Riemann solitons on perfect fluid within $f(r,T^2)$-gravity

This work is devoted to the study of Riemann solitons in the setting of the energy-momentum squared gravity framework, expressed as $f(r,T^2)$, regarded as a deformation of Einstein's general relativity. Our attention is placed on the particular model $f(r,T^2)=r+\lambda T^2$, coupled with a perfect fluid, where the dynamics naturally admit Riemann solitons. Within the steady formulation of such solitons, we obtain the corresponding fluid relation of state under $f(r,T^2)$-gravity. Moreover, by exploiting the solitonic structure, we further analyze the admissibility of energy conditions, the emergence of black hole geometries, and the manifestation of singularities in the presence of a perfect fluid for this modified gravitational scenario.