Banach bundle One can take a Banach bundle over one manifold and use the pull-back construction to define a new Banach bundle on a second manifold. Specifically, let π : E → N be a Banach bundle and f : M → N a differentiable map (as usual, everything is Cp). Then the pull-back of π : E → N is the Banach bundle f*π : f*E → M satisfying the following properties: for each x ∈ M, (f*E)x = Ef(x); there is a commutative diagram with the top horizontal map being the identity on each fibre; if E is trivial, i.e. equal to N × X for some Banach space X, then f*E is also trivial and equal to M × X, and is the projection onto the first coordinate; if V is an open subset of N and U = f−1(V), then and there is a commutative diagram where the maps at the "front" and "back" are the same as those in the previous diagram, and the maps from "back" to "front" are (induced by) the inclusions. References Banach spaces Differential geometry Generalized manifolds Manifolds Nonlinear functional analysis Structures on manifolds Vector bundles