Connections on principal fiber bundles ed

useful:

There are several equivalent definitions of connections on a principal bundle \[(P, \pi, M, G)\]...

Table of Contents

As horizontal subspaces ed

Definition ed

vertical subspaces
Each fiber has a natural definition of a vertical tangential space \[Th_p = \operatorname{ker} \pi^\star \subset T_p\]. We have \[\dim Th_p = \dim \mathfrak g\].

connections = horizontal spaces
A connection is a choice of horizontal space \[Tv_p\], i.e. \[T_p = Tv_p \oplus Th_p\], that is also right invariant:
\[ R^\star_g \, Th_p = Th_{gp} \]

Parallel transport ed

A curve \[\gamma\] on the base manifold \[M\] can be horizontally lifted onto \[P\] along the connection. For a given starting point \[p\in P\] of the lift, this is unique.

If \[\gamma\] is a closed loop, start point \[p\] and end point \[p'\] might be different but have to lie in the same fiber. Therefore there is a \[g \in G\] with \[p' = g \cdot p\].

A connection is called flat if always \[p=p'\] or \[g=e\]. Then the connection is integrable to a surface.

Connection forms ed

Categories: Mathematik