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
(where we are using the Einstein summation convention).
A spacetime with vanishing Ricci curvature is also called Ricci flat.
By a trick of Lanczos 1922, rediscovered by DeTurck & Kazdan 1981, in harmonic coordinates the Ricci tensor can be expressed as
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.
See at Cheeger-Gromoll theorem.
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
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
curvature in Riemannian geometry |
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Riemann curvature |
Ricci curvature |
scalar curvature |
sectional curvature |
p-curvature |
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]
See also:
On Lanczos’s trick:
Lanczos, Ein vereinfachendes Koordinatensystem für die Einsteinschen Gravitationsgleichungen, Phys. Z. 23 (1922) 537-539
Dennis DeTurck, Jerry Kazdan, Some regularity theorems in Riemannian geometry, Annales scientifiques de l’École Normale Supérieure, Serie 4, Volume 14 (1981) no. 3, pp. 249-260 [numdam:ASENS_1981_4_14_3_249_0]
On regularity results:
Michael T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990) 429-445 [doi:10.1007/BF01233434, pdf]
Jeff Cheeger, Aaron Naber: Lower bounds on Ricci curvature and quantitative behavior of singular sets, Invent. Math. 191 (2013) 321–339 [arXiv:1103.1819, doi:10.1007/s00222-012-0394-3]
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 July 30, 2024 at 13:34:42. See the history of this page for a list of all contributions to it.