Connection (differential geometry)

In the mathematical sub-area of differential geometry , a relationship is an aid to quantify changes in direction in the course of a movement and to relate directions at different points to one another.

This article essentially deals with the connection on a differentiable manifold or on a vector bundle . An excellent connection on a tensor bundle, a special vector bundle, is called a covariant derivative . More generally, there are also relationships on principal bundles with analogous defining properties.

motivation

In differential geometry, one is interested in the curvature of curves, especially of geodesics . In Euclidean spaces the curvature is simply given by the second derivative. The second derivative cannot be formed directly on differentiable manifolds. If there is a curve, then for the second derivative of this curve the difference quotient must be formed with the vectors and . However, these vectors are in different vector spaces, so you cannot simply calculate the difference between the two. To solve the problem, a mapping has been defined, which is called a connection. This mapping is intended to provide a connection between the vector spaces involved and therefore bears this name. ${\displaystyle \gamma }$ ${\displaystyle \gamma '(t)}$ ${\displaystyle \gamma '(t_{0})}$

Definitions

In this section denotes a smooth manifold , the tangent bundle and a vector bundle . The number of smooth cuts in the vector bundle is noted with. ${\displaystyle M}$ ${\displaystyle TM}$ ${\displaystyle \pi \colon E\to M}$ ${\displaystyle \Gamma (E)}$ ${\displaystyle E}$

context

By saying what the directional derivative of a vector field is in the direction of a tangential vector , one obtains a connection on a differentiable manifold . Accordingly, a relationship on a vector bundle is defined as a map ${\displaystyle M}$

${\displaystyle {\begin{aligned}\nabla \colon \Gamma (TM)\times \Gamma (E)&\rightarrow \Gamma (E)\\(X,s)&\mapsto \nabla _{X}s,\end{aligned}}}$

which assigns a cut in to a vector field on and a cut in the vector bundle , so that the following conditions are met: ${\displaystyle X}$ ${\displaystyle M}$ ${\displaystyle s}$ ${\displaystyle E}$ ${\displaystyle E}$

• ${\displaystyle \nabla _{X}s}$is in linear over , that is ${\displaystyle X\in \Gamma (TM)}$ ${\displaystyle C^{\infty }(M)}$

${\displaystyle \nabla _{fX_{1}+gX_{2}}s=f\cdot \nabla _{X_{1}}s+g\cdot \nabla _{X_{2}}s}$

for and ${\displaystyle f,g\in C^{\infty }(M)}$ ${\displaystyle X_{1},X_{2}\in \Gamma (TM).}$

• ${\displaystyle \nabla _{X}s}$is -linear in that is, it applies ${\displaystyle \mathbb {R} }$ ${\displaystyle s,}$

${\displaystyle \nabla _{X}(\lambda _{1}s_{1}+\lambda _{2}s_{2})=\lambda _{1}\cdot \nabla _{X}s_{1}+\lambda _{2}\cdot \nabla _{X}s_{2}}$

for . ${\displaystyle \lambda _{1},\lambda _{2}\in \mathbb {R} }$

• In addition, the product rule or Leibniz rule applies

${\displaystyle \nabla _{X}(fs)=Xf\cdot s+f\cdot \nabla _{X}s}$

for every function . ${\displaystyle f\in C^{\infty }(M)}$

Here the directional derivative of the function denotes in the direction (tangential vectors are thus understood as derivatives ). Another notation for is . ${\displaystyle Xf}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle Xf}$ ${\displaystyle \mathrm {d} f(X)}$

Alternatively, the connection can also be shown as an illustration

${\displaystyle \nabla \colon \Gamma (E)\to \Gamma (T^{*}M\otimes E)}$

define with the same properties.

Linear relationship

A linear or affine connection on is a connection on . That is, it is an illustration ${\displaystyle M}$ ${\displaystyle TM}$

${\displaystyle \nabla \colon \Gamma (TM)\times \Gamma (TM)\to \Gamma (TM),}$

which fulfills the three defining properties from the section above.

Induced relationships

There are different possibilities to naturally induce connections on other vector bundles.

Connection on a real submanifold

Let be the standard basis of , then the Euclidean relationship is defined by, where and are representations of the vector fields with respect to the standard basis. If there is a submanifold of , then one obtains a connection induced by. This is through ${\displaystyle \partial _{1},\ldots ,\partial _{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \nabla _{X}^{\mathbb {R} ^{n}}}$ ${\displaystyle \textstyle \nabla _{X}^{\mathbb {R} ^{n}}Y:=\sum _{i,j}(X^{i}\partial _{i}Y^{j})\partial _{j}}$ ${\displaystyle \textstyle X=\sum _{i}X^{i}\partial _{i}}$ ${\displaystyle \textstyle Y=\sum _{j}Y^{j}\partial _{j}}$ ${\displaystyle X,Y}$ ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle \nabla _{X}^{M}Y:=\pi (\nabla _{X}^{\mathbb {R} ^{n}}Y)}$

certainly. The orthogonal projection denotes . ${\displaystyle \pi \colon T_{p}\mathbb {R} ^{n}\to T_{p}M}$

Relationships on the tensor bundle

Let be a linear connection on the manifold . A clear connection can be induced on the tensor bundle , which is also noted and fulfills the following properties: ${\displaystyle \nabla }$ ${\displaystyle M}$ ${\displaystyle T_{l}^{k}M}$ ${\displaystyle \nabla }$

1. Up agrees with the given context. ${\displaystyle TM}$ ${\displaystyle \nabla }$
2. Auf is the common directional derivative of functions: ${\displaystyle T^{0}M}$ ${\displaystyle \nabla }$
   ${\displaystyle \nabla _{X}f=Xf.}$
3. The following product rule applies to ${\displaystyle \nabla }$
   ${\displaystyle \nabla _{X}(F\otimes G)=(\nabla _{X}F)\otimes G+F\otimes (\nabla _{X}G).}$
4. The relationship commutes with the tensor taper , that is ${\displaystyle \nabla }$ ${\displaystyle \operatorname {tr} }$
   ${\displaystyle \nabla _{X}(\operatorname {tr} F)=\operatorname {tr} (\nabla _{X}F).}$

This relationship is also called covariant derivation. ${\displaystyle T_{l}^{k}M}$

Compatibility with Riemannian metrics and symmetry

Let be a Riemannian or a pseudo-Riemannian manifold . A relationship is called compatible with the metric of this manifold, if ${\displaystyle (M,g)}$ ${\displaystyle \nabla }$ ${\displaystyle g}$

${\displaystyle X(g(Y,Z))=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z)}$

applies. The equation is obtained with the third property from the section Relationships on the tensor bundle

${\displaystyle (\nabla _{X}g)(Y,Z)=X(g(Y,Z))-g(\nabla _{X}Y,Z)-g(Y,\nabla _{X}Z)}$

and therefore the compatibility condition is equivalent to

${\displaystyle (\nabla _{X}g)(Y,Z)=0.}$

A relationship is called symmetric or torsion-free if the torsion tensor vanishes, that is, it holds

${\displaystyle \nabla _{X}Y-\nabla _{Y}X=[X,Y].}$

These two properties appear natural, since they are already fulfilled by an induced connection on a real submanifold. A connection on an (abstract) manifold which fulfills these two properties is uniquely determined. This statement is called the main theorem of Riemannian geometry and the uniquely determined relationship is called the Levi-Civita or Riemannian relationship. A relationship that is compatible with the Riemannian metric is called a metric relationship . A Riemannian manifold can generally have several different metric relationships.

properties

• Let and be two vector fields such that in a neighborhood of . Then follows for all vector fields ${\displaystyle p\in M}$ ${\displaystyle Y_{1},Y_{2}}$ ${\displaystyle M}$ ${\displaystyle Y_{1}=Y_{2}}$ ${\displaystyle U}$ ${\displaystyle p}$ ${\displaystyle X}$

${\displaystyle (\nabla _{X}Y_{1})(p)=(\nabla _{X}Y_{2})(p).}$

More generally need and not even be the same on an entire environment. Specifically: If there is a smooth curve are (for an appropriate ) so that and and, if all is valid, already follows . This means that the two vector fields and only need to match along a suitable smooth curve. ${\displaystyle Y_{1}}$ ${\displaystyle Y_{2}}$ ${\displaystyle \gamma :(-\epsilon ,\epsilon )\subset \mathbb {R} \to M}$ ${\displaystyle \epsilon >0}$ ${\displaystyle \gamma (0)=p}$ ${\displaystyle \gamma '(0)=X_{p}}$ ${\displaystyle (Y_{1})_{\gamma (t)}=(Y_{2})_{\gamma (t)}}$ ${\displaystyle |t|<\epsilon }$ ${\displaystyle (\nabla _{X}Y_{1})(p)=(\nabla _{X}Y_{2})(p)}$ ${\displaystyle Y_{1}}$ ${\displaystyle Y_{2}}$

• Analogous to the property just mentioned: Let two vector fields be such that . Then for all that . ${\displaystyle X_{1},X_{2}}$ ${\displaystyle M}$ ${\displaystyle (X_{1})_{p}=(X_{2})_{p}}$ ${\displaystyle Y}$ ${\displaystyle (\nabla _{X_{1}}Y)(p)=(\nabla _{X_{2}}Y)(p).}$

Representation in coordinates: Christoffel symbols

If the local vector fields form a basis of the tangent space at each point, the Christoffel symbols are defined by ${\displaystyle X_{1},\dots ,X_{n}}$

${\displaystyle \nabla _{X_{i}}X_{j}=\sum _{k=1}^{n}\Gamma _{ij}^{k}X_{k}\quad }$or in Einstein's summation convention . ${\displaystyle \quad \nabla _{X_{i}}X_{j}=\Gamma _{ij}^{k}X_{k}\quad }$

Have the vector fields and on these basis the figure and , then for the components of ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X=x^{i}X_{i}}$ ${\displaystyle Y=y^{j}X_{j}}$ ${\displaystyle z^{k}}$ ${\displaystyle \nabla _{X}Y=z^{k}X_{k}}$

${\displaystyle z^{k}=\Gamma _{ij}^{k}x^{i}y^{j}+x^{i}X_{i}(y^{k})}$ ,

where denotes the directional derivative of the function in the direction of the vector . ${\displaystyle X_{i}(y^{k})}$ ${\displaystyle y^{k}}$ ${\displaystyle X_{i}}$

If one specifically selects the vector fields given by a map as basic vector fields , then one obtains the coordinate representation ${\displaystyle \partial _{1},\dots ,\partial _{n}}$

${\displaystyle z^{k}=\Gamma _{ij}^{k}x^{i}y^{j}+x^{i}\partial _{i}y^{k}}$.

This result corresponds to the product rule: In the case of infinitesimal changes, both the basis vectors and the component functions change in the product and the sum of both changes arises. ${\displaystyle Y_{k}y^{k}}$ ${\displaystyle Y_{k}}$ ${\displaystyle y^{k}\,,}$

Applications

In physics, the central terms of this article relate to a. the general theory of relativity and gauge theories (eg. as quantum electrodynamics , quantum chromodynamics and Yang-Mills theory ) of high energy physics, as well as in solid state physics, the BCS theory of superconductivity . What these theories have in common is that “connection” and “covariant derivative” are generated by vector potentials that meet certain calibration conditions, and that they are explicitly included in the energy function of the system in a certain way.

See also

• Parallel transport
• Connection (principal bundle)

literature

• John M. Lee: Riemannian manifolds. An introduction to curvature (= Graduate texts in mathematics 176). Springer, New York NY a. a. 1997, ISBN 0-387-98322-8 .

Web links

Manifold Atlas