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 $$\pi\colon E\to B$$ such that the total space $$E$$ and the base space $$B$$ are both G-spaces (continuous or smooth, depending on the setting) and the projection map $$\pi$$ between them is equivariant: $$\pi \circ g = g \circ \pi$$ 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.