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

TL;DR

This paper introduces and studies weak contact metric structures, generalizing classical contact and K-contact geometries, and explores conditions under which these structures relate to Einstein metrics and Ricci solitons.

Contribution

It characterizes weak K-contact manifolds within weak contact metric structures and establishes conditions linking Ricci solitons and Einstein metrics in this context.

Findings

Weak K-contact manifolds are characterized among weak contact metric manifolds.

Conditions are identified under which these manifolds are Einstein.

The study connects Ricci solitons with Einstein metrics in weak contact geometry.

Abstract

We study metric structures on a smooth manifold (introduced in our recent works and called a weak contact metric structure and a weak K-structure) which generalize the metric contact and K-contact structures, and allow a new look at the classical theory. First, we characterize weak K-contact manifolds among all weak contact metric manifolds by the property well known for K-contact manifolds, and find when a Riemannian manifold endowed with a unit Killing vector field forms a weak K-contact structure. Second, we find sufficient conditions for a weak K-contact manifold with parallel Ricci tensor or with a generalized Ricci soliton structure to be an Einstein manifold.