# nLab Ricci curvature

Contents

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## Idea

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

$d\mu _{g}=\left[1-{\tfrac {1}{6}}Ric_{jk}x^{j}x^{k}+O\left(|x|^{3}\right)\right]d\mu _{Euclidean}$

(Einstein summation convention). A spacetime with vanishing Ricci curvature is also called Ricci flat.

## Properties

### Harmonic coordinate representation and regularity

By a trick of Lanczos, that was recovered by DeTurck and Kazdan, in harmonic coordinates the Ricci tensor can be expressed as

$Ric_{lm} = -\frac{1}{2} \sum_{j,k} g^{jk} \partial_j \partial_k g_{lm} + Q_{lm}(g, \nabla g)$

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.

### Cheeger-Gromoll theorem

• 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

• Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. (1990)
• Cheeger and Naber, Lower bounds on Ricci curvature and quantitative behavior of singular sets Invent. Math. (2013)

• Anderson and Cheeger, $C^\alpha$-compactness for manifolds with Ricci curvature and injectivity radius bounded below J. Diff. Geo. (1992)