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| Ngā kaituhi matua: | , |
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| Hōputu: | Preprint |
| I whakaputaina: |
2025
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| Ngā marau: | |
| Urunga tuihono: | https://arxiv.org/abs/2508.08021 |
| Ngā Tūtohu: |
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| _version_ | 1866916891267694592 |
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| author | Zlatanović, Milan Rovenski, Vladimir |
| author_facet | Zlatanović, Milan Rovenski, Vladimir |
| contents | Linear connections satisfying the Einstein metricity condition are important in the study of generalized Riemannian manifolds $(M,G=g+F)$, where the symmetric part $g$ of $G$ is a non-degenerate $(0,2)$-tensor, and $F$ is the skew-symmetric part. Such structures naturally arise in spacetime models in theoretical physics, where $F$ can be defined as an almost complex or almost contact metric (a.c.m.) structure. In the paper, we first study more general models, where $F$ has constant rank and is based on weak metric structures (introduced by the second author and R.~Wolak), which generalize almost complex and a.c.m. structures. We consider linear connections with totally skew-symmetric torsion that satisfy both the Einstein metricity condition and the $A$-torsion condition, where $A$ is a skew-symmetric (1,1)-tensor adjoint to~$F$. In the almost Hermitian case, we prove that the manifold with such a connection is weak nearly K\" ahler, the torsion is completely determined by the exterior derivative of the fundamental 2-form and the Nijenhuis tensor, and the structure tensors are parallel, while in the weak a.c.m. case, the contact distribution is involutive, the Reeb vector field is Levi-Civita parallel, and the structure tensors are also parallel with respect to both connections.
For rank$(F)=\dim M$, we apply weak almost Hermitian structures to fundamental results (by the first author and S. Ivanov) on generalized Riemannian manifolds and prove that the manifold equipped with an Einstein's connection is a weighted product of several nearly Kähler manifolds.
For~rank$(F)<\dim M$ we apply weak almost Hermitian and weak a.c.m. structures and obtain splitting results for generalized Riemannian manifolds equipped with Einstein's connections. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08021 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Applications of Weak Metric Structures to Non-Symmetrical Gravitational Theory Zlatanović, Milan Rovenski, Vladimir Differential Geometry Mathematical Physics 53C15, 53C25, 53B05 Linear connections satisfying the Einstein metricity condition are important in the study of generalized Riemannian manifolds $(M,G=g+F)$, where the symmetric part $g$ of $G$ is a non-degenerate $(0,2)$-tensor, and $F$ is the skew-symmetric part. Such structures naturally arise in spacetime models in theoretical physics, where $F$ can be defined as an almost complex or almost contact metric (a.c.m.) structure. In the paper, we first study more general models, where $F$ has constant rank and is based on weak metric structures (introduced by the second author and R.~Wolak), which generalize almost complex and a.c.m. structures. We consider linear connections with totally skew-symmetric torsion that satisfy both the Einstein metricity condition and the $A$-torsion condition, where $A$ is a skew-symmetric (1,1)-tensor adjoint to~$F$. In the almost Hermitian case, we prove that the manifold with such a connection is weak nearly K\" ahler, the torsion is completely determined by the exterior derivative of the fundamental 2-form and the Nijenhuis tensor, and the structure tensors are parallel, while in the weak a.c.m. case, the contact distribution is involutive, the Reeb vector field is Levi-Civita parallel, and the structure tensors are also parallel with respect to both connections. For rank$(F)=\dim M$, we apply weak almost Hermitian structures to fundamental results (by the first author and S. Ivanov) on generalized Riemannian manifolds and prove that the manifold equipped with an Einstein's connection is a weighted product of several nearly Kähler manifolds. For~rank$(F)<\dim M$ we apply weak almost Hermitian and weak a.c.m. structures and obtain splitting results for generalized Riemannian manifolds equipped with Einstein's connections. |
| title | Applications of Weak Metric Structures to Non-Symmetrical Gravitational Theory |
| topic | Differential Geometry Mathematical Physics 53C15, 53C25, 53B05 |
| url | https://arxiv.org/abs/2508.08021 |