Fiber bundles ed

Something that looks locally like a product \[M \times F\] of manifolds.

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.