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tangent bundle

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**tangent bundle**of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in

*M*. As a set, it is given by the disjoint unionThe 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. of the tangent spaces of

*M*. That is,

begin{array}{lcr} TM = bigsqcup_{x in M} T_xM

= bigcup_{x in M} left{xright} times T_xM

~~~~~~~ = bigcup_{x in M} left{(x, y) mid y in T_xMright} ~~~~~~~= left{ (x, y) mid x in M,, y in T_xM right}. end{array}where T_x M denotes the tangent space to M at the point x . So, an element of TM can be thought of as a 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
= bigcup_{x in M} left{xright} times T_xM

pi : TM twoheadrightarrow M

defined by pi(x, v) = x. This projection maps each tangent space T_xM to the single point x .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 (a fiber bundle whose fibers are vector spaces). A section of TM is a vector field on M, and the dual bundle to TM is the cotangent bundle, which is the disjoint union of the cotangent spaces 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 {{nowrap|1=

*TM*âŠ•

*E*}} is trivial. For example, the

*n*-dimensional sphere

*Sn*is framed for all

*n*, but parallelizable only for {{nowrap|1=

*n*= 1, 3, 7}} (by results of Bott-Milnor and Kervaire).

## 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*f*:

*M*â†’

*N*is a smooth function, with

*M*and

*N*smooth manifolds, its derivative is a smooth function

*Df*:

*TM*â†’

*TN*.

## 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

*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 subset of

*M*, then there is a diffeomorphism from

*TU*to

*U*Ã—

**R**

*n**which restricts to a linear isomorphism from each tangent space*T*'x***'U***to {*x*} Ã—**R**n*. As a manifold, however,

*TM*is not always diffeomorphic to the product manifold

*M*Ã—

**R**

*n**. When it is of the form*M*Ã—**R**n*, 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 modelled on Euclidean space, tangent bundles are locally modelled on

*U*Ã—

**R**

*n*, where

*U*is an open subset of Euclidean space.If

*M*is a smooth

*n*-dimensional manifold, then it comes equipped with an atlas of charts (

*U*Î±, Ï†Î±) where

*U*Î± is an open set in

*M*and

phi_alphacolon U_alpha to mathbf R^n

is a diffeomorphism. These local coordinates on *U*give rise to an isomorphism between

*T*and

**'x**'M**R**

*n*for each

*x*âˆˆ

*U*. We may then define a map

widetildephi_alphacolon pi^{-1}left(U_alpharight) to mathbf R^{2n}

by
widetildephi_alphaleft(x, v^ipartial_iright) = left(phi_alpha(x), v^1, cdots, v^nright)

We use these maps to define the topology and smooth structure on *TM*. A subset

*A*of

*TM*is open if and only if

widetildephi_alphaleft(Acap pi^{-1}left(U_alpharight)right)

is open in **R**2

*n*for each Î±. These maps are then homeomorphisms between open subsets of

*TM*and

**R**2

*n*and therefore serve as charts for the smooth structure on

*TM*. The transition functions on chart overlaps pi^{-1}left(U_alpha cap U_betaright) are induced by the Jacobian matrices of the associated coordinate transformation and are therefore smooth maps between open subsets of

**R**2

*n*.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

*n*-dimensional manifold

*M*may be defined as a rank

*n*vector bundle over

*M*whose transition functions are given by the Jacobian of the associated coordinate transformations.

## Examples

The simplest example is that of**R**

*n*. In this case the tangent bundle is trivial: each T_x mathbf R^n is canonically isomorphic to T_0 mathbf R^n via the map mathbf R^n to mathbf R^n which subtracts x , giving a diffeomorphism Tmathbf R^n to mathbf R^n times mathbf R^n.Another simple example is the unit circle,

*S*1 (see picture above). The tangent bundle of the circle is also trivial and isomorphic to

*S*1 Ã—

**R**. Geometrically, this is a cylinder of infinite height.The only tangent bundles that can be readily visualized are those of the real line

**R**and the unit circle

*S*1, 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

*S*2: 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

*M*is a smooth map

Vcolon M to TM

such that the image of *x*, denoted

*V*

**'x****, lies in**

*T**'x*

*M*, the tangent space at

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

*section*. 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 Gamma(TM). Vector fields can be added together pointwise

(V+W)_x = V_x + W_x,

and multiplied by smooth functions on *M*

(fV)_x = f(x)V_x,

to get other vector fields. The set of all vector fields Gamma(TM) then takes on the structure of a module over the commutative algebra of smooth functions on *M*, denoted C^{infty}(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 of real vector spaces on

*M*.Note that the above construction applies equally well to the cotangent bundle - the differential 1-forms on

*M*are precisely the sections of the cotangent bundle omega in Gamma(T^*M), omega: M to T^*Mthat associate to each point x in Ma 1-covector omega_x in T^*_xM, which map tangent vectors to real numbers: omega_x : T_xM to R. Equivalently, a differential 1-form omega in Gamma(T^*M)maps a smooth vector field X in Gamma(TM)to a smooth function omega(X) in C^{infty}(M).

## Higher-order tangent bundles

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

T^2 M = T(TM).,

In general, the *k*-th order tangent bundle T^k M can be defined recursively as Tleft(T^{k-1}Mright).A smooth map

*f*:

*M*â†’

*N*has an induced derivative, for which the tangent bundle is the appropriate domain and range

*Df*:

*TM*â†’

*TN*. Similarly, higher-order tangent bundles provide the domain and range for higher-order derivatives D^k f : T^k M to T^k N.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*TM,*considered as a manifold itself, one can define a

**canonical vector field**

*V*:

*TM*â†’

*TTM*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, TW cong W times W, since the vector space itself is flat, and thus has a natural diagonal map W to TW given by w mapsto (w, w) 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

*M*is curved, each tangent space at a point

*m*, T_m M approx mathbf{R}^n, is flat, so the tangent bundle manifold

*TM*is locally a product of a curved

*M*and a flat mathbf{R}^n. Thus the tangent bundle of the tangent bundle is locally (using approx for "choice of coordinates" and cong for "natural identification"):

T(TM) approx T(M times mathbf{R}^n) cong TM times T(mathbf{R}^n) cong TM times ( mathbf{R}^ntimesmathbf{R}^n)

and the map TTM to TM is the projection onto the first coordinates:
(TM to M) times (mathbf{R}^n times mathbf{R}^n to mathbf{R}^n).

Splitting the first map via the zero section and the second map by the diagonal yields the canonical vector field.If (*x*,

*v*) are local coordinates for

*TM*, the vector field has the expression

V = sum_i left. v^i frac{partial}{partial v^i} right|_{(x,v)}.

More concisely, (x, v) mapsto (x, v, 0, v) â€“ 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. Note that this expression for the vector field depends only on *v,*not on

*x,*as only the tangent directions can be naturally identified.Alternatively, consider the scalar multiplication function:

begin{cases}

mathbf{R} times TM to TM (t,v) longmapsto tvend{cases}The derivative of this function with respect to the variable **R**at time

*t*= 1 is a function

*V*:

*TM*â†’

*TTM*, which is an alternative description of the canonical vector field.The existence of such a vector field on

*TM*is analogous to the canonical one-form on the cotangent bundle. 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 LeÃ³n 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 of

*c*) is a curve in

*TM*. In contrast, without further assumptions on

*M*(say, a Riemannian metric), there is no similar lift into the cotangent bundle.The

*vertical lift*of a function

*f*:

*M*â†’

**R**is the function

*fv*:

*TM*â†’

**R**defined by f^v=fcirc pi, where Ï€ :

*TM*â†’

*M*is the canonical projection.

## See also

## Notes

## References

{{refimprove|date=July 2009}}{{Reflist}}- {{citation|first=Jeffrey M.|last=Lee|title=Manifolds and Differential Geometry|series=Graduate Studies in Mathematics|volume=Vol. 107 |publisher=American Mathematical Society|publication-place=Providence|year=2009}}. {{isbn|978-0-8218-4815-9}}
- John M. Lee,
*Introduction to Smooth Manifolds*, (2003) Springer-Verlag, New York. {{isbn|0-387-95495-3}}. - JÃ¼rgen Jost,
*Riemannian Geometry and Geometric Analysis*, (2002) Springer-Verlag, Berlin. {{isbn|3-540-42627-2}} - Ralph Abraham and Jerrold E. Marsden,
*Foundations of Mechanics*, (1978) Benjamin-Cummings, London. {{isbn|0-8053-0102-X}} - M. De LeÃ³n, E. Merino, J.A. OubiÃ±a, M. Salgado,
*A characterization of tangent and stable tangent bundles*, Annales de l'institut Henri PoincarÃ© (A) Physique thÃ©orique, Vol. 61, no. 1, 1994, 1-15 weblink

## External links

- {{springer|title=Tangent bundle|id=p/t092110}}
- Wolfram MathWorld: Tangent Bundle
- PlanetMath: Tangent Bundle

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