Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. As such, it 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. The Ricci tensor is defined on any pseudo-Riemannian manifold, as a trace of the Riemann curvature tensor. Like the metric itself, the Ricci tensor is a symmetric bilinear form on the tangent space of the manifold (Besse 1987, p. 43).

In relativity theory, the Ricci tensor is the part of the curvature of space-time that determines the degree to which matter will tend to converge or diverge in time (via the Raychaudhuri equation). It is related to the matter content of the universe by means of the Einstein field equation. In differential geometry, lower bounds on the Ricci tensor on a Riemannian manifold allow one to extract global geometric and topological information by comparison (cf. comparison theorem) with the geometry of a constant curvature space form. If the Ricci tensor satisfies the vacuum Einstein equation, then the manifold is an Einstein manifold, which have been extensively studied (cf. Besse 1987). In this connection, the Ricci flow equation governs the evolution of a given metric to an Einstein metric; the precise manner in which this occurs ultimately leads to the solution of the Poincaré conjecture.

Definition

Suppose that $(M,g)$ is an n-dimensional Riemannian manifold, equipped with its Levi-Civita connection $\nabla$. The Riemannian curvature tensor of $M$ is the $(1,3)$ tensor defined by

$R(X,Y)Z = \nabla_X\nabla_Y Z - \nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z$

on vector fields $X,Y,Z$. Let $T_pM$ denote the tangent space of M at a point p. For any pair of tangent vectors $\xi$ and $\eta$ in $T_pM$, the Ricci tensor $\operatorname{Ric}$ evaluated at $(\xi, \eta )$ is defined to be the trace of the linear map $T_pM\to T_pM$ given by

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

In local coordinates (using the Einstein summation convention), one has

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

where

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

In terms of the Riemann curvature tensor and the Christoffel symbols, one has

$R_{\alpha\beta} = {R^\rho}_{\alpha\rho\beta} = \partial_{\rho}{\Gamma^\rho_{\beta\alpha}} - \partial_{\beta}\Gamma^\rho_{\rho\alpha} + \Gamma^\rho_{\rho\lambda} \Gamma^\lambda_{\beta\alpha} - \Gamma^\rho_{\beta\lambda}\Gamma^\lambda_{\rho\alpha} = 2 \Gamma^{\rho}_{{\alpha[\beta,\rho]}} + 2 \Gamma^\rho_{\lambda [\rho} \Gamma^\lambda_{\beta]\alpha} .$

Properties

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 $\xi$ 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 generally contains less information. Indeed, if $\xi$ is a vector of unit length on a Riemannian n-manifold, then Ric(ξ, ξ) is precisely (n − 1) times the average value of the sectional curvature, taken over all the 2-planes containing $\xi$. There is an (n−2)-dimensional family of such 2-planes, and so only in dimensions 2 and 3 does the Ricci tensor determine the full curvature tensor. A notable exception is when the manifold is given a priori as a hypersurface of Euclidean space. The second fundamental form, which determines the full curvature via the Gauss–Codazzi equation, is itself determined by the Ricci tensor and the principal directions of the hypersurface are also the eigendirections of the Ricci tensor. The tensor was introduced by Ricci for this reason.

If the Ricci curvature function Ric(ξ, ξ) 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 Ric is a constant multiple of the metric tensor g.

The Ricci curvature is usefully thought of as a multiple of the Laplacian of the metric tensor (Chow & Knopf 2004, Lemma 3.32). Specifically, in harmonic local coordinates the components satisfy

$R_{ij} = -\frac{1}{2}\Delta (g_{ij}) + \text{lower order terms} ,$

where Δ is the Laplace–Beltrami operator regarded here as acting on the functions g_ij. This fact motivates, for instance, the introduction of the Ricci flow equation as a natural extension of the heat equation for the metric. Alternatively, in a normal coordinate system based at p, at the point p

$R_{ij} = -\frac{3}{2}\Delta (g_{ij}).$

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 so that geodesics through p correspond 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 well-approximated by the Euclidean metric, in the precise sense that

$g_{ij} = \delta_{ij} + O (|x|^2) .$

In fact, by taking the Taylor expansion of the metric applied to a Jacobi field along a radial geodesic in the normal coordinate system, one has

$g_{ij} = \delta_{ij} - \frac{1}{3}R_{ikj\ell}x^kx^\ell + O(|x|^3) .$

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

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

which follows by expanding the square root of the determinant of the metric.

Thus, if the Ricci curvature Ric(ξ, ξ) 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 with initial velocity inside a small cone around ξ will have smaller volume than the corresponding conical region in Euclidean space, just as the surface of a small spherical wedge has lesser area than a corresponding circular sector. 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.

The Ricci curvature is essentially an average of curvatures in the planes including ξ. Thus if a cone emitted with an initially circular (or spherical) cross-section becomes distorted into an ellipse (ellipsoid), it is possible for the volume distortion to vanish if the distortions along the principal axes counteract one another. The Ricci curvature would then vanish along ξ. In physical applications, the presence of a nonvanishing sectional curvature does not necessarily indicate the presence of any mass locally; if an initially circular cross-section of a cone of world-lines later becomes elliptical, without changing its volume, then this is due to tidal effects from a mass at some other location.

Applications

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 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 and topology

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 of a Riemannian manifold has strong topological consequences, while (for dimension at least 3), negative Ricci curvature has no topological implications. (The Ricci curvature is said to be positive if the Ricci curvature function Ric(ξ,ξ) is positive on the set of non-zero tangent vectors ξ.) Some results are also known for pseudo-Riemannian manifolds.

1. 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.
2. 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 less than 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.)
3. 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 $d(\gamma(u),\gamma(v))=|u-v|$ 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. The theorem is also true under some additional hypotheses for complete Lorentzian manifolds (of metric signature (+−−...)) with non-negative Ricci tensor (Galloway 2000).

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; Lohkamp (1994) has shown that any manifold of dimension greater than two admits a Riemannian 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 $e^{2f}$, the Ricci tensor of the new, conformally related metric $\tilde{g}= e^{2f}g$ is given (Besse 1987, p. 59) by

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

where Δ = d^∗d is the (positive spectrum) Hodge Laplacian, i.e., the opposite of the usual trace of the Hessian.

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 pointwise 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 ↦ e^2fg 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 $\operatorname{Ric}$ 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

$\operatorname{Ric} = \lambda g$

for some constant $\lambda$. 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.

Kähler manifolds

On a Kähler manifold X, the Ricci curvature determines the curvature form of the canonical line bundle (Moroianu 2007, Chapter 12). The canonical line bundle is the top exterior power of the bundle of holomorphic Kähler differentials:

$\kappa = \wedge^n \Omega_X.$

The Levi-Civita connection corresponding to the metric on X gives rise to a connection on κ. The curvature of this connection is the two form defined by

$\rho(X,Y)\,\stackrel{\text{def}}{=}\,\operatorname{Ric}(JX,Y)$

where J is the complex structure map on the tangent bundle determined by the structure of the Kähler manifold. The Ricci form is a closed two-form. Its cohomology class is, up to a real constant factor, the first Chern class of the canonical bundle, and is therefore a topological invariant of X (for X compact) in the sense that it depends only on the topology of X and the homotopy class of the complex structure.

Conversely, the Ricci form determines the Ricci tensor by

$\operatorname{Ric}(X,Y) = \rho(X,JY).$

In local holomorphic coordinates z^α, the Ricci form is given by

$\rho = -i\partial\overline{\partial}\log\det(g_{\alpha\overline{\beta}})$

where $\partial$ is the Dolbeault operator and

$g_{\alpha\overline{\beta}} = g\left(\frac{\partial}{\partial z^\alpha},\frac{\partial}{\partial \overline{z}{}^\beta}\right).$

If the Ricci tensor vanishes, then the canonical bundle is flat, so the structure group can be locally reduced to a subgroup of the special linear group SL(n, C). However, Kähler manifolds already possess holonomy in U(n), and so the (restricted) holonomy of a Ricci-flat Kähler manifold is contained in SU(n). Conversely, if the (restricted) holonomy of a 2n-dimensional Riemannian manifold is contained in SU(n), then the manifold is a Ricci-flat Kähler manifold (Kobayashi & Nomizu 1996, IX, §4).

Generalization to affine connections

The Ricci tensor can also be generalized to arbitrary affine connections, where it is an invariant that plays an especially important role in the study of projective geometry (geometry associated to unparameterized geodesics) (Nomizu & Sasaki 1994). If $\nabla$ denotes an affine connection, then the curvature tensor $R$ is the $(1,3)$ tensor defined by

$R(X,Y)Z = \nabla_X\nabla_Y Z - \nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z$

for any vector fields $X,Y,Z$. The Ricci tensor is defined to be the trace:

$\operatorname{ric}(X,Y) = \operatorname{tr}(Z\mapsto R(Z,X)Y).$

In this more general situation, the Ricci tensor is symmetric if and only if there exist locally a parallel volume form for the connection.

See also

• Curvature of Riemannian manifolds
• Scalar curvature
• Ricci calculus
• Ricci decomposition
• Ricci-flat manifold
• Christoffel symbols
• Basic introduction to the mathematics of curved spacetime

References

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External links

• Z. Shen, C. Sormani "The Topology of Open Manifolds with Nonnegative Ricci Curvature" (a survey)
• G. Wei, "Manifolds with A Lower Ricci Curvature Bound" (a survey)