Recurrent Curvature

Recurrent curvature is a tensorial property of a Riemannian manifold in which the curvature tensor, when parallel-transported along any closed loop, returns to itself multiplied by a scalar factor rather than being unchanged. Arthur Walker provided the foundational classification of such spaces, now called Walker spaces, showing that recurrent curvature imposes strong constraints on the manifold's holonomy and global topology. The property arises when a manifold possesses a parallel recurrent vector field, and it generalizes the better-known condition of constant curvature. Walker spaces have found applications in string theory and the study of spacetimes with special holonomy, where geometric constraints substitute for dynamical assumptions.