Riemannian manifold

In differential geometry, a Riemannian manifold (or Riemann space) is a geometric space where important ideas like distance, angles, length, volume, and curvature can be described. Examples include Euclidean space, the n-sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids. The concept was first introduced by the German mathematician Bernhard Riemann in 1854.

A Riemannian manifold is created by adding a special structure called a Riemannian metric to a smooth manifold. This metric lets mathematicians use tools from differential and integral calculus to study the shape and properties of the space. For instance, they can measure distances and understand how the space curves.

Riemannian geometry, the study of these manifolds, connects deeply with many areas of mathematics, such as geometric topology, complex geometry, and algebraic geometry. It also has important uses in physics, including general relativity, as well as in computer graphics, machine learning, and cartography.

History

In 1827, Carl Friedrich Gauss found that the curvature of a surface depends only on measurements made on the surface itself. This important discovery is called the Theorema Egregium.

Later, in 1854, Bernhard Riemann introduced the idea of Riemannian manifolds. These are spaces where we can measure distance, angles, and curves. The concept was formalized much later. Albert Einstein used a related idea called pseudo-Riemannian manifolds to develop his theory of general relativity.

Definition

A Riemannian manifold is a special kind of geometric space used in math. It helps us understand ideas like distance, angles, and shapes. You might have heard of Euclidean space—this is the flat space we live in, where things look the same in all directions. But Riemannian manifolds can be curved, like the surface of a sphere or a squished ball.

These spaces were first thought up by a mathematician named Bernhard Riemann in 1854. Examples include the surface of a ball, curved surfaces like ellipsoids, and even space itself in theories of gravity. In Riemannian geometry, we can measure how “bumpy” or “smooth” a surface is, and these ideas are important in physics and advanced math.

Examples

Riemannian manifolds are geometric spaces where concepts like distance and angles can be measured. Familiar examples include ordinary flat space, the surface of a sphere, and curved surfaces like ellipsoids. These spaces are named after the German mathematician Bernhard Riemann, who first described them in 1854.

Euclidean space, such as the flat space we live in, is a basic example of a Riemannian manifold. Other examples include the surface of a sphere and smooth curved shapes like ellipsoids. These surfaces carry their own natural ways of measuring distances and angles, making them important in the study of geometry.

Every smooth manifold admits a Riemannian metric

Every smooth shape can have a special measurement system called a Riemannian metric. This is important because it helps us understand distances and angles on different kinds of shapes.

To create this system, mathematicians use special rules that make sure the shape fits well together. They can also place the shape inside ordinary space, like the space we live in, and then copy the measurements from that space to the shape. Even though placing the shape in space might not always show all its special properties clearly, it still works to give the shape its measurement system.

Metric space structure

A Riemannian manifold allows us to measure distances and angles in a way similar to how we measure them in ordinary space. Think of it like a flexible grid that can wrap around shapes like spheres or hills.

One important idea is the diameter, which tells us the greatest possible distance between any two points on the shape. For example, the diameter of a sphere is the distance across its widest part. The Hopf–Rinow theorem helps us understand when a shape with finite diameter will also be compact, meaning it stays within certain bounds. This is useful in many areas of geometry and physics.

Main article: metric space

Main articles: diameter, Hopf–Rinow theorem, complete, if and only if

Connections, geodesics, and curvature

An affine connection is an additional structure on a Riemannian manifold that defines how to differentiate one vector field with respect to another. Connections contain geometric data, and two Riemannian manifolds with different connections have different geometry.

A Levi-Civita connection is a natural connection associated with a Riemannian manifold. It is torsion-free and preserves the metric. Once a Riemannian metric is fixed, there exists a unique Levi-Civita connection.

Geodesics are curves with no intrinsic acceleration. They are the generalization of straight lines in Euclidean space to arbitrary Riemannian manifolds. An ant walking straight ahead without making any effort to accelerate or turn would trace out a geodesic.

The Hopf–Rinow theorem characterizes geodesically complete manifolds. It states that a connected Riemannian manifold is geodesically complete if and only if its metric space is complete, all closed and bounded subsets are compact, or every maximal geodesic can be extended indefinitely.

Parallel transport is a way of moving vectors from one tangent space to another along a curve in the setting of a general Riemannian manifold. Given a fixed connection, there is a unique way to do parallel transport.

The Riemann curvature tensor measures how parallel transporting vectors around a small loop differs from the identity map. It is zero at every point if and only if the manifold is locally isometric to Euclidean space.

The Ricci curvature tensor is a contraction of the Riemann curvature tensor. It plays a key role in the theory of Einstein manifolds, which have applications in the study of gravity. An Einstein metric satisfies Einstein's equation, where the Ricci curvature tensor is proportional to the metric tensor.

Examples of Einstein manifolds include Euclidean space, the n-sphere, hyperbolic space, and complex projective space with the Fubini-Study metric.

Main article: Affine connection

Main article: Levi-Civita connection

Main article: Geodesic

Main article: Hopf–Rinow theorem

Main article: Parallel transport

Main article: Riemann curvature tensor

Main article: Ricci curvature

Main article: Einstein manifold

Main article: Scalar curvature

Constant curvature and space forms

A Riemannian manifold has constant curvature if its curvature measurements stay the same everywhere. This makes calculations easier and helps us understand the shape of the space. For example, a sphere, flat Euclidean space, and hyperbolic space all have constant curvature.

A Riemannian space form is a special kind of Riemannian manifold with constant curvature that is also connected and complete. Depending on whether the curvature is positive, zero, or negative, these space forms are called spherical, Euclidean, or hyperbolic space forms. These forms help mathematicians study the properties of different geometric spaces by using ideas from group theory.

Riemannian metrics on Lie groups

A Lie group, like the group of rotations in three-dimensional space, can have special kinds of Riemannian metrics called left-invariant metrics. These metrics use the group's structure to define distances and angles consistently across the whole space. Such metrics are useful because they simplify calculations and provide clear examples of Riemannian manifolds.

Many important examples of Riemannian manifolds come from Lie groups with these special metrics. For example, Berger spheres are created using left-invariant metrics on certain groups and show interesting geometric properties. Another example is hyperbolic space, which can also be viewed as a Lie group with a left-invariant metric. These examples help mathematicians understand more complex geometric ideas.

Infinite-dimensional manifolds

The statements and theorems above are for finite-dimensional manifolds—manifolds whose charts map to open subsets of Rⁿ. These can be extended, to a certain degree, to infinite-dimensional manifolds; that is, manifolds that are modeled after a topological vector space; for example, Fréchet, Banach, and Hilbert manifolds.

Riemannian metrics are defined in a way similar to the finite-dimensional case. However, there is a distinction between two types of Riemannian metrics. If a metric is strong, the manifold must be a Hilbert manifold. Examples include Hilbert spaces and certain groups of smooth maps on compact Riemannian manifolds.