $h$ on $M$. As main results, we establish the existence of a canonical connection associated to a generalized quaternionic and to a generalized para-quaternionic structure, i.e., a torsion-free generalized affine connection that parallelizes these structures. We show that, in the quaternionic case, the canonical connection is the generalized Obata connection and that on a quasi-statistical manifold $(M,h,\nabla)$, an integrable $h$-symmetric and $\nabla$-parallel $(1,1)$-tensor field gives rise to a generalized para-quaternionic structure whose canonical connection is precisely $\hat \nabla^*$. Finally we prove that the generalized affine connection that parallelizes certain families of generalized almost complex and almost product structures is preserved. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_05239 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Canonical connections attached to generalized quaternionic and para-quaternionic structures Blaga, Adara M. Nannicini, Antonella Differential Geometry We put into light some generalized almost quaternionic and almost para-quaternionic structures and characterize their integrability with respect to a $\nabla$-bracket on the generalized tangent bundle $TM\oplus T^*M$ of a smooth manifold $M$, defined by an affine connection $\nabla$ on $M$. Also, we provide necessary and sufficient conditions for these structures to be $\hat \nabla$-parallel and $\hat \nabla^*$-parallel, where $\hat \nabla$ is an affine connection on $TM\oplus T^*M$ induced by $\nabla$, and $\hat\nabla^*$ is its generalized dual connection with respect to a bilinear form $\check h$ on $TM\oplus T^*M$ induced by a non-degenerate symmetric or skew-symmetric $(0,2)$-tensor field $h$ on $M$. As main results, we establish the existence of a canonical connection associated to a generalized quaternionic and to a generalized para-quaternionic structure, i.e., a torsion-free generalized affine connection that parallelizes these structures. We show that, in the quaternionic case, the canonical connection is the generalized Obata connection and that on a quasi-statistical manifold $(M,h,\nabla)$, an integrable $h$-symmetric and $\nabla$-parallel $(1,1)$-tensor field gives rise to a generalized para-quaternionic structure whose canonical connection is precisely $\hat \nabla^*$. Finally we prove that the generalized affine connection that parallelizes certain families of generalized almost complex and almost product structures is preserved. |
| title | Canonical connections attached to generalized quaternionic and para-quaternionic structures |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2302.05239 |