Tangent bundle

In mathematics, the tangent bundle of a smooth (or differentiable) manifold "M", denoted by "T"("M") or just "TM", is the disjoint unionThe disjoint union assures 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 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 spaces of the points "x" of "M"

:$TM\; =\; coprod\_\{xin\; M\}T\_xM.$

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

:$picolon\; TM\; o\; M,\; ,\; (x,v)\; mapsto\; x,$

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 $f\; :\; M\; o\; N$ is a smooth function, with $M$ and $N$ smooth manifolds, its derivative is a smooth function $Df\; :\; TM\; o\; TN$.

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" × R^"n" which restricts to a linear isomorphism from each tangent space "T"_"x""U" to {"x"}× R^"n" . As a manifold, however, "TM" is not always diffeomorphic to the product manifold "M" × R^"n". When it is of the form "M" × 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" × 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"_α, φ_α) where "U"_α is an open set in "M" and:$phi\_alphacolon\; U\_alpha\; o\; mathbb\; R^n$is a diffeomorphism. These local coordinates on "U" give rise to an isomorphism between "T"_"x""M" and R^"n" for each "x" ∈ "U". We may then define a
$ildephi\_alphacolon\; pi^\{-1\}(U\_alpha)\; o\; mathbb\; R^\{2n\}$by:$ildephi\_alpha(x,\; v^ipartial\_i)\; =\; (phi\_alpha(x),\; v^1,\; cdots,\; v^n)$We use these maps to define the topology and smooth structure on "TM". A subset "A" of "TM" is open if and only if $ildephi\_alpha(Acap\; pi^\{-1\}(U\_alpha))$ is open in R^2"n" for each α. 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 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 and isomorphic to R^2"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 × R. Geometrically, this is a cylinder of infinite height (see the bottom picture).

Unfortunately, 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 not easily visualizable.

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.

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:$Vcolon\; M\; o\; TM$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 Γ("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 Γ("TM") then takes on the structure of a module over the 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 of real vector spaces on "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 inductively as $T(T^\{k-1\}M)$.

A smooth map $f\; :\; M\; o\; N$ has an induced derivative, for which the tangent bundle is the appropriate domain and range $Df\; :\; TM\; o\; TN$. Similarly, higher-order tangent bundles provide the domain and range for higher-order derivatives $D^k\; f\; :\; T^k\; M\; o\; T^k\; N$.

Canonical vector field on tangent bundle

On every tangent bundle "TM" one can define a canonical vector field $V\; :\; TM\; o\; TTM$. If ("x", "v") are local coordinates for "TM", the vector field has the expression:$V\; =\; sum\_i\; left.\; v^i\; frac\{partial\}\{partial\; v^i\}\; ight|\_\{(x,v)\}.$Alternatively, consider $mathbb\; R\; imes\; TM\; o\; TM$ be the scalar multiplication function $(t,v)\; longmapsto\; tv$. The derivative of this function with respect to the variable $mathbb\; R$ at time $t=1$ is a function $V\; :\; TM\; o\; TTM$, 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. 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 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.

The "vertical lift" of a function $fcolon\; M\; o\; mathbb\{R\}$ is the function $f^vcolon\; TM\; o\; mathbb\{R\}$ defined by$f^v=fcirc\; pi$, where $picolon\; TM\; o\; M$ is thecanonical projection.

Notes

ee also

* pushforward (differential)
* vector field
* cotangent bundle
* frame bundle
* Musical isomorphism

References

* John M. Lee, "Introduction to Smooth Manifolds", (2003) Springer-Verlag, New York. ISBN 0-387-95495-3.
* Jurgen 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 León, E. Merino, J.A. Oubiña, M. Salgado, "A characterization of tangent and stable tangent bundles", Annales de l'institut Henri Poincaré (A) Physique théorique, Vol. 61, no. 1, 1994, 1-15 [http://archive.numdam.org/ARCHIVE/AIHPA/AIHPA_1994__61_1/AIHPA_1994__61_1_1_0/AIHPA_1994__61_1_1_0.pdf]

External links

* [http://mathworld.wolfram.com/TangentBundle.html MathWorld: Tangent Bundle]
* [http://planetmath.org/encyclopedia/TangentBundle.html PlanetMath: Tangent Bundle]