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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the tangent bundle of a smooth (or differentiable) manifold
Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to Euclidean space to allow one to do calculus. This article deals with differentiability in different contexts including: smooth function, k times differentiable, and holomorphic function....
 M, denoted by T(M) or just TM, is the disjoint union
Disjoint union

In set theory, a disjoint union is a modified union operation which indexes the elements according to which set they originated in.Formally, let be a family of sets indexed by I....
 The disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle S1, see examples below: all tangents to a circle lie in the plane of the circle.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the tangent bundle of a smooth (or differentiable) manifold
Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to Euclidean space to allow one to do calculus. This article deals with differentiability in different contexts including: smooth function, k times differentiable, and holomorphic function....
 M, denoted by T(M) or just TM, is the disjoint union
Disjoint union

In set theory, a disjoint union is a modified union operation which indexes the elements according to which set they originated in.Formally, let be a family of sets indexed by I....
 The disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle S1, see examples below: 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 plane. of the tangent space
Tangent space

In mathematics, the tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
s of the points x of M

An element of TM is a pair (x,v) where x ? M and v ? TxM, the corresponding tangent spaces at x. There is a natural projection

which sends (x,v) to the base point x.

Role


The main role 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
Derivative (generalizations)

Derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry....
 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
Differential structure

In mathematics, an n-dimensional differential structure on a set M makes it into an n-dimensional differential manifold, which is a Topological manifold with some additional structure that allows us to do differential calculus on the manifold....
 so as to make it into a manifold in its own right. The dimension of TM is twice the dimension of M.

Each tangent space of an n-dimensional manifold is an n-dimensional vector space. If U is an open contractible
Contractible space

In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map....
 subset of M, then there is a diffeomorphism
Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that map s one differentiable manifold to another, such that both the function and its inverse are smooth function....
 from TU to U × Rn which restricts to a linear isomorphism from each tangent space TxU to × Rn . As a manifold, however, TM is not always diffeomorphic to the product manifold M × Rn. When it is of the form M × Rn, 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
Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
. 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 modelled on Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
, tangent bundles are locally modelled on U × Rn, where U is an open subset of Euclidean space.

If M is a smooth n-dimensional manifold, then it comes equipped with an atlas
Atlas (topology)

In mathematics, particularly topology, an atlas describes how a manifold is equipped with a differential structure. Each piece is given by a chart ....
 of charts (Ua, fa) where Ua is an open set in M and is a diffeomorphism
Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that map s one differentiable manifold to another, such that both the function and its inverse are smooth function....
. These local coordinates on U give rise to an isomorphism between TxM and Rn for each x ? U. We may then define a map by We use these maps to define the topology and smooth structure on TM. A subset A of TM is open if and only if is open in R2n for each a. These maps are then homeomorphisms between open subsets of TM and R2n and therefore serve as charts for the smooth structure on TM. 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 R2n.

The tangent bundle is an example of a more general construction called a vector bundle
Vector bundle

In mathematics, a vector bundle is a topology construction which makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together to form another space of the same kind as X , which is t...
 (which is itself a specific kind of fiber bundle
Fiber bundle

File:Roundhairbrush.JPGIn mathematics, and particularly topology, a fiber bundle is intuitively a space E which locally "looks" like a product space B ? F, but globally may have a different topological structure....
). Explicitly, the tangent bundle to an n-dimensional manifold M may be defined as a rank n vector bundle over M whose transition functions are given by the Jacobian
Jacobian

In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant.In algebraic geometry the Jacobian of a algebraic curve means the Jacobian variety: a group variety associated to the curve, in which the curve can be embedded....
 of the associated coordinate transformations.

Examples

The simplest example is that of Rn. In this case the tangent bundle is trivial.

Another simple example is the unit circle
Unit circle

In mathematics, a unit circle is a circle with a 1 radius, i.e., a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin in the Cartesian coordinate system in the Euclidean plane....
, S1 (see picture above). The tangent bundle of the circle is also trivial and isomorphic to S1 × R. Geometrically, this is a cylinder
Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....
 of infinite height (see the bottom picture).

The only tangent bundles that can be readily visualized are those of the real line R and the unit circle S1, both of which are trivial. For 2-dimensional manifolds the tangent bundle is 4-dimensional and hence not easily visualizable.

A simple example of a nontrivial tangent bundle is that of the unit sphere S2: this tangent bundle is nontrivial as a consequence of the hairy ball theorem
Hairy ball theorem

The hairy ball theorem of algebraic topology states that there is no nonvanishing continuous function tangent vector vector field on the sphere....
. 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
Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic field or gravity for...
. Specifically, a vector field on a manifold M is a smooth map such that the image of x, denoted Vx, lies in TxM, the tangent space at x. In the language of fiber bundles, such a map is called a section
Section (fiber bundle)

In the mathematical field of topology, a section of a fiber bundle, π: E → B, over a topological space, B, is a continuous map, s : B → E, such that π=x for all x in B....
. A vector field on M is therefore a section of the tangent bundle of M.

The set of all vector fields on M is denoted by G(TM). 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 G(TM) then takes on the structure of a module
Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalar to lie in a field , the "scalars" may lie in an arbitrary ring....
 over the commutative algebra
Commutative algebra

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideal , and module over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra....
 of smooth functions on M, denoted C8(M).

A local vector field on M is a local section of the tangent bundle. That is, a local vector field is defined only on some open set U in M and assigns to each point of U a vector in the associated tangent space. The set of local vector fields on M forms a structure known as a sheaf
Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one....
 of real vector spaces on M.

Higher-order tangent bundles

Since the tangent bundle is itself a smooth manifold, the second-order tangent bundle
Double tangent bundle

In differential topology, a subject in mathematics, the double tangent bundle is an object designed to hold the 2nd order derivative information of a differentiable manifold....
 can be defined via repeated application of the tangent bundle construction:

In general, the th-order tangent bundle can be defined inductively as .

A smooth map has an induced derivative, for which the tangent bundle is the appropriate domain and range . Similarly, higher-order tangent bundles provide the domain and range for higher-order derivatives .

A distinct but related construction are the jet bundle
Jet bundle

In differential geometry, the jet bundle is a certain construction which makes a new smooth_manifold fiber bundle out of a given smooth fiber bundle....
s on a manifold, which are vector bundles consisting of jets
Jet (mathematics)

In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain....
.

Canonical vector field on tangent bundle

On every tangent bundle TM one can define a canonical vector field . If (x, v) are local coordinates for TM, the vector field has the expression Alternatively, consider be 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 TM can be compared with the existence of a canonical 1-form on the cotangent bundle
Cotangent bundle

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold....
. Sometimes V is also called the Liouville vector field, or radial vector field. Using V one can characterize the tangent bundle. Essentially, V 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 M into objects on TM. For example, if c is a curve in M, then c (the tangent
Tangent

In geometry, the tangent line to a curve at a given Point is the straight line that "just touches" the curve at that point . As it passes through the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point....
 of c) is a curve in TM. Let us point out that without further assumptions on M (say, a Riemannian metric), there is no similar lift into the cotangent bundle
Cotangent bundle

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold....
.

The vertical lift of a function is the function defined by , where is the canonical projection.

See also

  • pushforward (differential)
  • vector field
    Vector field

    In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic field or gravity for...
  • cotangent bundle
    Cotangent bundle

    In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold....
  • frame bundle
    Frame bundle

    In mathematics, a frame bundle is a principal fiber bundle F associated to any vector bundle E. The fiber of F over a point x is the set of all ordered basis, or frames, for Ex....
  • Musical isomorphism
    Musical isomorphism

    In mathematics, the musical isomorphism is an isomorphism between the tangent bundle TM and the cotangent bundle of a Riemannian manifold given by its Riemannian metric....


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