nLab Ricci curvature

Contents

Context

Riemannian geometry

Idea
Properties

Harmonic coordinate representation and regularity
Cheeger-Gromoll theorem

Examples
Related concepts
References

Idea

Formally, the 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 vectors $v$ 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}

(where we are using the 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 1922, rediscovered by DeTurck & Kazdan 1981, in harmonic coordinates the Ricci tensor can be expressed as

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

where $g^{j k}$ 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 is especially useful in two ways:

There are theorems that give bounds on the regularity of the metric tensor in harmonic coordinates under geometric assumptions (Anderson 1990, Anderson & Cheeger 1992, Cheeger & Naber 2013).
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

See at Cheeger-Gromoll theorem.

Examples

Example

For $n \in \mathbb{N}_{\gt 0}$ and $r \in \mathbb{R}_{\gt 0}$ , the Ricci tensor of the round $n$ -sphere $S^n$ of radius $r$ satisfies

Ric(v,v) \;=\; \frac{n-1}{r^2}

for all unit-length tangent vectors $v \in T S^n$ , ${\vert v \vert} = 1$ .

Accordingly, the scalar curvature of the round $n$ -sphere of radius $r$ is the constant function with value

\mathrm{R} \;=\; \frac{n(n-1)}{r^2} \,.

(e.g. Lee 2018, Cor. 11.20)

Related concepts

curvature in Riemannian geometry
Riemann curvature
Ricci curvature
scalar curvature
sectional curvature
p-curvature

References

See most references listed at Riemannian geometry, for instance:

John M. Lee, Riemannian manifolds. An introduction to curvature, Graduate Texts in Mathematics 176 Springer (1997) [ISBN: 0-387-98271-X]

second edition (retitled):

John M. Lee, Introduction to Riemannian Manifolds, Springer (2018) [ISBN:978-3-319-91754-2, doi:10.1007/978-3-319-91755-9]