Equivariant bundle

In geometry and topology, given a group G (which may be a topological or Lie group), an equivariant bundle is a fiber bundle ${\displaystyle \pi \colon E\to B}$ such that the total space ${\displaystyle E}$ and the base space ${\displaystyle B}$ are both G-spaces (continuous or smooth, depending on the setting) and the projection map ${\displaystyle \pi }$ between them is equivariant: ${\displaystyle \pi \circ g=g\circ \pi }$ with some extra requirement depending on a typical fiber.

For example, an equivariant vector bundle is an equivariant bundle such that the action of G restricts to a linear isomorphism between fibres.