In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold.
The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space.
In general relativity, which involves the pseudo-Riemannian setting, this is reflected by the presence of the Ricci tensor in the Raychaudhuri equation.
Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship between the Ricci tensor and the matter content of the universe.
In part, this simplicity allows for the application of many geometric and analytic tools, which led to the solution of the Poincaré conjecture through the work of Richard S. Hamilton and Grigori Perelman.
In differential geometry, the determination of lower bounds on the Ricci tensor on a Riemannian manifold would allow one to extract global geometric and topological information by comparison (cf.
This is since lower bounds on the Ricci tensor can be successfully used in studying the length functional in Riemannian geometry, as first shown in 1941 via Myers's theorem.
For example, this formula explains why the gradient estimates due to Shing-Tung Yau (and their developments such as the Cheng-Yau and Li-Yau inequalities) nearly always depend on a lower bound for the Ricci curvature.
In 2007, John Lott, Karl-Theodor Sturm, and Cedric Villani demonstrated decisively that lower bounds on Ricci curvature can be understood entirely in terms of the metric space structure of a Riemannian manifold, together with its volume form.
[2] This established a deep link between Ricci curvature and Wasserstein geometry and optimal transport, which is presently the subject of much research.
It is a standard exercise of (multi)linear algebra to verify that this definition does not depend on the choice of the basis
, then relative to any smooth coordinates one has 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.
Arguably, the definitions directly using local coordinates are preferable, since the "crucial property" of the Riemann tensor mentioned above requires
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.
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
, will have smaller volume than the corresponding conical region in Euclidean space, at least provided that
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.
Ricci curvature plays an important role in general relativity, where it is the key term in the Einstein field equations.
In harmonic local coordinates the Ricci tensor can be expressed as (Chow & Knopf 2004, Lemma 3.32).
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.
Here is a short list of global results concerning manifolds with positive Ricci curvature; see also classical theorems of Riemannian geometry.
These results, particularly Myers' and Hamilton's, 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 complete Riemannian metric of negative Ricci curvature.
There are very few two-dimensional manifolds which fail to admit Riemannian metrics of negative Gaussian curvature.
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.
In particular, the vanishing of trace-free Ricci tensor characterizes Einstein manifolds, as defined by the condition
In this more general situation, the Ricci tensor is symmetric if and only if there exists locally a parallel volume form for the connection.
Notions of Ricci curvature on discrete manifolds have been defined on graphs and networks, where they quantify local divergence properties of edges.
[4] A different (and earlier) notion, Forman's Ricci curvature, is based on topological arguments.