Generalized Ricci solitons and Einstein metrics on weak $ K $-contact manifolds

Vladimir Rovenski; Vladimir Rovenski

doi:10.3934/cam.2023010

Communications in Analysis and Mechanics

2023, Volume 15, Issue 2: 177-188. doi: 10.3934/cam.2023010

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Generalized Ricci solitons and Einstein metrics on weak $ K $-contact manifolds

Vladimir Rovenski ^,

Department of Mathematics, University of Haifa, Mount Carmel, 3498838 Haifa, Israel

Received: 09 January 2023 Revised: 03 April 2023 Accepted: 27 April 2023 Published: 06 May 2023
53C15, 53C25, 53D15

We study so-called "weak" metric structures on a smooth manifold, which generalize the metric contact and $ K $-contact structures and allow a new look at the classical theory. We characterize weak $ K $-contact manifolds among all weak contact metric manifolds using the property well known for $ K $-contact manifolds, as well as find when a Riemannian manifold endowed with a unit Killing vector field is a weak $ K $-contact manifold. We also find sufficient conditions for a weak $ K $-contact manifold with a parallel Ricci tensor or with a generalized Ricci soliton structure to be an Einstein manifold.
- weak $ K $-contact manifold,
- unit Killing vector field,
- generalized Ricci soliton,
- Einstein metric,
- curvature
Citation: Vladimir Rovenski. Generalized Ricci solitons and Einstein metrics on weak $ K $-contact manifolds[J]. Communications in Analysis and Mechanics, 2023, 15(2): 177-188. doi: 10.3934/cam.2023010

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Abstract

We study so-called "weak" metric structures on a smooth manifold, which generalize the metric contact and $ K $-contact structures and allow a new look at the classical theory. We characterize weak $ K $-contact manifolds among all weak contact metric manifolds using the property well known for $ K $-contact manifolds, as well as find when a Riemannian manifold endowed with a unit Killing vector field is a weak $ K $-contact manifold. We also find sufficient conditions for a weak $ K $-contact manifold with a parallel Ricci tensor or with a generalized Ricci soliton structure to be an Einstein manifold.

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