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Wikipedia / Tangent bundle

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. As a set, it is given by the disjoint union. That is,

denotes the tangent space can be thought of as a pair.

There is a natural projection

.

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 is a vector field, and the dual bundle is the cotangent bundle, which is the disjoint union of the cotangent spaces is parallelizable if and only if the tangent bundle is trivial is framed the Whitney sum is trivial. For example, the n-dimensional sphere Sn is framed for all n, but parallelizable only for (by results of Bott-Milnor and Kervaire).

Role

smooth manifolds, its derivative.

Topology and smooth structure

The tangent bundle comes equipped with a natural topology (not the disjoint union topology) and smooth structure.

Each tangent space of an n-dimensional manifold is an n is an open contractible, then there is a diffeomorphism, then the tangent bundle is said to be trivial. 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. Just as manifolds are locally modeled on Euclidean space is an open subset of Euclidean space.

If M is a smooth n-dimensional manifold, then it comes equipped with an atlas and

is a diffeomorphism. We may then define a map

by

is open if and only if

are induced by the Jacobian matrices.

The tangent bundle is an example of a more general construction called a vector bundle (which is itself a specific kind of fiber bundle whose transition functions are given by the Jacobian of the associated coordinate transformations.

Examples

.

Another simple example is the unit circle. Geometrically, this is a cylinder of infinite height.

, both of which are trivial. For 2-dimensional manifolds the tangent bundle is 4-dimensional and hence difficult to visualize.

: this tangent bundle is nontrivial as a consequence of the hairy ball theorem. Therefore, the sphere is not parallelizable.

Vector fields

A smooth assignment of a tangent vector to each point of a manifold is called a vector field is a smooth map

. In the language of fiber bundles, such a map is called a section.

. Vector fields can be added together pointwise

and multiplied by smooth functions on M

then takes on the structure of a module over the commutative algebra of smooth functions on M.

is a local section forms a structure known as a sheaf.

.

Higher-order tangent bundles

is itself a smooth manifold, the second-order tangent bundle can be defined via repeated application of the tangent bundle construction:

.

.

A distinct but related construction are the jet bundles on a manifold, which are bundles consisting of jets.

Canonical vector field on tangent bundle

, considered as a manifold itself, one can define a canonical vector field as the diagonal map on the tangent space at each point. This is possible because the tangent space of a vector space W for "natural identification"):

is the projection onto the first coordinates:

Splitting the first map via the zero section and the second map by the diagonal yields the canonical vector field.

, the vector field has the expression

, as only the tangent directions can be naturally identified.

Alternatively, consider the scalar multiplication function:

, which is an alternative description of the canonical vector field.

is analogous to the canonical one-form on the cotangent bundle is also called the Liouville vector field, or radial vector field can be characterized using 4 axioms, and if a manifold has a vector field satisfying these axioms, then the manifold is a tangent bundle and the vector field is the canonical vector field on it. See for example, De León et al.

Lifts

There are various ways to lift (the tangent (say, a Riemannian metric), there is no similar lift into the cotangent bundle.

The vertical lift is the canonical projection.

See also

Notes

References