Persistent AI agents face a fundamental problem: between sessions, their factual memory can be restored, but not their acquired personality. We introduce cognitive curvature: the set of persistent behavioral deformations in an agent induced by its past experiences. By analogy with spacetime curvature in general relativity, an agent's experiences deform its decisional geometry. We present a concrete implementation—a deformation profile file (CURVATURE.md)—and discuss implications for designing AI agents with identity continuity. Also available in French: Courbure cognitive : la déformation persistante du comportement des agents IA par l'expérience