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

Alternatively, one can define

Definition ed

A connection form \(A\) is a \(\mathfrak g\)-valued 1-form on \(P\) with

The first condition completely defines the vertical behavior (\(A_v = \mu_G\)). The remaining degrees of freedom are equivalent to choosing a horizontal subspace via \(Th_p = \operatorname{ker} A_p\).

The second condition sets the equivariant transformation behavior along the fibers. Informally, moving \(A\) transforms its vector-input and \(\mathfrak g\)-output. To mimic the horizontal spaces, the input transforms trivially, while the output transforms like the fundamental vector fields.

Curvature ed

\[ F^A = dA + \frac{1}{2} [A \wedge A] \]

Categories: Mathematik