nLab
Ricci curvature

Contents

Contents

Idea

Formally, 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 vv and ww. This is made explicit by the following formula for the volume element around some point

dμ g=[116Ric 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}.

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=12 j,kg jk j kg lm+Q lm(g,g) Ric_{lm} = -\frac{1}{2} \sum_{j,k} g^{jk} \partial_j \partial_k g_{lm} + Q_{lm}(g, \nabla g)

where g jkg^{jk} 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 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 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 Cheeger-Gromoll theorem

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

References

  • 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)

For weaker but more general regularity results see also:

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

A conjecture that all compact Ricci flat manifolds either have special holonomy or else are “unstable”:

Last revised on June 26, 2019 at 11:41:47. See the history of this page for a list of all contributions to it.