The base manifold in a bundle is analogous to the domain for a set of functions. In fact,
a bundle, by definition, comes with a map
to the base manifold, often called 
or projection.
For example, the base manifold to the tangent bundle of a manifold 
is the manifold 
. A vector field is a function
from the manifold to the tangent
bundle, with the restriction that every point gets
mapped to a manifold tangent vector at
that point. In general, a bundle
with nonempty fibers has
bundle sections, at least locally,
which are maps from the base manifold to the bundle.
