![cover image](https://wikiwandv2-19431.kxcdn.com/_next/image?url=https://upload.wikimedia.org/wikipedia/commons/thumb/8/87/Tangent_bundle.svg/640px-Tangent_bundle.svg.png&w=640&q=50)
Tangent bundle
Tangent spaces of a manifold / From Wikipedia, the free encyclopedia
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold is a manifold
which assembles all the tangent vectors in
. As a set, it is given by the disjoint union[note 1] of the tangent spaces of
. That is,
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/8/87/Tangent_bundle.svg/320px-Tangent_bundle.svg.png)
where denotes the tangent space to
at the point
. So, an element of
can be thought of as a pair
, where
is a point in
and
is a tangent vector to
at
.
There is a natural projection
defined by . This projection maps each element of the tangent space
to the single point
.
The tangent bundle comes equipped with a natural topology (described in a section below). With this topology, the tangent bundle to a manifold is the prototypical example of a vector bundle (which is a fiber bundle whose fibers are vector spaces). A section of is a vector field on
, and the dual bundle to
is the cotangent bundle, which is the disjoint union of the cotangent spaces of
. By definition, a manifold
is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold
is framed if and only if the tangent bundle
is stably trivial, meaning that for some trivial bundle
the Whitney sum
is trivial. For example, the n-dimensional sphere Sn is framed for all n, but parallelizable only for n = 1, 3, 7 (by results of Bott-Milnor and Kervaire).