Connection (mathematics)

In the mathematical subfield of differential geometry, a context is a tool to quantify changes of direction in the course of a movement and to set directions in different points relate to each other.

This article is essentially the connection on a differentiable manifold or on a vector bundle. An excellent connection on a Tensorbündel, a special vector bundle is called covariant derivative. More generally, there are also connections on principal bundles with similar defining characteristics.

• 3.1 context of a real submanifold
• 3.2 contexts on the Tensorbündel

Motivation

In differential geometry one is interested in the curvature of curves, particularly of geodesics. In Euclidean spaces the curvature is simply given by the second derivative. On differentiable manifolds, the second derivative is to be formed directly. Is a curve, it is necessary for the second derivative of the curve of the difference quotient with the vectors, and form. These vectors are located but in different vector spaces, so you can not just make up the difference between the two. To solve the problem, there is defined an image which is called the context. This illustration is intended to provide a connection between the involved vector spaces and therefore also bears this name.

Definitions

This section describes a smooth manifold, the tangent bundle and a vector bundle. With the set of smooth sections is listed on the vector bundle.

Context

By saying what is the directional derivative of a vector field in the direction of a tangent vector, you get a connection on a differentiable manifold. Accordingly, we define a connection on a vector bundle as a mapping

The vector field on a for, and a section in the vector bundle again a cut, so that the following conditions are met in associates:

• Is linear over, ie

• Is linear in other words, it is

• In addition, the product rule, or Leibniz rule

Alternatively, the context as Figure

Defined with the same properties.

Linear relationship

A linear or affine context is a context. That is, it is a picture

Which satisfies the three defining characteristics of the above section.

Induced correlations

There are different ways on other vector bundles to induce in a natural way relationships.

Context of a real submanifold

Is the standard basis, is then related to the Euclidean by, said representations, and the vector fields are related to the standard basis. Is a submanifold of, we obtain a relation of induced. This is by

Determined. This refers to the orthogonal projection.

Correlations on the Tensorbündel

Be a linear connection on the manifold. On the Tensorbündel can induce, which is also listed with and satisfies the following properties is a clear relationship:

This context is also called the covariant derivative.

Compatibility with the Riemannian metric and symmetry

Be a Riemannian or pseudo - Riemannian manifold. A connection is called compatible with the metric of this manifold, if

Applies. The third property from the section links on the Tensorbündel one obtains the equation

And therefore, the compatibility condition is equivalent to

A connection is called symmetric or torsion when the Torsionstensor disappears, which means that it applies

These two properties are in this way, of course, that they are met by an induced connection on a real submanifold. A connection on a ( abstract ) manifold, which satisfies these two properties is uniquely determined. This statement is called the fundamental theorem of Riemannian geometry and the uniquely determined connection is called the Levi-Civita or Riemannian context. A connection that is compatible with the Riemannian metric is called a metric connection. A Riemannian manifold can generally have several different metric correlations.

Properties

• Let and be two vector fields, such that in a neighborhood of. Then follows for all vector fields

General and need not even be the same on a whole environment. More precisely: If there is a smooth curve ( for a suitable ) so that and and if for all, then it follows already. This means that the two vector fields and must match only along a suitable smooth curve.

• Analogous to the aforementioned property: Let two vector fields on such that. Then for all that.

Representation in coordinates: Christoffel symbols

Constitute the local vector fields at each point a base of the tangent space, the Christoffel symbols are defined by

If the vector fields and with respect to this basis, the shape and so is true for the components of

The directional derivative of the function in the direction of the vector respectively.

If you choose the basis vector fields especially the vector fields given by a map, we obtain the coordinate representation

This result corresponds to the product rule in the product change during infinitesimal changes in both the base vectors and the components function and there is the sum of the two changes.

Applications

The central concepts of this article concern in physics, inter alia, the general theory of relativity and gauge theories (eg, quantum electrodynamics, quantum chromodynamics and Yang-Mills theory ) of High Energy Physics, as well as in solid-state physics, the BCS theory of superconductivity. The Common to these theories is that " context " and " covariant derivative " by vector potentials, called, are generated that satisfy certain calibration conditions, and that they are entering explicitly in a certain way in the energy function of the system.