Formally, Ricci curvature of a Riemannian manifold is a symmetric rank-2 tensor obtained by contraction from the Riemann curvature. Geometrically one may think as the first order approximation of the infinitesimal behavior of the surface spanned by and . This is made explicit by the following formula for the volume element around some point
(Einstein summation convention). A spacetime with vanishing Ricci curvature is also called Ricci flat.
By a trick of Lanczos, that was recovered by DeTurck and Kazdan, in harmonic coordinates the Ricci tensor can be expressed as
where denotes the inverse of the metric tensor and is a quadratic form in with coefficients that are rational expressions in which numerators are polynomials and the denominator depends only on . Note that this formula describes the metric tensor as a quasilinear elliptic PDE. This representation is especially useful in two ways: First, there are theorems that give bounds on the regularity of the metric tensor in harmonic coordinates under geometric assumptions (Anderson, Cheeger, and Naber). Second, as this expression is a quasilinear elliptic PDE, one can conclude on regularity bounds for the metric tensor from regularity estimates for the Ricci tensor. This argument allows for a regularity bootstrap in case of Einstein manifolds: given a rough Einstein metric with derivatives, the regularity theory for quasilinear PDEs gives -regularity of the metric tensor. But the Einstein property implies the same regularity for the Ricci tensor. Hence one can apply the argument again and add infinitum.
curvature in Riemannian geometry |
---|
Riemann curvature |
Ricci curvature |
scalar curvature |
sectional curvature |
p-curvature |
Wikipedia, Ricci curvature
Lanczos, Ein vereinfachendes Koordinatensystem für die Einsteinschen Gravitationsgleichungen Phys. Z. 23, 537-539 (1922)
DeTurck and Kazdan, Some regularity theorems in Riemannian geometry Ann. scient. Éc. Norm. Sup. (1981)
For regularity result see
For weaker but more general regularity results see also:
A conjecture that all compact Ricci flat manifolds either have special holonomy or else are “unstable”:
Last revised on May 2, 2024 at 13:07:13. See the history of this page for a list of all contributions to it.