- Ricci curvature
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 metricmight differ from that of ordinary Euclidean "n-"space. Like the metric itself, the Ricci tensor is a symmetric bilinear formon the tangent spaceof 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.
Suppose that is an "n-"dimensional
Riemannian manifold, and let denote the tangent spaceof "M" at "p". For any pair of tangent vectors at "p", the Ricci tensor evaluated at is defined to be the trace of the linear map given by:where "R" is the Riemann curvature tensor. In local coordinates(using the summation convention), one has:where:
As a consequence of the
Bianchi identities, the Ricci tensor of a Riemannian manifold is "symmetric", in the sense that :It thus follows 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 simply calledthe Ricci curvature, since knowing it is equivalent to knowing the Ricci curvature tensor.
TheRicci curvature is determined by the
sectional curvatures ofa Riemannian manifold, but contains less information. Indeed, if is a vector of unit lengthon a Riemannian "n-"manifold,then is precisely ("n"−1) times the average value of thesectional curvature, taken over all the 2-planes containing . (There isan ("n"−2)-dimensional family of such 2-planes.)
If the Ricci curvature function is constant on the set of unit tangent vectors ,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 is a constant multiple of the metric tensor .
If the Ricci curvature function is positive (nonnegative) on the set of unit tangent vectors ,the Riemannian manifold is said to have positive (nonnegative) Ricci curvature. See below fortopological 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 connectionis 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 , one can define preferred local coordinates, called
geodesic normal coordinates. These are adaptedto 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 :In these coordinates, the metric volume formthen has the following Taylor expansionat "p": :
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 short geodesic segments emanating from p and roughly pointing in the direction of will have smaller volumethan the corresponding conical region in Euclidean space. 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 volumethan 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 flowequation, 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 firstintroduced 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 fora manifold for which the Ricci curvature is constant. Recent contributions to the subject due to Grigori Perelmannow seem to show that this program works well enough in dimension three to lead to a complete classification of compact 3-manifolds, along linesfirst conjectured by William Thurstonin the 1970s.
Kähler manifold, the Ricci curvature determines the first Chern classof 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 , then the manifold has diameter , 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
Bishop–Gromov inequalitystates 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 denotes the volume of the ball with center "p" and radius in the manifold and denotes the volume of the ball of radius "R" in Euclidean "m"-space then function 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.)
Splitting theoremstates that if a complete Riemannian manifold with contains a "line", meaning a geodesic γ such that for all , then it is isometric to a product space . 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 Lohkamphas 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 pointis 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 , the Ricci tensor of the new, conformally related metric is given by Fact|date=February 2007:where is the geometric Laplacian.
If we let , then this can be rewritten as
In particular, given a point p in a Riemannianmanifold, 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 does not change the Ricci curvature.
Trace-free Ricci tensor
Riemannian geometryand general relativity, the trace-free Ricci tensor of a pseudo-Riemannian manifold is the tensor defined by
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:
If n 3, the trace-free Ricci tensor vanishes identically if and only if :for some constant .In mathematics, this is the condition for to be an
Einstein manifold. In physics, this equationstates that is a solution of Einstein's vacuum fieldequations with cosmological constant.
Curvature of Riemannian manifolds
Basic introduction to the mathematics of curved spacetime
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