Ricci tensor

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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. It can be associated to any affine connection; it does not require a metric or pseudometric.

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 Trace-free Ricci tensor
7 See also
8 References

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

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

That is

$R_{\sigma\nu} = {R^\rho}_{\sigma\rho\nu} = {\Gamma^\rho_{\nu\sigma}}_{,\rho} - \Gamma^\rho_{\rho\sigma ,\nu} + \Gamma^\rho_{\rho\lambda} \Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\rho\sigma} .$

As a consequence of the Bianchi identities, 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 $g$ .

If the Ricci curvature function $\operatorname{Ric} (\xi , \xi )$ is positive (nonnegative) on the set of unit tangent vectors $ξ$ , the Riemannian manifold is said to have positive (nonnegative) Ricci curvature. See below for topological consequences.

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 List of formulas in Riemannian geometry. It is valid in pseudo-Riemannian geometry as well.

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 through p corresponds to straight lines through the origin, in such a manner that the geodesic distance from p corresponds to the Euclidean distance from the origin. In these coordinates, the metric tensor is nicely approximated by the Euclidean metric, in the precise sense that

g i j = δ i j + O ( | x | 2).

In these coordinates, the metric volume form then has the following Taylor expansion at p:

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

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.

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 Kähler 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.

Global geometry/topology and Ricci curvature

Here is a short list of global results concerning manifolds with positive Ricci curvature; see also classical theorems of Riemannian geometry.

Briefly, positive Ricci curvature has strong topological consequences, while (for dimension at least 3), negative Ricci curvature has no topological implications.

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. Moreover, 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.)

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^{[citation needed]}

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

where $Δ = (d * + d) 2$ is the geometric Laplacian.

If we let $F = e - f$ , then this can be rewritten as

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

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.

For two dimensional manifolds, the above formula shows that if f is a harmonic function, then the conformal scaling $g\mapsto e^{2f} g$ does not change the Ricci curvature.

Trace-free Ricci tensor

In Riemannian geometry and general relativity, the trace-free Ricci tensor of a pseudo-Riemannian manifold $(M, g)$ is the tensor defined by

$Z =\operatorname{Ric}- \frac{S}{n}g$

where "Ric" is the Ricci tensor, "S" is the scalar curvature, "g" is the metric tensor, and n is the dimension of M. The name of this object reflects the fact that its trace automatically vanishes:

$Z_{ab}g^{ab}=\, 0$

If n $\geq$ 3, the trace-free Ricci tensor vanishes identically if and only if

R i c = λ g

for some constant $λ$ . In mathematics, this is the condition for $(M, g)$ to be an Einstein manifold. In physics, this equation states that $(M, g)$ is a solution of Einstein's vacuum field equations with cosmological constant.

References

A.L. Besse, Einstein manifolds, Springer (1987)
L.A. Sidorov (2001), "Ricci tensor", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
L.A. Sidorov (2001), "Ricci curvature", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
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)
Z. Shen ,C. Sormani "The Topology of Open Manifolds with Nonnegative Ricci Curvature" (a survey)[1]
G. Wei, "Manifolds with A Lower Ricci Curvature Bound" (a survey)[2]

Different notions of curvature defined in differential geometry

Differential geometry of curves	Curvature · Torsion of curve · Frenet-Serret formulas · Radius of curvature (applications) · Affine curvature · Total curvature

Differential geometry of surfaces	Principal curvatures · Gaussian curvature · Mean curvature · Darboux frame · Gauss–Codazzi equations · First fundamental form · Second fundamental form

Riemannian geometry	Curvature of Riemannian manifolds · Riemann curvature tensor · Ricci curvature · Scalar curvature · Sectional curvature

Curvature of connections	Curvature form · Torsion tensor · Cocurvature · Holonomy