Tangent bundle

Tangent bundle

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In differential geometry, the tangent bundle of a differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

 M is the disjoint union
Disjoint union
In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...

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 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. of the tangent space
Tangent space
In mathematics, the tangent space of a manifold 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 M. That is,
where TxM denotes the tangent space
Tangent space
In mathematics, the tangent space of a manifold 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....

 to M at the point x. So, an element of TM can be thought of as a pair
Ordered pair
In mathematics, an ordered pair is a pair of mathematical objects. In the ordered pair , the object a is called the first entry, and the object b the second entry of the pair...

 (x, v), where x is a point in M and v is a tangent vector to M at x. There is a natural projection
Projection (mathematics)
Generally speaking, in mathematics, a projection is a mapping of a set which is idempotent, which means that a projection is equal to its composition with itself. A projection may also refer to a mapping which has a left inverse. Bot notions are strongly related, as follows...


defined by π(x, v) = x. This projection maps each tangent space TxM to the single point x.

The tangent bundle to a manifold is the prototypical example of a vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that 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...

 (a fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...

 whose fibers are vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

s). A section
Section (fiber bundle)
In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...

 of TM is a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

 on M, and the dual bundle
Dual bundle
In mathematics, the dual bundle of a vector bundle π : E → X is a vector bundle π* : E* → X whose fibers are the dual spaces to the fibers of E...

 to TM is 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...

, which is the disjoint union of the cotangent space
Cotangent space
In differential geometry, one can attach to every point x of a smooth manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions...

s of M. By definition, a manifold M is parallelizable if and only if the tangent bundle is trivial.
By definition, a manifold M is framed if and only if the tangent bundle TM is stably trivial, meaning that for some trivial bundle E the Whitney sum
Vector bundle
In mathematics, a vector bundle is a topological construction that 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...

  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).

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)
The derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry.- Derivatives in analysis :...

 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
Smooth structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis 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. Intuitively, a contractible space is one that can be continuously shrunk to a point....

 subset of M, then there is a diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...

 from TU to U × Rn which restricts to a linear isomorphism from each tangent space TxU to {x}× 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 smooth 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
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

, 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, one describesa manifold using an atlas. An atlas consists of individualcharts that, roughly speaking, describe individual regionsof the manifold. If the manifold is the surface of the Earth,...

 of charts (Uα, φα) where Uα is an open set in M and
is a diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...

. 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 α. 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 topological construction that 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...

 (which is itself a specific kind of fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, 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 matrix is the matrix of all first-order partial derivatives of a vector- or scalar-valued function with respect to another vector. Suppose F : Rn → Rm is a function from Euclidean n-space to Euclidean m-space...

 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 radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one 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 basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a 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 difficult to visualize.

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 tangent vector field on an even-dimensional n-sphere. An ordinary sphere is a 2-sphere, so that this theorem will hold for an ordinary 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 vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

. 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 is a continuous right inverse of the function π...

. 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 Γ(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 Γ(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, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

 over the commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...

 of smooth functions on M, denoted C(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...

 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 mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle of the total space TM of the tangent bundle of a smooth manifold M...

 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, 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 fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form...

s on a manifold, which are 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 to 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 plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

 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)
  • unit tangent bundle
    Unit tangent bundle
    In Riemannian geometry, a branch of mathematics, the unit tangent bundle of a Riemannian manifold , denoted by UT or simply UTM, is the unit sphere bundle for the tangent bundle T...

  • 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 bases, 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 T*M of a Riemannian manifold given by its metric. There are similar isomorphisms on symplectic manifolds....


External links