Formally, Ricci curvature $Ric$ of a Riemannian manifold is a symmetric rank-2 tensor obtained by contraction from the Riemann curvature. Geometrically one may think $Ric(v, w)$ as the first order approximation of the infinitesimal behavior of the surface spanned by $v$ and $w$. 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 $g^{jk}$ denotes the inverse of the metric tensor and $Q_{lm}$ is a quadratic form in $\nabla g$ with coefficients that are rational expressions in which numerators are polynomials $g$ and the denominator depends only on $\sqrt{\det g}$. 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 $k$ derivatives, the regularity theory for quasilinear PDEs gives $k+2$-regularity of the metric tensor. But the Einstein property $g = \lambda Ric$ implies the same regularity for the Ricci tensor. Hence one can apply the argument again and add infinitum.
curvature in Riemannian geometry |
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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.