Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, helps us understand how curved space differs from flat space. It measures how nearby paths, called geodesics, move apart or come together. This idea is important in studying shapes and spaces.

The Ricci curvature tensor plays a key role in general relativity, where it is part of the Einstein field equations. These equations describe how space and time change due to mass and energy. Ricci curvature is also essential in Riemannian geometry and geometric analysis.

In three dimensions, the Ricci tensor can tell us everything about the full curvature of space. This idea is important in solving big problems in geometry, like the Poincaré conjecture. Scientists use Ricci curvature to study how spaces can change over time and to connect geometry with other areas of mathematics.

Definition

The Ricci curvature tensor, named after Gregorio Ricci-Curbastro, helps us understand how a curved space differs from a flat one. It measures how nearby paths spread apart or come together.

In simple terms, Ricci curvature is a way to average the way space curves in different directions. It is linked to the Riemann curvature tensor, which describes the overall curvature of space, by focusing on a specific part of that information.

Properties

The Ricci curvature tensor, named after Gregorio Ricci-Curbastro, measures how curved space differs from flat space. It is symmetric, meaning the order of its inputs does not change the result.

This tensor is linked to the sectional curvatures of a space. In simple terms, it gives an average of these curvatures for planes that contain a certain direction. Only in two or three dimensions does the Ricci tensor fully describe the space's curvature. The tensor was introduced by Ricci because it helps describe the curvature of certain shapes in Euclidean space.

Direct geometric meaning

Near any point in a curved space, we can choose special coordinates where straight lines through the point represent paths called geodesics. In these coordinates, the space looks almost flat close to the point.

The Ricci curvature tells us how the volume of a small cone of geodesics changes compared to flat space. If the Ricci curvature is positive in a certain direction, the cone’s volume will be smaller than in flat space. If it is negative, the volume will be larger. The Ricci curvature averages the curvatures in all planes that include that direction.

Examples

Space forms

In special types of spaces called space forms, the Ricci curvature is very simple. For an n-dimensional space with the same curvature everywhere (called constant sectional curvature K), the Ricci curvature is given by (n − 1) times K. This means that flat Euclidean space has zero Ricci curvature, a round sphere has positive Ricci curvature, and hyperbolic space has negative Ricci curvature.

Surfaces

For two-dimensional surfaces, the Ricci curvature is directly tied to the Gaussian curvature. In this case, the Ricci curvature is just the Gaussian curvature multiplied by the metric. The overall curvature of the surface, called scalar curvature, is twice the Gaussian curvature.

Products

When you combine two different spaces into a product, the Ricci curvature of the combined space is the sum of the Ricci curvatures of the individual spaces. This means the combined space can show different Ricci curvature behaviors in different directions.

Applications

Ricci curvature is very important in general relativity. It is a main part of the Einstein field equations, which help us understand how space and time change with mass and energy.

It is also used in something called the Ricci flow, a way to study shapes that was first suggested by Richard S. Hamilton in 1982. This flow helps us learn more about the shapes of spaces and how they can change over time.

Global geometry and topology

Positive Ricci curvature has important effects on the shape and structure of space. For example, if a space has positive Ricci curvature everywhere, it cannot be too large — it must have a limited size. This helps us understand how spaces can be shaped.

Negative Ricci curvature, however, does not place the same restrictions on a space’s structure, except in very special cases. This means spaces with negative Ricci curvature can be very different in shape and size. Main article: classical theorems of Riemannian geometry

Myers's theorem

The Bishop–Gromov inequality

The Cheeger–Gromoll splitting theorem

Hamilton's first convergence theorem

Lohkamp (1994)

Behavior under conformal rescaling

When we change the way we measure distances in space by multiplying the original measurement by a special factor, the Ricci curvature changes in a specific way. This change depends on the factor used and the point in space we are looking at.

In two-dimensional spaces, if we use a special kind of function for scaling, the Ricci curvature does not change at all, even though the way we measure distances does.

Trace-free Ricci tensor

In pseudo-Riemannian geometry, the trace-free Ricci tensor is a special way to look at space. It shows how space curves by removing a part of the Ricci curvature that repeats in all directions. This helps us understand the shape of space better.

The trace-free Ricci tensor is linked to Einstein manifolds, which are special spaces where the Ricci curvature meets certain conditions. In general relativity, this idea helps describe how space behaves without extra forces pulling on it.

Kähler manifolds

On a Kähler manifold, the Ricci curvature helps us understand the curvature form of something called the canonical line bundle. This bundle is linked to special mathematical objects known as Kähler differentials.

The Ricci form, which comes from the Ricci curvature, is a closed 2-form. Its cohomology class is tied to the first Chern class of the canonical bundle, making it a topological feature of the manifold. This means it only depends on the basic shape and structure of the space.

If the Ricci tensor is zero, the canonical bundle becomes flat. In such cases, the holonomy group of the manifold is special and can be described in a particular way.

Generalization to affine connections

The Ricci tensor can also be used with special kinds of connections, called affine connections. These are important in studying a type of geometry called projective geometry, which looks at paths called geodesics without worrying about how they are measured.

When we use an affine connection, we can create a special tensor called the curvature tensor. The Ricci tensor comes from taking a certain part of this curvature tensor, called its trace. In simple terms, this helps us understand more about the geometry we are studying.

Discrete Ricci curvature

Ricci curvature can also be studied on graphs and networks. It helps us understand how edges in these structures spread apart or come together. There are different ways to define this, such as using ideas from movement of objects or looking at the shape of the network itself.