Symmetry inheritance in Riemannian manifolds with physical applications

We study Riemannian manifolds, subject to a prescribed symmetry inheritance, defined by L=2, where , ga, and L are geometric/physical object, function, and Lie derivative operator with respect to a vector field . In this paper, we set =Riemann curvature tensor or Ricci tensor and obtain several new results relevant to physically significant material curves, proper conformai and proper nonconformal symmetries. In particular, we concentrate on a time-like Ricci inheritance vector parallel to the velocity vector of a perfect fluid spaced me. We claim new and physically relevant equations of state. All key results are supported by physical examples, including the Friedman-Robertson-Walker universe models. In general, this paper opens a new area of research on symmetry inheritance with a potential for further applications in mathematical physics.