Einstein-type metrics and generalized Ricci solitons on weak $f$-K-contact manifolds
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866916144128983040 |
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| author | Rovenski, Vladimir |
| author_facet | Rovenski, Vladimir |
| contents | A weak metric $f$-structure $(f,Q,ξ_i,η^i,g)\ (i=1,\ldots,s)$, generalizes the metric $f$-structure on a smooth manifold, i.e., the complex structure on the contact distribution is replaced with a nonsingular skew-symmetric tensor. We study geometry of a weak $f$-K-contact structure, which is a weak $f$-contact structure, whose characteristic vector fields are Killing. We show that $\ker f$ of a weak $f$-contact manifold defines a $\mathfrak{g}$-foliation with an abelian Lie algebra. Then we characterize weak $f$-K-contact manifolds among all weak metric $f$-manifolds by the property known for $f$-K-contact manifolds, and find when a Riemannian manifold endowed with a set of orthonormal Killing vector fields is a weak $f$-K-contact manifold. We show that for $s>1$, an Einstein weak $f$-K-contact manifold is Ricci flat, then find sufficient conditions for a weak $f$-K-contact manifold with parallel Ricci tensor or with a generalized gradient Ricci soliton structure to be Ricci flat or a quasi Einstein manifold. We prove positive definiteness of the Jacobi operators in the characteristic directions and use this to deform a weak $f$-K-contact structure to an $f$-K-contact structure. We define an $η$-Ricci soliton and $η$-Einstein structures on a weak metric $f$-manifold (which for $s=1$, give the well-known structures on contact metric manifolds) and find sufficient conditions for a compact weak $f$-K-contact manifold with an $η$-Ricci soliton structure of constant scalar curvature to be $η$-Einstein. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_07102 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Einstein-type metrics and generalized Ricci solitons on weak $f$-K-contact manifolds Rovenski, Vladimir Differential Geometry A weak metric $f$-structure $(f,Q,ξ_i,η^i,g)\ (i=1,\ldots,s)$, generalizes the metric $f$-structure on a smooth manifold, i.e., the complex structure on the contact distribution is replaced with a nonsingular skew-symmetric tensor. We study geometry of a weak $f$-K-contact structure, which is a weak $f$-contact structure, whose characteristic vector fields are Killing. We show that $\ker f$ of a weak $f$-contact manifold defines a $\mathfrak{g}$-foliation with an abelian Lie algebra. Then we characterize weak $f$-K-contact manifolds among all weak metric $f$-manifolds by the property known for $f$-K-contact manifolds, and find when a Riemannian manifold endowed with a set of orthonormal Killing vector fields is a weak $f$-K-contact manifold. We show that for $s>1$, an Einstein weak $f$-K-contact manifold is Ricci flat, then find sufficient conditions for a weak $f$-K-contact manifold with parallel Ricci tensor or with a generalized gradient Ricci soliton structure to be Ricci flat or a quasi Einstein manifold. We prove positive definiteness of the Jacobi operators in the characteristic directions and use this to deform a weak $f$-K-contact structure to an $f$-K-contact structure. We define an $η$-Ricci soliton and $η$-Einstein structures on a weak metric $f$-manifold (which for $s=1$, give the well-known structures on contact metric manifolds) and find sufficient conditions for a compact weak $f$-K-contact manifold with an $η$-Ricci soliton structure of constant scalar curvature to be $η$-Einstein. |
| title | Einstein-type metrics and generalized Ricci solitons on weak $f$-K-contact manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2306.07102 |