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Ricci curvature in nLab

Contents

Context

Riemannian geometry

Riemannian geometry

Basic definitions

Further concepts

Theorems

Poincaré conjecture-theorem

Applications

gravity
- Einstein-Hilbert action, Einstein equations, supergravity

Idea

Formally, the Ricci curvature RicRic of a Riemannian manifold is a symmetric rank-2 tensor obtained by contraction from the Riemann curvature. Geometrically one may think Ric(v,w)Ric(v, w) as the first order approximation of the infinitesimal behavior of the surface spanned by vectors vv and ww. This is made explicit by the following formula for the volume element around some point

dμ g=[1−16Ric jkx jx k+O(|x| 3)]dμ Euclidean 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=−12∑ j,kg jk∂ j∂ kg lm+Q lm(g,∇g), Ric_{lm} \;=\; -\frac{1}{2} \sum_{j,k} g^{j k} \partial_j \partial_k g_{lm} + Q_{lm}(g, \nabla g) \,,

where g jkg^{j k} denotes the inverse of the metric tensor and Q lmQ_{lm} is a quadratic form in ∇g\nabla g with coefficients that are rational expressions in which numerators are polynomials gg and the denominator depends only on detg\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 kk derivatives, the regularity theory for quasilinear PDEs gives k+2k+2-regularity of the metric tensor. But the Einstein property g=λRicg = \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

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

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]