Ricci curvature

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. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.

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.

Like the metric tensor, the Ricci tensor assigns to each tangent space of the manifold a symmetric bilinear form (Besse 1987, p. 43).[1] Broadly, one could analogize the role of the Ricci curvature in Riemannian geometry to that of the Laplacian in the analysis of functions; in this analogy, the Riemann curvature tensor, of which the Ricci curvature is a natural by-product, would correspond to the full matrix of second derivatives of a function. However, there are other ways to draw the same analogy.

In three-dimensional topology, the Ricci tensor contains all of the information which in higher dimensions is encoded by the more complicated Riemann curvature tensor. 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 Grigory Perelman.

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. 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.

One common source of the Ricci tensor is that it arises whenever one commutes the covariant derivative with the tensor Laplacian. This, for instance, explains its presence in the Bochner formula, which is used ubiquitously in Riemannian geometry. 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.[citation needed]

The first subsection here is meant as an indication of the definition of the Ricci tensor for readers who are comfortable with linear algebra and multivariable calculus. The later subsections use more sophisticated terminology.

Let U be an open subset of n, and for each pair of numbers i and j between 1 and n, let gij : U → ℝ be a smooth function, subject to the condition that, for each p in U, the matrix

is symmetric and invertible. For each p in U, let [gij(p)] be the inverse of the above matrix [gij(p)]. The functions Rij are defined explicitly by the following formulae:

where xi is the ith coordinate of n. It can be seen directly from inspection of this formula that Rij must equal Rji for any i and j. So one can view the functions Rij as associating to any point p of U a symmetric n × n matrix. This matrix-valued map on U is called the Ricci curvature associated to the collection of functions gij.

As presented, there is nothing intuitive or natural about the definition of Ricci curvature. It is singled out as an object for study only because it satisfies the following remarkable property. Let V ⊂ ℝn be another open set and let y : VU be a smooth map whose matrix of first derivatives

is invertible for any choice of qV. Define gij : V → ℝ by the matrix product

and let [Rij(q)] be the Ricci curvature associated with [gij(q)]. Then one can compute, using the product rule and the chain rule, the following relationship between the Ricci curvature of the collection of functions gij and the Ricci curvature of the collection of functions gij: for any q in V, one has

This is quite unexpected since, directly plugging the formula which defines gij into the formula defining Rij, one sees that one will have to consider up to third derivatives of y, arising when the second derivatives in the first four terms of the definition of Rij act upon the components of J. The "miracle" is that the imposing collection of first derivatives, second derivatives, and inverses comprising the definition of the Ricci curvature is perfectly set up so that all of these higher derivatives of y cancel out, and one is left with the remarkably clean matrix formula above which relates Rij and Rij. It is even more remarkable that this cancellation of terms is such that the matrix formula relating Rij to Rij is identical to the matrix formula relating gij to gij.

With the use of some sophisticated terminology, the definition of Ricci curvature can be summarized as saying:

Let U be an open subset of n. Given a smooth mapping g on U which is valued in the space of invertible symmetric n × n matrices, one can define (by a complicated formula involving various partial derivatives of the components of g) the Ricci curvature of g to be a smooth mapping from U into the space of symmetric n × n matrices.

The remarkable and unexpected property of Ricci curvature can be summarized as:

Let J denote the Jacobian matrix of a diffeomorphism y from some other open set V to U. The Ricci curvature of the matrix-valued function given by the matrix product JT(gy)J is given by the matrix product JT(Ry)J, where R denotes the Ricci curvature of g.

In mathematics, this property is referred to by saying that the Ricci curvature is a "tensorial quantity", and marks the formula defining Ricci curvature, complicated though it may be, as of outstanding significance in the field of differential geometry.[3] In physical terms, this property is a manifestation of "general covariance" and is a primary reason that Albert Einstein made use of the formula defining Rij when formulating general relativity. In this context, the possibility of choosing the mapping y amounts to the possibility of choosing between reference frames; the "unexpected property" of the Ricci curvature is a reflection of the broad principle that the equations of physics do not depend on reference frame.

This is discussed from the perspective of differentiable manifolds in the following subsection, although the underlying content is virtually identical to that of this subsection.

Let (M, g) be a smooth Riemannian or pseudo-Riemannian n-manifold. Given a smooth chart (U, Greek phi Didot.svg) one then has functions gij : Greek phi Didot.svg(U) → ℝ and gij : Greek phi Didot.svg(U) → ℝ for each i and j between 1 and n which satisfy

for all x in Greek phi Didot.svg(U). The functions gij are defined by evaluating g on coordinate vector fields, while the functions gij are defined so that, as a matrix-valued function, they provide an inverse to the matrix-valued function xgij(x).

Now let (U, Greek phi Didot.svg) and (V, ψ) be two smooth charts for which U and V have nonempty intersection. Let Rij : Greek phi Didot.svg(U) → ℝ be the functions computed as above via the chart (U, Greek phi Didot.svg) and let rij : ψ(V) → ℝ be the functions computed as above via the chart (V, ψ). Then one can check by a calculation with the chain rule and the product rule that

This shows that the following definition does not depend on the choice of (U, Greek phi Didot.svg). For any p in U, define a bilinear map Ricp : TpM × TpM → ℝ by

where X1, ..., Xn and Y1, ..., Yn are the components of X and Y relative to the coordinate vector fields of (U, Greek phi Didot.svg).

It is common to abbreviate the above formal presentation in the following style:

Let M be a smooth manifold, and let g be a Riemannian or pseudo-Riemannian metric. In local smooth coordinates, define the Christoffel symbols

so that Rij define a (0,2)-tensor field on M. In particular, if X and Y are vector fields on M 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.

Suppose that (M, g) is an n-dimensional Riemannian or pseudo-Riemannian manifold, equipped with its Levi-Civita connection . The Riemann curvature of M is a map which takes smooth vector fields X, Y, and Z, and returns the vector field

on vector fields X, Y, Z. The crucial property of this mapping is that if X, Y, Z and X', Y', and Z' are smooth vector fields such that X and X' define the same element of some tangent space TpM, and Y and Y' also define the same element of TpM, and Z and Z' also define the same element of TpM, then the vector fields R(X,Y)Z and R(X′,Y′)Z also define the same element of TpM.

The implication is that the Riemann curvature, which is a priori a mapping with vector field inputs and a vector field output, can actually be viewed as a mapping with tangent vector inputs and a tangent vector output. That is, it defines for each p in M a (multilinear) map

That is, having fixed Y and Z, then for any basis v1, ..., vn of the vector space TpM, one defines

where for any fixed i the numbers ci1, ..., cin are the coordinates of Rmp(vi,Y,Z) relative to the basis v1, ..., vn. It is a standard exercise of (multi)linear algebra to verify that this definition does not depend on the choice of the basis v1, ..., vn.

As can be seen from the Bianchi identities, the Ricci tensor of a Riemannian manifold is symmetric, in the sense that

The Ricci curvature is determined by the sectional curvatures of a Riemannian manifold, but generally contains less information. Indeed, if ξ 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 ξ. 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.

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

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

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 p:

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

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 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.

Ricci curvature also appears in the Ricci flow equation, where certain one-parameter families of Riemannian metrics are singled out as solutions of a geometrically-defined partial differential equation. This system of equations can be thought of as a geometric analog of the heat equation, and was first introduced by Richard S. Hamilton in 1982. 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 which is Einstein or of constant curvature. However, such a clean "convergence" picture cannot be achieved since many manifolds cannot support much 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.

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.

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. In the case of two-dimensional manifolds, negativity of the Ricci curvature is synonymous with negativity of the Gaussian curvature, which has very clear topological implications. There are very few two-dimensional manifolds which fail to admit Riemannian metrics of negative Gaussian curvature.

If the metric g is changed by multiplying it by a conformal factor e2f, the Ricci tensor of the new, conformally-related metric = e2fg is given (Besse 1987, p. 59) by

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 ge2fg does not change the Ricci tensor (although it still changes its trace with respect to the metric unless f = 0).

In Riemannian geometry and pseudo-Riemannian geometry, the trace-free Ricci tensor (also called traceless Ricci tensor) of a Riemannian or pseudo-Riemannian n-manifold (M, g) is the tensor defined by

It is less immediately obvious that the two terms on the right hand side are orthogonal to each other:

An identity which is intimately connected with this (but which could be proved directly) is that

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:

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

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 2-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 compact X) in the sense that it depends only on the topology of X and the homotopy class of the complex structure.

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).

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 denotes an affine connection, then the curvature tensor R is the (1,3)-tensor defined by

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

In this more general situation, the Ricci tensor is symmetric if and only if there exist 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. Olliver's Ricci curvature is defined using optimal transport theory.[citation needed] A second notion, Forman's Ricci curvature, is based on topological arguments.[citation needed]