Ricci curvature

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In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n-space. Like the metric itself, the Ricci tensor is a symmetric bilinear form on the tangent space of the Riemannian manifold. Roughly speaking, the Ricci tensor is a measure of volume distortion; that is, it encapsulates the degree to which n-dimensional volumes of regions in the given n-dimensional manifold differ from the volumes of comparable regions in Euclidean n-space. This is made more precise in the direct geometric interpretation section below.

1 Formal definition
2 Direct geometric meaning
3 Applications of the Ricci curvature tensor
4 Global geometry/topology and Ricci curvature
5 Behavior under conformal rescaling
6 See also
7 References

[edit] Formal definition

Suppose that $(M, g)$ is an n-dimensional Riemannian manifold, and let $T p M$ denote the tangent space of M at p. For any pair $\xi, \eta\in T_pM$ of tangent vectors at p, the Ricci tensor $Ric(ξ,η)$ evaluated at $(ξ,η)$ is defined to be the trace of the linear map $T_pM\to T_pM$ given by

$\zeta \mapsto R(\zeta,\eta) \xi$

where R is the Riemann curvature tensor. In local coordinates (using the summation convention), one has

$\operatorname{Ric} = R_{ij}\,dx^i \otimes dx^j$

where

$R_{ij} = {R^k}_{ikj}.$

Because the Levi-Civita connection is torsion-free, the Ricci tensor of a Riemannian manifold is symmetric, in the sense that

$\operatorname{Ric}(\xi ,\eta) = \operatorname{Ric}(\eta ,\xi)$

It thus follows that the Ricci tensor is completely determined by knowing the quantity $\operatorname{Ric} (\xi , \xi )$ for all vectors $ξ$ of unit length. This function on the set of unit tangent vectors is often simply called the Ricci curvature, since knowing it is equivalent to knowing the Ricci curvature tensor.

The Ricci curvature is determined by the sectional curvatures of a Riemannian manifold, but contains less information. Indeed, if $ξ$ is a vector of unit length on a Riemannian n-manifold, then $\operatorname{Ric} (\xi , \xi)$ is precisely (n−1) times the average value of the sectional curvature, taken over all the 2-planes containing $ξ$ . (There is an (n−2)-dimensional family of such 2-planes.)

If the Ricci curvature function $\operatorname{Ric} (\xi , \xi )$ is constant on the set of unit tangent vectors $ξ$ , the Riemannian manifold is said to have constant Ricci curvature, or to be an Einstein manifold. This happens if and only if the Ricci tensor $\operatorname{Ric}$ is a constant multiple of the metric tensor $g$ .

In dimensions 2 and 3 Ricci curvature algebraically determines the entire curvature tensor, but in higher dimensions Ricci curvature contains less information. For instance, Einstein manifolds do not have to have constant curvature in dimensions 4 and up.

An explicit expression for the Ricci tensor in terms of the Levi-Civita connection is given in the article on Christoffel symbols.

[edit] Direct geometric meaning

Near any point p in a Riemannian manifold $(M, g)$ , one can define preferred local coordinates, called geodesic normal coordinates. These are adapted to the metric such that geodesics from p corresponds to straight lines. Furthermore, at p, the metric tensor is Euclidean. That is, $g i j (p) = δ i j$ . In these specific coordinates, the metric volume form has the following Taylor expansion at p ^{[citation needed]}:

$d\mu_g = \Big[ 1 - \frac{1}{6}R_{jk}x^jx^k+ O(|x|^3)] d\mu_{{\rm Euclidean}}$

Thus, if the Ricci curvature $\operatorname{Ric} (\xi , \xi )$ is positive in the direction of a vector $ξ$ , the conical region in $M$ swept out by a tightly focused family of short geodesic segments emanating from p and roughly pointing in the direction of $ξ$ will have smaller volume than the corresponding conical region in Euclidean space. Similarly, if the Ricci curvature is negative in the direction of a given vector $ξ$ , such a conical region in the manifold will instead have larger volume than it would in Euclidean space.

[edit] Applications of the Ricci curvature tensor

Ricci curvature plays an important role in general relativity, where it is the key term in the Einstein field equations.

Ricci curvature also appears in the Ricci flow equation, where a time-dependent Riemannian metric is deformed in the direction of minus its Ricci curvature. This system of partial differential equations is a non-linear analog of the heat equation, and was first introduced by Richard Hamilton in the early 1980s. Since heat tends to spread through a solid until the body reaches an equilibrium state of constant temperature, Ricci flow may be hoped to produce an equilibrium geometry for a manifold for which the Ricci curvature is constant. Recent contributions to the subject due to Grigori Perelman now seem to show that this program works well enough in dimension three to lead to a complete classification of compact 3-manifolds, along lines first conjectured by William Thurston in the 1970s.

On a Kaehler manifold, the Ricci curvature determines the first Chern class of the manifold (mod torsion). However, the Ricci curvature has no analogous topological interpretation on a generic Riemannian manifold.

[edit] Global geometry/topology and Ricci curvature

Here is a short list of global results concerning manifolds with positive Ricci curvature.

Myers' Theorem states that if the Ricci curvature is bounded from below on a complete Riemannian manifold by $\left(n-1\right)k > 0 \,\!$ , then the manifold has diameter $\le \pi/\sqrt{k}$ , with equality only if the manifold is isometric to a sphere of a constant curvature k. By a covering-space argument, it follows that any compact manifold of positive Ricci curvature must have finite fundamental group.
The Bishop-Gromov inequality states that if a complete m-dimensional Riemannian manifold has non-negative Ricci curvature, then the volume of a ball is smaller or equal to the volume of a ball of the same radius in Euclidean m-space. More over if $v p (R)$ denotes the volume of the ball with center p and radius $R$ in the manifold and $V (R) = c m R m$ denotes the volume of the ball of radius R in Euclidean m-space then function $v p (R) / V (R)$ is nonincreasing. (The last inequality can be generalized to arbitrary curvature bound and is the key point in the proof of Gromov's compactness theorem.)
The Cheeger-Gromoll Splitting theorem states that if a complete Riemannian manifold with $\operatorname{Ric} \ge 0$ contains a line, meaning a geodesic γ such that $d (γ(u),γ(v)) = | u - v |$ for all $v,u\in\mathbb{R}$ , then it is isometric to a product space $\mathbb{R}\times L$ . Consequently, a complete manifold of positive Ricci curvature can have at most one topological end.

These results show that positive Ricci curvature has strong topological consequences. By contrast, excluding the case of surfaces, negative Ricci curvature is now known to have no topological implications; Joachim Lohkamp has shown (Annals of Mathematics, 1994) that any manifold of dimension greater than two admits a metric of negative Ricci curvature. (For surfaces, negative Ricci curvature implies negative sectional curvature; but the point is that this fails rather dramatically in all higher dimensions.)

[edit] Behavior under conformal rescaling

If you change the metric g by multiplying it by a conformal factor $e 2 f$ , the Ricci tensor of the new, conformally related metric $\tilde{g}= e^{2f}g$ is given by

$\tilde{\operatorname{Ric}}=\operatorname{Ric}+(2-n)[ \nabla df-df\otimes df]+[\Delta f -(n-2)\|df\|^2]g ,$

where $Δ = d * d$ is the geometric Laplacian. In particular, given a point p in a Riemannian manifold, it is always possible to find metrics conformal to the given metric g for which the Ricci tensor vanishes at p. Note, however, that this is only point-wise assertion; it is usually impossible to make the Ricci curvature vanish identically on the entire manifold by a conformal rescaling.

[edit] See also

[edit] References

A.L. Besse, Einstein manifolds, Springer (1987)
L.A. Sidorov, "Ricci tensor" SpringerLink Encyclopaedia of Mathematics (2001)
L.A. Sidorov, "Ricci curvature" SpringerLink Encyclopaedia of Mathematics (2001)
G. Ricci, Atti R. Inst. Venelo , 53 : 2 (1903–1904) pp. 1233–1239
L.P. Eisenhart, Riemannian geometry , Princeton Univ. Press (1949)
S. Kobayashi, K. Nomizu, Foundations of differential geometry , 1 , Interscience (1963)