In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, measures how a curved space locally differs from flat space by tracking how nearby geodesics spread apart or converge. Formally, it is a symmetric rank-two tensor obtained by taking a trace of the Riemann curvature tensor of a Riemannian or pseudo-Riemannian metric. In Riemannian geometry, the Ricci curvature in a given tangent direction may be interpreted as an average of the sectional curvatures of planes containing that direction. A further trace of the Ricci curvature yields the scalar curvature.
In general relativity, the Ricci curvature tensor enters the Einstein field equations through the Einstein tensor, formed from the Ricci tensor, the scalar curvature, and the metric. The Ricci tensor is also fundamental in Riemannian geometry and geometric analysis. Bounds on Ricci curvature imply strong global geometric and topological consequences, as in Myers's theorem and related comparison theorems.
In dimension three, the Ricci tensor determines the full Riemann curvature tensor, a simplification that is important in the study of three-dimensional geometry. Ricci curvature is the curvature term driving the Ricci flow, introduced by Richard S. Hamilton and used by Grigori Perelman in the proof of the Poincaré conjecture. More recently, lower bounds for Ricci curvature have been extended to certain metric-measure spaces, connecting the subject with optimal transport and Wasserstein geometry.
Definition Suppose that is an -dimensional Riemannian or pseudo-Riemannian manifold, equipped with its Levi-Civita connection . The Riemann curvature of is a map that takes smooth vector fields , , and , and returns the vector field on vector fields , , . Since is a tensor field, for each point , it gives rise to a (multilinear) map:
Define for each point the map by
That is, having fixed and , then for any orthonormal basis of the vector space , one has
It is a standard exercise of (multi)linear algebra to verify that this definition does not depend on the choice of the basis .
In abstract index notation,
Sign conventions. Note that some sources define to be what would here be called ; they would then define as . Although sign conventions differ about the Riemann tensor, they do not differ about the Ricci tensor.
Definition via local coordinates on a smooth manifold Let be a smooth Riemannian or pseudo-Riemannian -manifold. Given a smooth chart one then has functions and for each , which satisfy
for all . The latter shows that, expressed as matrices, . The functions are defined by evaluating on coordinate vector fields, while the functions are defined so that, as a matrix-valued function, they provide an inverse to the matrix-valued function .
Now define, for each , , , , and between 1 and , the functions
as maps .
Now let and be two smooth charts with . Let be the functions computed as above via the chart and let be the functions computed as above via the chart . Then one can check by a calculation with the chain rule and the product rule that
where is the first derivative along th direction of . This shows that the following definition does not depend on the choice of . For any , define a bilinear map by
where and are the components of the tangent vectors at in and relative to the coordinate vector fields of .
It is common to abbreviate the above formal presentation in the following style:
The final line includes the demonstration that the bilinear map Ric is well-defined, which is much easier to write out with the informal notation.
Comparison of the definitions The two above definitions are identical. The formulas defining and in the coordinate approach have an exact parallel in the formulas defining the Levi-Civita connection, and the Riemann curvature via the Levi-Civita connection. Arguably, the definitions directly using local coordinates are preferable, since the "crucial property" of the Riemann tensor mentioned above requires to be Hausdorff in order to hold. By contrast, the local coordinate approach only requires a smooth atlas. It is also somewhat easier to connect the "invariance" philosophy underlying the local approach with the methods of constructing more exotic geometric objects, such as spinor fields.
The complicated formula defining in the introductory section is the same as that in the following section. The only difference is that terms have been grouped so that it is easy to see that .
Properties As can be seen from the symmetries of the Riemann curvature tensor, the Ricci tensor of a Riemannian manifold is symmetric, in the sense that
for all .
It thus follows linear-algebraically that the Ricci tensor is completely determined by knowing the quantity for all vectors of unit length. This function on the set of unit tangent vectors is often also 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 is a vector of unit length on a Riemannian -manifold, then is precisely times the average value of the sectional curvature, taken over all the 2-planes containing . There is an -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.
As can be seen from the second Bianchi identity, one has
where is the scalar curvature, defined in local coordinates as . This is often called the contracted second Bianchi identity.
Direct geometric meaning Near any point in a Riemannian manifold , one can define preferred local coordinates, called geodesic normal coordinates. These are adapted to the metric so that geodesics through correspond to straight lines through the origin, in such a manner that the geodesic distance from 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
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
In these coordinates, the metric volume element then has the following expansion at :
which follows by expanding the square root of the determinant of the metric.
Thus, if the Ricci curvature is positive in the direction of a vector , the conical region in swept out by a tightly focused family of geodesic segments of length emanating from , with initial velocity inside a small cone about , will have smaller volume than the corresponding conical region in Euclidean space, at least provided that is sufficiently small. Similarly, if the Ricci curvature is negative in the direction of a given vector , 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 . 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 . 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 worldlines later becomes elliptical, without changing its volume, then this is due to tidal effects from a mass at some other location.
Examples
Space forms If an -dimensional Riemannian manifold has constant sectional curvature , then
and
Thus Euclidean space has zero Ricci curvature, the round sphere has positive Ricci curvature, and hyperbolic space has negative Ricci curvature.
Surfaces On a two-dimensional Riemannian manifold, the Ricci tensor is completely determined by the Gaussian curvature:
The scalar curvature is therefore
Products For a product Riemannian manifold with the product metric, the Ricci tensor splits as the sum of the Ricci tensors of the factors. In particular, a product may have mixed Ricci behavior in different tangent directions.
Applications Ricci curvature plays an important role in general relativity, where it is the key term in the Einstein field equations. Through the Raychaudhuri equation, it also governs the expansion of geodesic congruences.
Ricci curvature also appears in the Ricci flow equation, first introduced by Richard S. Hamilton in 1982, where certain one-parameter families of Riemannian metrics are singled out as solutions of a geometrically defined partial differential equation. In harmonic local coordinates the Ricci tensor can be expressed as
where are the components of the metric tensor and is the Laplace–Beltrami operator. This fact motivates the introduction of the Ricci flow equation as a natural extension of the heat equation for the metric. Since heat tends to spread through a solid until the body reaches an equilibrium state of constant temperature, if one is given a manifold, the Ricci flow may be hoped to produce an 'equilibrium' Riemannian metric that is Einstein or of constant curvature. However, such a clean "convergence" picture cannot be achieved since many manifolds cannot support such metrics. A detailed study of the nature of solutions of the Ricci flow, due principally to Hamilton and Grigori Perelman, shows that the types of "singularities" that occur along a Ricci flow, corresponding to the failure of convergence, encodes deep information about 3-dimensional topology. The culmination of this work was a proof of the geometrization conjecture first proposed by William Thurston in the 1970s, which can be thought of as a classification of compact 3-manifolds.
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 is positive on the set of non-zero tangent vectors .) Some results are also known for pseudo-Riemannian manifolds. Myers's theorem (1941) states that if the Ricci curvature is bounded from below on a complete Riemannian n-manifold by , then the manifold has diameter . By a covering-space argument, it follows that any compact manifold of positive Ricci curvature must have finite fundamental group. Cheng (1975) showed that, in this setting, equality in the diameter inequality occurs if only if the manifold is isometric to a sphere of a constant curvature .