Plane wave limits of Riemannian manifolds

Abstract

Utilizing the covariant formulation of Penrose's plane wave limit by Blau et al., we construct for any Riemannian metric $g$ a family of "plane wave limits" of one higher dimension. These limits are taken along geodesics of $g$, yield simpler metrics of Lorentzian signature, and are isometric invariants. They can also be seen to arise locally from a suitable expansion of $g$ in Fermi coordinates, and they directly encode much of $g$'s geometry. For example, normal Jacobi fields of $g$ are encoded as geodesics of its plane wave limits. Furthermore, $g$ will have constant sectional curvature if and only if each of its plane wave limits is locally conformally flat. In fact $g$ will be flat, or Ricci-flat, or geodesically complete, if and only if all of its plane wave limits are, respectively, the same. Many other curvature properties are preserved in the limit, including certain inequalities, such as signed Ricci curvature.