Tangent bundle

Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom).

Atangent bundleis the collection of all of thetangent spacesfor all points on amanifold,structured in a way that it forms a new manifold itself. Formally, indifferential geometry,the tangent bundle of adifferentiable manifold$M$is a manifold$TM$which assembles all the tangent vectors in$M$.As a set, it is given by thedisjoint unionof the tangent spaces of$M$.That is,

${\begin{aligned}TM&=\bigsqcup _{x\in M}T_{x}M\\&=\bigcup _{x\in M}\left\{x\right\}\times T_{x}M\\&=\bigcup _{x\in M}\left\{(x,y)\mid y\in T_{x}M\right\}\\&=\left\{(x,y)\mid x\in M,\,y\in T_{x}M\right\}\end{aligned}}$

where$T_{x}M$denotes thetangent spaceto$M$at the point$x$.So, an element of$TM$can be thought of as apair$(x,v)$,where$x$is a point in$M$and$v$is a tangent vector to$M$at$x$.

There is a naturalprojection

$\pi:TM\twoheadrightarrow M$

defined by$\pi (x,v)=x$.This projection maps each element of the tangent space$T_{x}M$to the single point$x$.

The tangent bundle comes equipped with anatural topology(described in a sectionbelow). With this topology, the tangent bundle to a manifold is the prototypical example of avector bundle(which is afiber bundlewhose fibers arevector spaces). Asectionof$TM$is avector fieldon$M$,and thedual bundleto$TM$is thecotangent bundle,which is the disjoint union of thecotangent spacesof$M$.By definition, a manifold$M$isparallelizableif and only if the tangent bundle istrivial.By definition, a manifold$M$isframedif and only if the tangent bundle$TM$is stably trivial, meaning that for some trivial bundle$E$theWhitney sum$TM\oplus E$is trivial. For example, then-dimensional sphereSⁿis framed for alln,but parallelizable only forn= 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$f:M\rightarrow N$is a smooth function, with$M$and$N$smooth manifolds, itsderivativeis a smooth function$Df:TM\rightarrow TN$.

Topology and smooth structure

The tangent bundle comes equipped with a natural topology (notthedisjoint union topology) andsmooth structureso 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 ann-dimensional manifold is ann-dimensional vector space. If$U$is an opencontractiblesubset of$M$,then there is adiffeomorphism$TU\to U\times \mathbb {R} ^{n}$which restricts to a linear isomorphism from each tangent space$T_{x}U$to$\{x\}\times \mathbb {R} ^{n}$.As a manifold, however,$TM$is not always diffeomorphic to the product manifold$M\times \mathbb {R} ^{n}$.When it is of the form$M\times \mathbb {R} ^{n}$,then the tangent bundle is said to betrivial.Trivial tangent bundles usually occur for manifolds equipped with a 'compatible group structure'; for instance, in the case where the manifold is aLie 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 calledparallelizable.Just as manifolds are locally modeled onEuclidean space,tangent bundles are locally modeled on$U\times \mathbb {R} ^{n}$,where$U$is an open subset of Euclidean space.

IfMis a smoothn-dimensional manifold, then it comes equipped with anatlasof charts$(U_{\ Alpha },\phi _{\ Alpha })$,where$U_{\ Alpha }$is an open set in$M$and

$\phi _{\ Alpha }:U_{\ Alpha }\to \mathbb {R} ^{n}$

is adiffeomorphism.These local coordinates on$U_{\ Alpha }$give rise to an isomorphism$T_{x}M\rightarrow \mathbb {R} ^{n}$for all$x\in U_{\ Alpha }$.We may then define a map

${\widetilde {\phi }}_{\ Alpha }:\pi ^{-1}\left(U_{\ Alpha }\right)\to \mathbb {R} ^{2n}$

by

${\widetilde {\phi }}_{\ Alpha }\left(x,v^{i}\partial _{i}\right)=\left(\phi _{\ Alpha }(x),v^{1},\cdots,v^{n}\right)$

We use these maps to define the topology and smooth structure on$TM$.A subset$A$of$TM$is open if and only if

${\widetilde {\phi }}_{\ Alpha }\left(A\cap \pi ^{-1}\left(U_{\ Alpha }\right)\right)$

is open in$\mathbb {R} ^{2n}$for each$\ Alpha.$These maps are homeomorphisms between open subsets of$TM$and$\mathbb {R} ^{2n}$and therefore serve as charts for the smooth structure on$TM$.The transition functions on chart overlaps$\pi ^{-1}\left(U_{\ Alpha }\cap U_{\beta }\right)$are induced by theJacobian matricesof the associated coordinate transformation and are therefore smooth maps between open subsets of$\mathbb {R} ^{2n}$.

The tangent bundle is an example of a more general construction called avector bundle(which is itself a specific kind offiber 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 theJacobianof the associated coordinate transformations.

Examples

The simplest example is that of$\mathbb {R} ^{n}$.In this case the tangent bundle is trivial: each$T_{x}\mathbf {\mathbb {R} } ^{n}$is canonically isomorphic to$T_{0}\mathbb {R} ^{n}$via the map$\mathbb {R} ^{n}\to \mathbb {R} ^{n}$which subtracts$x$,giving a diffeomorphism$T\mathbb {R} ^{n}\to \mathbb {R} ^{n}\times \mathbb {R} ^{n}$.

Another simple example is theunit circle,$S^{1}$(see picture above). The tangent bundle of the circle is also trivial and isomorphic to$S^{1}\times \mathbb {R}$.Geometrically, this is acylinderof infinite height.

The only tangent bundles that can be readily visualized are those of the real line$\mathbb {R}$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 thehairy ball theorem.Therefore, the sphere is notparallelizable.

Vector fields

A smooth assignment of a tangent vector to each point of a manifold is called avector field.Specifically, a vector field on a manifold$M$is asmooth map

$V\colon M\to TM$

such that$V(x)=(x,V_{x})$with$V_{x}\in T_{x}M$for every$x\in M$.In the language of fiber bundles, such a map is called asection.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 onM

$(fV)_{x}=f(x)V_{x}$

to get other vector fields. The set of all vector fields$\Gamma (TM)$then takes on the structure of amoduleover thecommutative algebraof smooth functions onM,denoted$C^{\infty }(M)$.

A local vector field on$M$is alocal sectionof the tangent bundle. That is, a local vector field is defined only on some open set$U\subset M$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 asheafof real vector spaces on$M$.

The above construction applies equally well to the cotangent bundle – the differential 1-forms on$M$are precisely the sections of the cotangent bundle$\ Omega \in \Gamma (T^{*}M)$,$\ Omega:M\to T^{*}M$that associate to each point$x\in M$a 1-covector$\ Omega _{x}\in T_{x}^{*}M$,which map tangent vectors to real numbers:$\ Omega _{x}:T_{x}M\to \mathbb {R}$.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, thesecond-order tangent bundlecan 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$T\left(T^{k-1}M\right)$.

A smooth map$f:M\rightarrow N$has an induced derivative, for which the tangent bundle is the appropriate domain and range$Df:TM\rightarrow TN$.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 thejet bundleson a manifold, which are bundles consisting ofjets.

Canonical vector field on tangent bundle

On every tangent bundle$TM$,considered as a manifold itself, one can define acanonical vector field$V:TM\rightarrow T^{2}M$as thediagonal mapon the tangent space at each point. This is possible because the tangent space of a vector spaceWis 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$x$,$T_{x}M\approx \mathbb {R} ^{n}$,is flat, so the tangent bundle manifold$TM$is locally a product of a curved$M$and a flat$\mathbb {R} ^{n}.$Thus the tangent bundle of the tangent bundle is locally (using$\approx$for "choice of coordinates" and$\cong$for "natural identification" ):

$T(TM)\approx T(M\times \mathbb {R} ^{n})\cong TM\times T(\mathbb {R} ^{n})\cong TM\times (\mathbb {R} ^{n}\times \mathbb {R} ^{n})$

and the map$TTM\to TM$is the projection onto the first coordinates:

$(TM\to M)\times (\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} ^{n}).$

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,$(x,v)\mapsto (x,v,0,v)$– 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$v$,not on$x$,as only the tangent directions can be naturally identified.

Alternatively, consider the scalar multiplication function:

${\begin{cases}\mathbb {R} \times TM\to TM\\(t,v)\longmapsto tv\end{cases}}$

The derivative of this function with respect to the variable$\mathbb {R}$at time$t=1$is a function$V:TM\rightarrow T^{2}M$,which is an alternative description of the canonical vector field.

The existence of such a vector field on$TM$is analogous to thecanonical one-formon thecotangent bundle.Sometimes$V$is also called theLiouville vector field,orradial 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 toliftobjects on$M$into objects on$TM$.For example, if$\gamma$is a curve in$M$,then$\gamma '$(thetangentof$\gamma$) is a curve in$TM$.In contrast, without further assumptions on$M$(say, aRiemannian metric), there is no similar lift into thecotangent bundle.

Thevertical liftof a function$f:M\rightarrow \mathbb {R}$is the function$f^{\vee }:TM\rightarrow \mathbb {R}$defined by$f^{\vee }=f\circ \pi$,where$\pi:TM\rightarrow M$is the canonical projection.

See also

• Pushforward (differential)
• Unit tangent bundle
• Cotangent bundle
• Frame bundle
• Musical isomorphism

References

• Lee, Jeffrey M. (2009),Manifolds and Differential Geometry,Graduate Studies in Mathematics,vol. 107, Providence: American Mathematical Society.ISBN978-0-8218-4815-9
• Lee, John M. (2012).Introduction to Smooth Manifolds.Graduate Texts in Mathematics. Vol. 218.doi:10.1007/978-1-4419-9982-5.ISBN978-1-4419-9981-8.
• Jürgen Jost,Riemannian Geometry and Geometric Analysis,(2002) Springer-Verlag, Berlin.ISBN3-540-42627-2
• Ralph AbrahamandJerrold E. Marsden,Foundations of Mechanics,(1978) Benjamin-Cummings, London.ISBN0-8053-0102-X
• 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/S0723-0869(02)80027-5.

External links

• "Tangent bundle",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• Wolfram MathWorld: Tangent Bundle
• PlanetMath: Tangent Bundle