Tangent bundle
As a set, it is given by the disjoint union[note 1] of the tangent spaces of
This projection maps each element of the tangent space
The tangent bundle comes equipped with a natural topology (described in a section below).
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).
One of the main roles of the tangent bundle is to provide a domain and range for the derivative of a smooth function.
which restricts to a linear isomorphism from each tangent space
Trivial tangent bundles usually occur for manifolds equipped with a 'compatible group structure'; for instance, in the case where the manifold is a Lie group.
The tangent bundle of the unit circle is trivial because it is a Lie group (under multiplication and its natural differential structure).
It is not true however that all spaces with trivial tangent bundles are Lie groups; manifolds which have a trivial tangent bundle are called parallelizable.
If M is a smooth n-dimensional manifold, then it comes equipped with an atlas of charts
are induced by the Jacobian matrices of the associated coordinate transformation and are therefore smooth maps between open subsets of
whose transition functions are given by the Jacobian of the associated coordinate transformations.
The tangent bundle of the circle is also trivial and isomorphic to
The only tangent bundles that can be readily visualized are those of the real line
A simple example of a nontrivial tangent bundle is that of the unit sphere
: this tangent bundle is nontrivial as a consequence of the hairy ball theorem.
In the language of fiber bundles, such a map is called a section.
then takes on the structure of a module over the commutative algebra of smooth functions on M, denoted
That is, a local vector field is defined only on some open set
forms a structure known as a sheaf of real vector spaces on
The above construction applies equally well to the cotangent bundle – the differential 1-forms on
is itself a smooth manifold, the second-order tangent bundle can be defined via repeated application of the tangent bundle construction: In general, the
has an induced derivative, for which the tangent bundle is the appropriate domain and range
, considered as a manifold itself, one can define a canonical vector field
as the diagonal map on the tangent space at each point.
since the vector space itself is flat, and thus has a natural diagonal map
Applying this product structure to the tangent space at each point and globalizing yields the canonical vector field.
, which is an alternative description of the canonical vector field.
is analogous to the canonical one-form on the cotangent bundle.
(say, a Riemannian metric), there is no similar lift into the cotangent bundle.
