Tangent bundle

{{Short description|Tangent spaces of a manifold}} {{Use American English|date = March 2019}} [[Image:Tangent bundle.svg|right|thumb|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).<ref group=note name="disjoint"/>]]

A '''tangent bundle''' is the collection of all of the [[tangent space]]s for all points on a [[manifold]], structured in a way that it forms a new manifold itself. Formally, in [[differential geometry]], the tangent bundle of a [[differentiable manifold]] <math> M </math> is a manifold <math>TM</math> which assembles all the tangent vectors in <math> M </math>. As a set, it is given by the [[disjoint union]]<ref group="note" name="disjoint">The disjoint union ensures that for any two points {{math|''x''1}} and {{math|''x''2}} of manifold {{math|''M''}} the tangent spaces {{math|''T''1}} and {{math|''T''2}} have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle {{math|''S''1}}, see [[tangent bundle#Examples|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.</ref> of the tangent spaces of <math> M </math>. That is,

:<math> \begin{align} TM &= \bigsqcup_{x \in M} T_xM \\ &= \bigcup_{x \in M} \left\{x\right\} \times T_xM \\ &= \bigcup_{x \in M} \left\{(x, y) \mid y \in T_xM\right\} \\ &= \left\{ (x, y) \mid x \in M,\, y \in T_xM \right\} \end{align} </math>

where <math> T_x M</math> denotes the [[tangent space]] to <math> M </math> at the point <math> x </math>. So, an element of <math> TM</math> can be thought of as a [[ordered pair|pair]] <math> (x,v)</math>, where <math> x </math> is a point in <math> M </math> and <math> v </math> is a tangent vector to <math> M </math> at <math> x </math>.

There is a natural [[projection (mathematics)|projection]] :<math> \pi : TM \twoheadrightarrow M </math>

defined by <math> \pi(x, v) = x</math>. This projection maps each element of the tangent space <math> T_xM</math> to the single point <math> x </math>.

The tangent bundle comes equipped with a [[natural topology]] (described in a section [[#Topology and smooth structure|below]]). With this topology, the tangent bundle to a manifold is the prototypical example of a [[vector bundle]] (which is a [[fiber bundle]] whose fibers are [[vector space]]s). A [[Section (fiber bundle)|section]] of <math> TM</math> is a [[vector field]] on <math> M</math>, and the [[dual bundle]] to <math> TM</math> is the [[cotangent bundle]], which is the disjoint union of the [[cotangent space]]s of <math> M </math>. By definition, a manifold <math> M </math> is [[Parallelizable manifold|parallelizable]] if and only if the tangent bundle is [[trivial bundle|trivial]]. By definition, a manifold <math>M</math> is [[Framed (mathematics)|framed]] if and only if the tangent bundle <math>TM</math> is stably trivial, meaning that for some trivial bundle <math>E</math> the [[Whitney sum]] <math> TM\oplus E</math> 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 <math> f:M\rightarrow N </math> is a smooth function, with <math> M </math> and <math> N </math> smooth manifolds, its [[derivative (generalizations)|derivative]] is a smooth function <math> Df:TM\rightarrow TN </math>.

==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 <math> TM</math> is twice the dimension of <math> M</math>.

Each tangent space of an ''n''-dimensional manifold is an ''n''-dimensional vector space. If <math>U</math> is an open [[contractible space|contractible]] subset of <math>M</math>, then there is a [[diffeomorphism]] <math> TU\to U\times\mathbb R^n</math> which restricts to a linear isomorphism from each tangent space <math> T_xU</math> to <math> \{x\}\times\mathbb R^n</math>. As a manifold, however, <math> TM</math> is not always diffeomorphic to the product manifold <math>M\times\mathbb R^n</math>. When it is of the form <math> M\times\mathbb R^n</math>, 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 modeled on [[Euclidean space]], tangent bundles are locally modeled on <math>U\times\mathbb R^n</math>, where <math>U</math> is an open subset of Euclidean space.

If ''M'' is a smooth ''n''-dimensional manifold, then it comes equipped with an [[atlas (topology)|atlas]] of charts <math>(U_\alpha,\phi_\alpha)</math>, where <math> U_\alpha</math> is an open set in <math>M</math> and :<math>\phi_\alpha: U_\alpha \to \mathbb R^n</math>

is a [[diffeomorphism]]. These local coordinates on <math> U_\alpha </math> give rise to an isomorphism <math> T_xM\rightarrow\mathbb R^n</math> for all <math> x\in U_\alpha</math>. We may then define a map

:<math>\widetilde\phi_\alpha:\pi^{-1}\left(U_\alpha\right) \to \mathbb R^{2n}</math>

by :<math>\widetilde\phi_\alpha\left(x, v^i\partial_i\right) = \left(\phi_\alpha(x), v^1, \cdots, v^n\right)</math>

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

:<math>\widetilde\phi_\alpha\left(A\cap \pi^{-1}\left(U_\alpha\right)\right)</math>

is open in <math>\mathbb R^{2n}</math> for each <math> \alpha.</math> These maps are homeomorphisms between open subsets of <math>TM</math> and <math>\mathbb R^{2n}</math> and therefore serve as charts for the smooth structure on <math>TM</math>. The transition functions on chart overlaps <math>\pi^{-1}\left(U_\alpha \cap U_\beta\right)</math> are induced by the [[Jacobian matrix|Jacobian matrices]] of the associated coordinate transformation and are therefore smooth maps between open subsets of <math>\mathbb R^{2n}</math>.

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 <math>n</math>-dimensional manifold <math>M</math> may be defined as a rank <math>n</math> vector bundle over <math>M</math> whose transition functions are given by the [[Jacobian matrix and determinant|Jacobian]] of the associated coordinate transformations.

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

Another simple example is the [[unit circle]], <math> S^1 </math> (see picture above). The tangent bundle of the circle is also trivial and isomorphic to <math> S^1\times\mathbb R </math>. Geometrically, this is a [[cylinder (geometry)|cylinder]] of infinite height.

The only tangent bundles that can be readily visualized are those of the real line <math>\mathbb R </math> and the unit circle <math>S^1</math>, 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 <math> S^2 </math>: this tangent bundle is nontrivial as a consequence of the [[hairy ball theorem]]. Therefore, the sphere is not [[Parallelizable_manifold|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 <math> M </math> is a [[smooth map]] :<math>V\colon M \to TM</math>

such that <math>V(x) = (x,V_x)</math> with <math>V_x\in T_xM</math> for every <math>x\in M</math>. In the language of fiber bundles, such a map is called a ''[[section (fiber bundle)|section]]''. A vector field on <math>M</math> is therefore a section of the tangent bundle of <math>M</math>.

The set of all vector fields on <math>M</math> is denoted by <math>\Gamma(TM)</math>. Vector fields can be added together pointwise

:<math>(V+W)_x = V_x + W_x</math> and multiplied by smooth functions on ''M''

:<math>(fV)_x = f(x)V_x</math>

to get other vector fields. The set of all vector fields <math>\Gamma(TM)</math> then takes on the structure of a [[module (mathematics)|module]] over the [[associative algebra|commutative algebra]] of smooth functions on ''M'', denoted <math>C^{\infty}(M)</math>.

A local vector field on <math>M</math> is a ''local section'' of the tangent bundle. That is, a local vector field is defined only on some open set <math>U\subset M</math> and assigns to each point of <math>U</math> a vector in the associated tangent space. The set of local vector fields on <math>M</math> forms a structure known as a [[sheaf (mathematics)|sheaf]] of real vector spaces on <math>M</math>.

The above construction applies equally well to the cotangent bundle – the differential 1-forms on <math>M</math> are precisely the sections of the cotangent bundle <math>\omega \in \Gamma(T^*M)</math>, <math>\omega: M \to T^*M</math> that associate to each point <math>x \in M</math> a 1-covector <math>\omega_x \in T^*_xM</math>, which map tangent vectors to real numbers: <math>\omega_x : T_xM \to \R</math>. Equivalently, a differential 1-form <math>\omega \in \Gamma(T^*M)</math> maps a smooth vector field <math>X \in \Gamma(TM)</math> to a smooth function <math>\omega(X) \in C^{\infty}(M)</math>.

==Higher-order tangent bundles== Since the tangent bundle <math>TM</math> is itself a smooth manifold, the [[double tangent bundle|second-order tangent bundle]] can be defined via repeated application of the tangent bundle construction:

:<math>T^2 M = T(TM).\,</math>

In general, the <math>k</math>th order tangent bundle <math>T^k M</math> can be defined recursively as <math>T\left(T^{k-1}M\right)</math>.

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

A distinct but related construction are the [[jet bundle]]s on a manifold, which are bundles consisting of [[jet (mathematics)|jets]].

==Canonical vector field on tangent bundle== On every tangent bundle <math>TM</math>, considered as a manifold itself, one can define a '''canonical vector field''' <math>V:TM\rightarrow T^2M </math> 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, <math>TW \cong W \times W,</math> since the vector space itself is flat, and thus has a natural diagonal map <math>W \to TW</math> given by <math>w \mapsto (w, w)</math> 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 <math>M</math> is curved, each tangent space at a point <math>x</math>, <math>T_x M \approx \mathbb{R}^n</math>, is flat, so the tangent bundle manifold <math>TM</math> is locally a product of a curved <math>M</math> and a flat <math>\mathbb{R}^n.</math> Thus the tangent bundle of the tangent bundle is locally (using <math>\approx</math> for "choice of coordinates" and <math>\cong</math> for "natural identification"):

:<math>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)</math> and the map <math>TTM \to TM</math> is the projection onto the first coordinates: :<math>(TM \to M) \times (\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n).</math> Splitting the first map via the zero section and the second map by the diagonal yields the canonical vector field.

If <math>(x,v)</math> are local coordinates for <math>TM</math>, the vector field has the expression

:<math> V = \sum_i \left. v^i \frac{\partial}{\partial v^i} \right|_{(x,v)}.</math>

More concisely, <math>(x, v) \mapsto (x, v, 0, v)</math> – 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 <math>v</math>, not on <math>x</math>, as only the tangent directions can be naturally identified.

Alternatively, consider the scalar multiplication function: :<math>\begin{cases} \mathbb{R} \times TM \to TM \\ (t,v) \longmapsto tv \end{cases}</math>

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

The existence of such a vector field on <math> TM </math> is analogous to the [[canonical one-form]] on the [[cotangent bundle]]. Sometimes <math> V </math> is also called the '''Liouville vector field''', or '''radial vector field'''. Using <math> V </math> one can characterize the tangent bundle. Essentially, <math> V </math> 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 (mathematics)|lift]] objects on <math> M </math> into objects on <math> TM </math>. For example, if <math> \gamma </math> is a curve in <math> M </math>, then <math> \gamma' </math> (the [[tangent]] of <math> \gamma </math>) is a curve in <math> TM </math>. In contrast, without further assumptions on <math> M </math> (say, a [[Riemannian metric]]), there is no similar lift into the [[cotangent bundle]].

The ''vertical lift'' of a function <math> f:M\rightarrow\mathbb R </math> is the function <math> f^\vee:TM\rightarrow\mathbb R </math> defined by <math>f^\vee=f\circ \pi</math>, where <math> \pi:TM\rightarrow M </math> is the canonical projection.

==See also== * [[Pushforward (differential)]] * [[Unit tangent bundle]] * [[Cotangent bundle]] * [[Frame bundle]] * [[Musical isomorphism]] * [[Holomorphic tangent bundle]]

==Notes== <references group=note/>

==References== {{more citations needed|date=July 2009}} {{Reflist}} * {{citation|first=Jeffrey M.|last=Lee|title=Manifolds and Differential Geometry|series=[[Graduate Studies in Mathematics]]|volume=107 |publisher=American Mathematical Society|publication-place=Providence|year=2009}}. {{isbn|978-0-8218-4815-9}} *{{cite book |last1=Lee |first1=John M.|doi=10.1007/978-1-4419-9982-5 |title=Introduction to Smooth Manifolds |series=Graduate Texts in Mathematics |date=2012 |volume=218 |isbn=978-1-4419-9981-8 }} * [[Jürgen Jost]], ''Riemannian Geometry and Geometric Analysis'', (2002) Springer-Verlag, Berlin. {{isbn|3-540-42627-2}} * [[Ralph Abraham (mathematician)|Ralph Abraham]] and [[Jerrold E. Marsden]], ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London. {{isbn|0-8053-0102-X}} *{{cite journal |url=http://archive.numdam.org/article/AIHPA_1994__61_1_1_0.pdf |title=A characterization of tangent and stable tangent bundles |journal=Annales de l'I.H.P.: Physique Théorique |date=1994 |volume=61 |issue=1 |pages=1–15 |last1=León |first1=M. De |last2=Merino |first2=E. |last3=Oubiña |first3=J. A. |last4=Salgado |first4=M. }} *{{cite journal |doi=10.1016/S0723-0869(02)80027-5 |title=On the geometry of tangent bundles |date=2002 |last1=Gudmundsson |first1=Sigmundur |last2=Kappos |first2=Elias |journal=Expositiones Mathematicae |volume=20 |pages=1–41 }} *{{cite book |last1=Salimov |first1=Arif|doi=10.1007/978-981-99-1296-4 |title=Applications of Holomorphic Functions in Geometry |series=Frontiers in Mathematics, Birkhäuser |date=2023 |isbn=978-1-4419-9981-8 }}

==External links== * {{springer|title=Tangent bundle|id=p/t092110}} * [https://mathworld.wolfram.com/TangentBundle.html Wolfram MathWorld: Tangent Bundle] * [http://planetmath.org/tangentbundle PlanetMath: Tangent Bundle]

[[Category:Differential topology]] [[Category:Vector bundles]]