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,
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 ndimensional sphere S^{n} is framed for all n, but parallelizable only for n = 1, 3, 7 (by results of BottMilnor and Kervaire).
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Construction of the tangent bundle  Lec 10  Frederic Schuller

What is a Manifold? Lesson 13: The tangent bundle  an illustration.

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Differential structures: the pivotal concept of tangent vector spaces  Lec 09  Frederic Schuller
Transcription
Role
One of the main roles of the tangent bundle is to provide a domain and range for the derivative of a smooth function. Namely, if is a smooth function, with and smooth manifolds, its derivative is a smooth function .
Topology and smooth structure
The tangent bundle comes equipped with a natural topology (not the disjoint union topology) and smooth structure so as to make it into a manifold in its own right. The dimension of is twice the dimension of .
Each tangent space of an ndimensional manifold is an ndimensional vector space. If is an open contractible subset of , then there is a diffeomorphism which restricts to a linear isomorphism from each tangent space to . As a manifold, however, is not always diffeomorphic to the product manifold . When it is of the form , 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, tangent bundles are locally modeled on , where is an open subset of Euclidean space.
If M is a smooth ndimensional manifold, then it comes equipped with an atlas of charts , where is an open set in and
is a diffeomorphism. These local coordinates on give rise to an isomorphism for all . We may then define a map
by
We use these maps to define the topology and smooth structure on . A subset of is open if and only if
is open in for each These maps are homeomorphisms between open subsets of and and therefore serve as charts for the smooth structure on . The transition functions on chart overlaps are induced by the Jacobian matrices of the associated coordinate transformation and are therefore smooth maps between open subsets of .
The tangent bundle is an example of a more general construction called a vector bundle (which is itself a specific kind of fiber bundle). Explicitly, the tangent bundle to an dimensional manifold may be defined as a rank vector bundle over whose transition functions are given by the Jacobian of the associated coordinate transformations.
Examples
The simplest example is that of . In this case the tangent bundle is trivial: each is canonically isomorphic to via the map which subtracts , giving a diffeomorphism .
Another simple example is the unit circle, (see picture above). The tangent bundle of the circle is also trivial and isomorphic to . Geometrically, this is a cylinder of infinite height.
The only tangent bundles that can be readily visualized are those of the real line and the unit circle , both of which are trivial. For 2dimensional manifolds the tangent bundle is 4dimensional and hence difficult to visualize.
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. 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. Specifically, a vector field on a manifold is a smooth map
such that with for every . In the language of fiber bundles, such a map is called a section. A vector field on is therefore a section of the tangent bundle of .
The set of all vector fields on is denoted by . Vector fields can be added together pointwise
and multiplied by smooth functions on M
to get other vector fields. The set of all vector fields then takes on the structure of a module over the commutative algebra of smooth functions on M, denoted .
A local vector field on is a local section of the tangent bundle. That is, a local vector field is defined only on some open set and assigns to each point of a vector in the associated tangent space. The set of local vector fields on forms a structure known as a sheaf of real vector spaces on .
The above construction applies equally well to the cotangent bundle – the differential 1forms on are precisely the sections of the cotangent bundle , that associate to each point a 1covector , which map tangent vectors to real numbers: . Equivalently, a differential 1form maps a smooth vector field to a smooth function .
Higherorder tangent bundles
Since the tangent bundle is itself a smooth manifold, the secondorder tangent bundle can be defined via repeated application of the tangent bundle construction:
In general, the th order tangent bundle can be defined recursively as .
A smooth map has an induced derivative, for which the tangent bundle is the appropriate domain and range . Similarly, higherorder tangent bundles provide the domain and range for higherorder derivatives .
A distinct but related construction are the jet bundles on a manifold, which are bundles consisting of jets.
Canonical vector field on tangent bundle
On every 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 is naturally a product, since the vector space itself is flat, and thus has a natural diagonal map given by under this product structure. Applying this product structure to the tangent space at each point and globalizing yields the canonical vector field. Informally, although the manifold is curved, each tangent space at a point , , is flat, so the tangent bundle manifold is locally a product of a curved and a flat Thus the tangent bundle of the tangent bundle is locally (using for "choice of coordinates" and for "natural identification"):
and the map 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.
If are local coordinates for , the vector field has the expression
More concisely, – the first pair of coordinates do not change because it is the section of a bundle and these are just the point in the base space: the last pair of coordinates are the section itself. This expression for the vector field depends only on , not on , as only the tangent directions can be naturally identified.
Alternatively, consider the scalar multiplication function:
The derivative of this function with respect to the variable at time is a function , which is an alternative description of the canonical vector field.
The existence of such a vector field on is analogous to the canonical oneform on the cotangent bundle. Sometimes is also called the Liouville vector field, or radial vector field. Using one can characterize the tangent bundle. Essentially, 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 objects on into objects on . For example, if is a curve in , then (the tangent of ) is a curve in . In contrast, without further assumptions on (say, a Riemannian metric), there is no similar lift into the cotangent bundle.
The vertical lift of a function is the function defined by , where is the canonical projection.
See also
Notes
 ^ ^{a} ^{b} The disjoint union ensures that for any two points x_{1} and x_{2} of manifold M the tangent spaces T_{1} and T_{2} have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle S^{1}, see Examples section: all tangents to a circle lie in the plane of the circle. In order to make them disjoint it is necessary to align them in a plane perpendicular to the plane of the circle.
References
 Lee, Jeffrey M. (2009), Manifolds and Differential Geometry, Graduate Studies in Mathematics, vol. 107, Providence: American Mathematical Society. ISBN 9780821848159
 Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218. doi:10.1007/9781441999825. ISBN 9781441999818.
 Jürgen Jost, Riemannian Geometry and Geometric Analysis, (2002) SpringerVerlag, Berlin. ISBN 3540426272
 Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) BenjaminCummings, London. ISBN 080530102X
 León, M. De; Merino, E.; Oubiña, J. A.; Salgado, M. (1994). "A characterization of tangent and stable tangent bundles" (PDF). Annales de l'I.H.P.: Physique Théorique. 61 (1): 1–15.
 Gudmundsson, Sigmundur; Kappos, Elias (2002). "On the geometry of tangent bundles". Expositiones Mathematicae. 20: 1–41. doi:10.1016/S07230869(02)800275.
External links
 "Tangent bundle", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 Wolfram MathWorld: Tangent Bundle
 PlanetMath: Tangent Bundle