Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, helps us understand how space can be curved instead of flat. It shows how nearby paths, called geodesics, move away from or towards each other. This idea is useful when studying shapes and spaces.

The Ricci curvature tensor is very important in general relativity, where it is part of the Einstein field equations. These equations explain how space and time are affected by mass and energy. Ricci curvature is also used in Riemannian geometry and geometric analysis.

In three dimensions, the Ricci tensor can tell us all about the full curvature of space. This helps solve big problems in geometry, like the Poincaré conjecture. Scientists use Ricci curvature to study how spaces change over time and to link geometry with other parts of mathematics.

Definition

The Ricci curvature tensor, named after Gregorio Ricci-Curbastro, helps us understand how curved space is different from flat space. It shows how paths move away from or towards each other.

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

Properties

The Ricci curvature tensor, named after Gregorio Ricci-Curbastro, measures how curved space is compared to 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 pick special coordinates where straight lines through the point act like paths called geodesics. In these coordinates, the space looks almost flat near the point.

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

Examples

Space forms

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

Surfaces

For two-dimensional surfaces, Ricci curvature is linked to Gaussian curvature. Here, the Ricci curvature is the Gaussian curvature multiplied by the metric. The overall curvature, called scalar curvature, is twice the Gaussian curvature.

Products

When you combine two different spaces into one, the Ricci curvature of the new space is the sum of the Ricci curvatures of the original spaces. This means the new space can show different Ricci curvature 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 the Ricci flow. This flow helps us learn about the shapes of spaces and how they can change over time. It was first suggested by Richard S. Hamilton in 1982.

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 how we measure distances in space by multiplying the original size by a special number, the Ricci curvature changes in a special way. This change depends on the number we use and the point in space we are looking at.

In two-dimensional spaces, if we use a special kind of math for scaling, the Ricci curvature stays the same, even though how we measure distances changes.

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 is the same 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.

Kähler manifolds

On a Kähler manifold, the Ricci curvature helps us understand the curvature form of the canonical line bundle.

The Ricci form comes from the Ricci curvature. Its cohomology class is tied to the first Chern class of the canonical bundle. This makes it a topological feature of the manifold.

If the Ricci tensor is zero, the canonical bundle becomes flat. In these cases, the holonomy group of the manifold has special properties.

Generalization to affine connections

The Ricci tensor can be used with special kinds of connections, called affine connections. These are important in studying a type of geometry called projective geometry. Projective geometry looks at paths called geodesics.

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

Discrete Ricci curvature

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