in a fibre E_{m_1} of the vector bundle E, and this fibre is mapped down to the point m_1 by the
projection \pi: E \to M. A
real vector bundle consists of: • topological spaces X (
base space) and E (
total space) • a
continuous surjection \pi:E\to X (
bundle projection) • for every x in X, the structure of a
finite-dimensional real vector space on the
fiber \pi^{-1}(\{x\}) where the following compatibility condition is satisfied: for every point p in X, there is an
open neighborhood U\subseteq X of p, a
natural number k, and a
homeomorphism \varphi:U \times \R^k \to \pi^{-1}(U) such that for all x in U, • (\pi \circ \varphi)(x,v) = x for all
vectors v in \R^k, and • the map v \mapsto \varphi (x, v) is a
linear isomorphism between the vector spaces \R^k and \pi^{-1}(\{x\}). The open neighborhood U together with the homeomorphism \varphi is called a
local trivialization of the vector bundle. The local trivialization shows that
locally the map \pi "looks like" the projection of U\times\R^k on U. Every fiber \pi^{-1}(\{x\}) is a finite-dimensional real vector space and hence has a
dimension k_x. The local trivializations show that the
function x\to k_x is
locally constant, and is therefore constant on each
connected component of X. If k_x is equal to a constant k on all of X, then k is called the
rank of the vector bundle, and E is said to be a
vector bundle of rank k. Often the definition of a vector bundle includes that the rank is well defined, so that k_x is constant. Vector bundles of rank 1 are called
line bundles, while those of rank 2 are less commonly called plane bundles. The
Cartesian product X\times\R^k, equipped with the projection X\times\R^k\to X, is called the
trivial bundle of rank k over X.
Transition functions s U_\alpha and U_\beta may be
glued over the intersection U_{\alpha\beta} by transition functions g_{\alpha \beta} which serve to stick the shaded grey regions together after applying a
linear transformation to the fibres (note the transformation of the blue
quadrilateral under the effect of g_{\alpha\beta}). Different choices of transition functions may result in different vector bundles which are non-trivial after gluing is complete. can be constructed by a non-trivial gluing of two trivial bundles on open
subsets
U and
V of the
circle S1. When glued trivially (with
gUV=1) one obtains the trivial bundle, but with the non-trivial gluing of
gUV=1 on one overlap and
gUV=-1 on the second overlap, one obtains the non-trivial bundle
E, the Möbius strip. This can be visualised as a "twisting" of one of the local
charts. Given a vector bundle E\to X of rank k, and a pair of neighborhoods U and V over which the bundle trivializes via :\begin{align} \varphi_U\colon U\times \R^k &\mathrel{\xrightarrow{\cong}} \pi^{-1}(U), \\ \varphi_V\colon V\times \R^k &\mathrel{\xrightarrow{\cong}} \pi^{-1}(V) \end{align} the
composite function :\varphi_U^{-1}\circ\varphi_V \colon (U\cap V)\times\R^k\to (U\cap V)\times\R^k is well-defined on the overlap, and satisfies :\varphi_U^{-1}\circ\varphi_V (x,v) = \left (x,g_{UV}(x)v \right) for some
\text{GL}(k)-valued function :g_{UV}\colon U\cap V\to \operatorname{GL}(k). These are called the
transition functions (or the
coordinate transformations) of the vector bundle. The
set of transition functions forms a
Čech cocycle in the sense that :g_{UU}(x) = I, \quad g_{UV}(x)g_{VW}(x)g_{WU}(x) = I for all U,V,W over which the bundle trivializes satisfying U\cap V\cap W\neq \emptyset. Thus the data (E,X,\pi,\R^k) defines a
fiber bundle; the additional data of the g_{UV} specifies a \text{GL}(k) structure group in which the
action on the fiber is the standard action of \text{GL}(k). Conversely, given a fiber bundle (E,X,\pi,\R^k) with a \text{GL}(k) cocycle acting in the standard way on the fiber \R^k, there is
associated a vector bundle. This is an example of the
fibre bundle construction theorem for vector bundles, and can be taken as an alternative definition of a vector bundle.
Subbundles One simple method of constructing vector bundles is by taking subbundles of other vector bundles. Given a vector bundle \pi: E\to X over a topological space, a subbundle is simply a
topological subspace F\subset E for which the
restriction \left.\pi\right|_F of \pi to F gives \left.\pi\right|_F: F \to X the structure of a vector bundle also. In this case the fibre F_x\subset E_x is a vector subspace for every x\in X. A subbundle of a trivial bundle need not be trivial, and indeed every real vector bundle over a compact space can be viewed as a subbundle of a trivial bundle of sufficiently high rank. For example, the
Möbius band, a non-trivial
line bundle over the circle, can be seen as a subbundle of the trivial rank 2 bundle over the circle. ==Vector bundle morphisms==