A vector bundle is a 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 vector space of
tangent vectors at a point 
of an 
-dimensional manifold
is isomorphic to 
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 invertible linear transformations of the fiber.
Since the zero vector 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 has continuous local trivializations that
are linear on each fiber.
A smooth vector bundle over a smooth manifold
has local trivializations
that are smooth maps and whose restrictions to the
fibers are linear
transformations. A holomorphic vector
bundle over a complex manifold has complex
vector spaces as fibers and local trivializations that are holomorphic
maps and linear over 
on each fiber. The corresponding
projections have the same regularity. A smooth complex
vector bundle need not be a holomorphic
vector bundle; holomorphic vector bundles
have complex fibers.
Vector bundles can have inner products on their fibers, with Hermitian inner products in the complex case, and vector bundle connections.
