Rigidity and structural constraints for $\rho$-Einstein solitons on twisted warped product manifolds

We investigate the geometric structure of $\rho$-Einstein solitons on twisted warped product manifolds, establishing fundamental rigidity phenomena and structural constraints. Our main result proves that the existence of a $\rho$-Einstein soliton on a twisted warped product $M_1{\times }_f{M}_2$ with $\dim (M_2)\ge 2$ forces the warping function to be multiplicatively separable, $f(x_1,x_2)={\varphi }_1(x_1)\cdot {\varphi }_2(x_2)$, thereby excluding genuinely twisted structures. We derive complete decomposition formulas showing how the soliton equation separates into base and fiber components, with the mixed component imposing severe restrictions. For gradient solitons with separated potentials, we prove additional constraints linking the warping function to the geometry of factor manifolds. Applications to generalized Robertson–Walker and static space-times demonstrate the physical significance of these results. Our findings reveal an inherent geometric obstruction: non-separable warping functions are incompatible with $\rho$-Einstein soliton structures, suggesting deep connections between soliton geometry and product manifold topology.