Geometry of δ-almost gradient Yamabe solitons on pseudo-Riemannian manifolds
Rajdip Biswas, Santu Dey, Arindam Bhattacharyya
TL;DR
The paper investigates δ-almost Yamabe solitons on para-contact and related pseudo-Riemannian manifolds, focusing on the interaction between the soliton data and underlying paracontact geometry. By analyzing the soliton equation $\frac{\delta}{2}\mathfrak{L}_{Z}g=(r-\lambda)g$ across $K$-paracontact, para-Sasakian, $(k,\mu)$-paracontact, and para-cosymplectic settings, it derives rigidity results for the potential vector field $Z$ (e.g., when $Z$ is an infinitesimal contact transformation, or parallel to the Reeb field $\xi$) and connects scalar curvature relations to soliton parameters. The work establishes that in several cases $Z$ must be aligned with $\xi$, yielding conditions under which the manifold is a δ-Yamabe soliton or has vanishing scalar curvature, and it provides two concrete examples illustrating δ-almost Yamabe solitons on $K$-paracontact manifolds. These results advance the understanding of self-similar Yamabe-type flows in para-contact geometry and furnish explicit models for further study.
Abstract
In this article, we studied δ-almost Yamabe solitons within the framework of para- contact metric manifolds. First, we proved that for a paracontact metric manifold {M}, if a paracontact metric g represents a δ-almost Yamabe soliton associated with the potential vector field {Z} being an infinitesimal contact transformation, then {Z} is Killing and if the potential vector field {Z} is collinear with ξ, then the manifold {M} is {K}-paracontact. Next, if we take a {K}-paracontact metric mani- fold admitting δ-almost Yamabe soliton with the potential vector field {Z} parallel to the characteristic vector field and with constant scalar curvature then either scalar curvature will vanish or {g} becomes a δ-Yamabe soliton under a certain condition. We established some results on {K}-paracontact manifold admitting δ-almost gradient Yamabe soliton. Moreover, we consider a (k, μ)-paracontact metric manifold admitting a non-trivial δ-almost gradient Yamabe soliton. We shown that the potential vector field Z is parallel to ξ. We have also discussed about δ-almost gradient Yamabe soliton on the para-Sasakian manifold. Finally, we consider a para-cosymplectic manifold with a δ-almost Yamabe soliton. In the end, we construct two examples of K-paracontact metric manifolds with δ-almost Yamabe soliton.