A vector bundle is special class of fiber bundle in which the fiber is a vector
space 
.
Technically, a little more is required; namely, if 
is a bundle with fiber 
, to be a vector bundle, all of the
fibers 
for 
need to have a coherent vector
space structure. One way to say this is that the "trivializations"
, are fiber-for-fiber vector space isomorphisms.
A vector bundle is a total space 
along with a surjective map
to a base manifold 
. Any fiber 
is a vector space isomorphic to 
.
The simplest nontrivial vector bundle is a line bundle on the circle, and is analogous to the Möbius strip.
One use for vector bundles is a generalization of vector functions. For instance, the tangent vectors of an 
-dimensional manifold are isomorphic to 
at a point 
in a coordinate chart.
But the isomorphism with 
depends on the choice of coordinate chart. Nearby
, the vector fields look like functions.
To define vector fields on the whole manifold requires the tangent
bundle, which is a special case of a vector bundle.
A bundle section of a vector bundle 
is a map 
whose projection, 
is the identity map on 
. For instance, on a trivial
bundle 
,
a section 
corresponds to a function 
by 
.
Near every point in a vector bundle, there is a trivialization. The structure of the vector bundle, as in all bundles, is that it is locally trivial. In the case of a vector bundle, the transition functions between the trivializations take values in linear invertible transformations of the fiber.
Since the element zero in 
is fixed by any linear transformation, the zero section always
exists. By "nontrivial section," it is meant that it is not the zero section.
There are several adjectives that can specify properties of a vector bundle. A complex vector bundle has a fiber 
that is a complex vector
space. A real vector bundle has a fiber
that is a real vector space, which is the default
kind of vector bundle. A line bundle has a fiber that
is one dimensional.
A continuous vector bundle is a manifold 
with a continuous
projection map 
.
A smooth vector bundle is a smooth manifold
with a smooth projection 
. Finally, a holomorphic
vector bundle is a complex manifold 
with a holomorphic
projection 
.
In this last case, the fiber must be a complex vector space. So there could be a
smooth complex vector bundle, but not a holomorphic real vector bundle.
Vector bundles can have metrics on their fibers, either Riemannian or Hermitian, and vector bundle connections.
