Riemannian manifold

In differential geometry, a Riemannian manifold (or Riemann space) is a special kind of space where we can talk about distance, angles, length, area, and how much the space curves. Examples include Euclidean space, the n-sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids. The idea was first introduced by the German mathematician Bernhard Riemann in 1854.

A Riemannian manifold is made by adding something called a Riemannian metric to a smooth manifold. This metric helps mathematicians use tools from differential and integral calculus to study the shape and properties of the space. For example, they can measure distances and see how the space bends.

Riemannian geometry, the study of these spaces, connects with many parts of mathematics, such as geometric topology, complex geometry, and algebraic geometry. It is also useful in physics, especially in general relativity, as well as in computer graphics, machine learning, and cartography.

History

In 1827, Carl Friedrich Gauss discovered that the shape of a surface depends only on measurements made on that surface. This 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 idea was used much later by Albert Einstein in 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 help us in physics and advanced math.

Examples

Riemannian manifolds are special spaces where we can measure distance and angles. Common examples are flat space, like the space we live in, the surface of a sphere, and other curved shapes like ellipsoids. These spaces are named after the German mathematician Bernhard Riemann, who first described them in 1854.

Flat space, such as the space around us, is a simple example of a Riemannian manifold. Other examples include the surface of a sphere and smooth curved shapes like ellipsoids. These surfaces have their own natural ways of measuring distances and angles, which are important in the study of geometry.

Every smooth manifold admits a Riemannian metric

Every smooth shape can have a special way to measure distances and angles. This is called a Riemannian metric.

Mathematicians use special rules to make sure the shape fits well together. They can also put the shape inside regular space, like the space we live in, and then use those space measurements for the shape. Even if this doesn't show all the shape's special properties, it still works to give the shape its measurement system.

Metric space structure

A Riemannian manifold helps us measure distances and angles, just like we do in everyday space. Imagine a flexible grid that can wrap around shapes such as spheres or hills.

One key idea is the diameter. This tells us the farthest 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 a finite diameter will also stay within certain limits. This idea is useful in 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 extra part added to a Riemannian manifold. It helps us figure out how to change one vector field compared to another. Connections hold important geometric information, so two Riemannian manifolds with different connections will look different.

A Levi-Civita connection is a special connection that comes with a Riemannian manifold. It has two important properties: it has no twisting and it keeps the size and angles the same. Once we pick a Riemannian metric, there is only one Levi-Civita connection.

Geodesics are special curves that have no natural speed changes. They are like straight lines in flat space, but they work in any curved space. If an ant walked straight ahead without trying to speed up or turn, its path would be a geodesic.

The Hopf–Rinow theorem tells us about geodesically complete manifolds. It says a connected Riemannian manifold is geodesically complete when its metric space is complete, all closed and bounded areas can be made smaller, or every longest geodesic can keep going forever.

Parallel transport is a way to move vectors from one point to another along a curve in a Riemannian manifold. With a fixed connection, there is only one way to do this.

The Riemann curvature tensor measures how moving vectors around a small loop changes them. It is zero everywhere only if the space looks like flat Euclidean space nearby.

The Ricci curvature tensor is a simpler version of the Riemann curvature tensor. It is important in the study of Einstein manifolds, which help us understand things like gravity. An Einstein metric follows a special rule where the Ricci curvature tensor is linked to the metric tensor.

Examples of Einstein manifolds include flat 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 shape stays the same everywhere. This makes it easier to study. For example, a sphere, flat space, and hyperbolic space all have constant curvature.

A Riemannian space form is a special type of manifold with constant curvature. It is also connected and complete. Depending on the curvature, these forms are called spherical, Euclidean, or hyperbolic space forms. They help mathematicians learn about different geometric spaces 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 measurements called left-invariant metrics. These metrics use the group's structure to define distances and angles the same way everywhere. This makes calculations easier and provides 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 shapes. Another example is hyperbolic space, which can also be seen as a Lie group with a left-invariant metric. These examples help mathematicians learn about more complex geometry.

Infinite-dimensional manifolds

The ideas we talked about before work for special kinds of spaces that have infinite dimensions. These spaces are based on special math objects called topological vector spaces. Examples include Fréchet, Banach, and Hilbert manifolds.

We can also define Riemannian metrics for these infinite-dimensional spaces, much like we do for regular spaces. There are two types of these metrics. If a metric is strong, the space must be a Hilbert manifold. Examples include Hilbert spaces and some groups of smooth maps.