In this talk, I will introduce the class of geodesically rich representations. These are representations of (real or complex) hyperbolic lattices that preserve a significant amount of the geometric structure of the associated quotient manifold. When the quotient manifold has robust geometric structure, these representations exhibit rigidity phenomena. In particular, a recent superrigidity theorem for rich representations was used to prove that finite-volume hyperbolic manifolds with infinitely many maximal totally geodesic submanifolds are arithmetic (Bader-Fisher-Miller-Stover). I will discuss a new superrigidity theorem for rich representations that efficiently recovers existing results and addresses target groups that were previously inaccessible.