A new approach to the classification of almost contact metric manifolds via intrinsic endomorphisms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912832434470912 |
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| author | Agricola, Ilka Di Pinto, Dario Dileo, Giulia Kuhrt, Marius |
| author_facet | Agricola, Ilka Di Pinto, Dario Dileo, Giulia Kuhrt, Marius |
| contents | In 1990, D. Chinea and C. Gonzalez gave a classification of almost contact metric manifolds into $2^{12}$ classes, based on the behaviour of the covariant derivative $\nabla^gΦ$ of the fundamental $2$-form $Φ$. This large number makes it difficult to deal with this class of manifolds. We propose a new approach to almost contact metric manifolds by introducing two intrinsic endomorphisms $S$ and $h$, which bear their name from the fact that they are, basically, the entities appearing in the intrinsic torsion. We present a new classification scheme for them by providing a simple flowchart based on algebraic conditions involving $S$ and $h$, which then naturally leads to a regrouping of the Chinea-Gonzalez classes, and, in each step, to a further refinement, eventually ending in the single classes. This method allows a more natural exposition and derivation of both known and new results, like a new characterization of almost contact metric manifolds admitting a characteristic connection in terms of intrinsic endomorphisms. We also describe in detail the remarkable (and still very large) subclass of $\mathcal{H}$-parallel almost contact manifolds, defined by the condition $(\nabla^g_XΦ)(Y,Z)=0$ for all horizontal vector fields, $X,Y,Z\in\mathcal{H}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_16900 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new approach to the classification of almost contact metric manifolds via intrinsic endomorphisms Agricola, Ilka Di Pinto, Dario Dileo, Giulia Kuhrt, Marius Differential Geometry 53C15, 53C10, 53D15, 53C25 In 1990, D. Chinea and C. Gonzalez gave a classification of almost contact metric manifolds into $2^{12}$ classes, based on the behaviour of the covariant derivative $\nabla^gΦ$ of the fundamental $2$-form $Φ$. This large number makes it difficult to deal with this class of manifolds. We propose a new approach to almost contact metric manifolds by introducing two intrinsic endomorphisms $S$ and $h$, which bear their name from the fact that they are, basically, the entities appearing in the intrinsic torsion. We present a new classification scheme for them by providing a simple flowchart based on algebraic conditions involving $S$ and $h$, which then naturally leads to a regrouping of the Chinea-Gonzalez classes, and, in each step, to a further refinement, eventually ending in the single classes. This method allows a more natural exposition and derivation of both known and new results, like a new characterization of almost contact metric manifolds admitting a characteristic connection in terms of intrinsic endomorphisms. We also describe in detail the remarkable (and still very large) subclass of $\mathcal{H}$-parallel almost contact manifolds, defined by the condition $(\nabla^g_XΦ)(Y,Z)=0$ for all horizontal vector fields, $X,Y,Z\in\mathcal{H}$. |
| title | A new approach to the classification of almost contact metric manifolds via intrinsic endomorphisms |
| topic | Differential Geometry 53C15, 53C10, 53D15, 53C25 |
| url | https://arxiv.org/abs/2504.16900 |