Impact of Ambient Conformal Vector Fields on Yamabe Solitons on Riemannian Hypersurfaces

We investigate Yamabe solitons on Riemannian hypersurfaces induced by conformal vector fields in Riemannian and Lorentzian manifolds, with an emphasis on the tangential component. We show that these hypersurfaces are totally umbilical, and when the ambient manifold is Einstein, a rigidity condition emerges connecting the mean and scalar curvatures. Using this, we classify compact Yamabe solitons: each hypersurface is either totally geodesic or an extrinsic sphere. Additionally, we prove the non-existence of trivial Yamabe solitons on oriented hypersurfaces of higher dimension in Einstein manifolds. These results highlight the classification of compact hypersurfaces and rigidity phenomena in the ambient spaces, providing a clear understanding of the geometric structures associated with Yamabe solitons.