Fiber bundles ed

Something that looks locally like a product \(M \times F\) of manifolds.

useful: Lie groups

General ed

definition

Manifolds \(E, M, F\) with (\(M\) base space, \(E\) total space, \(F\) fiber)

a surjective projection \( \pi : E \rightarrow M \)

\(M\) has a cover of regions \(U \subset M\) with maps \(\psi : \pi^{-1}(U) \rightarrow U \times F\), with \( \psi(x) = (x,f) \) (local trivialisation)

sections

maps \( \sigma : M \rightarrow E \) with \( \pi \circ \sigma = \operatorname{id} \)

Vector bundle ed

Special case, that the fiber \(F\) is a vectorspace \(V\). Now, sections are vector fields on \(M\).

Famous example is the tangent bundle \(TM\).

Principal fiber bundle ed

Using a Lie group \(G\) as the fiber and usually calling the total space \(P\). But we have an additional action of \(G\) on \(P\), i.e. a multiplication \(p \cdot g \). This action should be

• preserving fibers: \(p \cdot g \in \pi^{-1}(p) \)
• respecting group structure: \( (p g) h = p (g h) \)
• free: \( p g = p \Rightarrow g = \operatorname{id} \)
• fiber-transitive: \(p, q\) in the same fiber \( \Rightarrow \exists g: p g = q \)

Informally, each fiber is the group \(G\), but forgetting the information, which element is the identity.