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| [[Image:Tangent bundle.svg|right|thumb|Informally, the tangent bundle of a manifold (in this case 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"/>]]
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| In [[differential geometry]], the '''tangent bundle''' of a [[differentiable manifold]] ''M'' is the [[disjoint union]]<ref group=note name="disjoint">The disjoint union assures that for any two points ''x''<sub>1</sub> and ''x''<sub>2</sub> of manifold ''M'' the tangent spaces ''T''<sub>1</sub> and ''T''<sub>2</sub> have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle ''S''<sup>1</sup>, 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 space]]s of ''M''. That is,
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| :<math>TM =\bigsqcup_{x\in M}T_xM=\bigcup_{x\in M} \left\{x\right\}\times T_xM | |
| =\bigcup_{x\in M} \left\{(x, y)\vert\; y\in T_xM\right\}.</math>
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| where ''T''<sub>''x''</sub>''M'' denotes the [[tangent space]] to ''M'' at the point ''x''. So, an element of ''TM'' can be thought of as a [[ordered pair|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 (mathematics)|projection]]
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| :<math> \pi : TM \twoheadrightarrow M </math>
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| defined by π(''x'', ''v'') = ''x''. This projection maps each tangent space ''T''<sub>''x''</sub>''M'' to the single point ''x''.
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| The tangent bundle to a manifold is the prototypical example of a [[vector bundle]] (a [[fiber bundle]] whose fibers are [[vector space]]s). A [[Section (fiber bundle)|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 space]]s of ''M''. By definition, a manifold ''M'' is [[Parallelizable manifold|parallelizable]] if and only if the tangent bundle is [[trivial bundle|trivial]]. | |
| By definition, a manifold ''M'' is [[Framed (mathematics)|framed]] if and only if the tangent bundle ''TM'' is stably trivial, meaning that for some trivial bundle ''E'' the [[Vector bundle|Whitney sum]] {{nowrap|1=''TM'' ⊕ ''E''}} is trivial. For example, the ''n''-dimensional sphere ''S<sup>n</sup>'' is framed for all ''n'', but parallelizable only for ''n''=1,3,7 (by results of Bott-Milnor and Kervaire).
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| ==Role==
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| 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'' → ''N'' is a smooth function, with ''M'' and ''N'' smooth manifolds, its [[derivative (generalizations)|derivative]] is a smooth function ''Df'' : ''TM'' → ''TN''.
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| ==Topology and smooth structure==
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| 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''.
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| Each tangent space of an ''n''-dimensional manifold is an ''n''-dimensional vector space. If ''U'' is an open [[contractible space|contractible]] subset of ''M'', then there is a [[diffeomorphism]] from ''TU'' to ''U'' × '''R'''<sup>''n''</sup> which restricts to a linear isomorphism from each tangent space ''T''<sub>''x''</sub>''U'' to {''x''} × '''R'''<sup>''n''</sup> . As a manifold, however, ''TM'' is not always diffeomorphic to the product manifold ''M'' × '''R'''<sup>''n''</sup>. When it is of the form ''M'' × '''R'''<sup>''n''</sup>, 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'''<sup>''n''</sup>, where ''U'' is an open subset of Euclidean space.
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| If ''M'' is a smooth ''n''-dimensional manifold, then it comes equipped with an [[atlas (topology)|atlas]] of charts (''U''<sub>α</sub>, φ<sub>α</sub>) where ''U''<sub>α</sub> is an open set in ''M'' and
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| :<math>\phi_\alpha\colon U_\alpha \to \mathbf R^n</math>
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| is a [[diffeomorphism]]. These local coordinates on ''U'' give rise to an isomorphism between ''T''<sub>''x''</sub>''M'' and '''R'''<sup>''n''</sup> for each ''x'' ∈ ''U''. We may then define a map
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| :<math>\widetilde\phi_\alpha\colon \pi^{-1}(U_\alpha) \to \mathbf R^{2n}</math>
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| by
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| :<math>\widetilde\phi_\alpha(x, v^i\partial_i) = (\phi_\alpha(x), v^1, \cdots, v^n)</math>
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| We use these maps to define the topology and smooth structure on ''TM''. A subset ''A'' of ''TM'' is open if and only if
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| :<math>\widetilde\phi_\alpha(A\cap \pi^{-1}(U_\alpha))</math>
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| is open in '''R'''<sup>2''n''</sup> for each α. These maps are then homeomorphisms between open subsets of ''TM'' and '''R'''<sup>2''n''</sup> and therefore serve as charts for the smooth structure on ''TM''. The transition functions on chart overlaps <math>\pi^{-1}(U_\alpha\cap U_\beta)</math> are induced by the [[Jacobian matrix|Jacobian matrices]] of the associated coordinate transformation and are therefore smooth maps between open subsets of '''R'''<sup>2''n''</sup>.
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| 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.
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| ==Examples==
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| The simplest example is that of '''R'''<sup>''n''</sup>. In this case the tangent bundle is trivial.
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| Another simple example is the [[unit circle]], ''S''<sup>1</sup> (see picture above). The tangent bundle of the circle is also trivial and isomorphic to ''S''<sup>1</sup> × '''R'''. Geometrically, this is a [[cylinder (geometry)|cylinder]] of infinite height.
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| The only tangent bundles that can be readily visualized are those of the real line '''R''' and the unit circle ''S''<sup>1</sup>, both of which are trivial. For 2-dimensional manifolds the tangent bundle is 4-dimensional and hence difficult to visualize.
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| A simple example of a nontrivial tangent bundle is that of the unit sphere ''S''<sup>2</sup>: this tangent bundle is nontrivial as a consequence of the [[hairy ball theorem]]. Therefore, the sphere is not parallelizable.
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| ==Vector fields==
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| 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]]
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| :<math>V\colon M \to TM</math>
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| such that the image of ''x'', denoted ''V''<sub>''x''</sub>, lies in ''T''<sub>''x''</sub>''M'', the tangent space at ''x''. In the language of fiber bundles, such a map is called a ''[[section (fiber bundle)|section]]''. A vector field on ''M'' is therefore a section of the tangent bundle of ''M''.
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| The set of all vector fields on ''M'' is denoted by Γ(''TM''). Vector fields can be added together pointwise
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| :<math>(V+W)_x = V_x + W_x\,</math> <!-- "\," improves the display of this formula. Do not delete.-->
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| and multiplied by smooth functions on ''M''
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| :<math>(fV)_x = f(x)V_x\,</math> <!-- "\," improves the display of this formula. Do not delete.-->
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| to get other vector fields. The set of all vector fields Γ(''TM'') then takes on the structure of a [[module (mathematics)|module]] over the [[commutative algebra]] of smooth functions on ''M'', denoted ''C''<sup>∞</sup>(''M'').
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| 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 (mathematics)|sheaf]] of real vector spaces on ''M''.
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| ==Higher-order tangent bundles==
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| Since the tangent bundle ''TM'' is itself a smooth manifold, the [[double tangent bundle|second-order tangent bundle]] can be defined via repeated application of the tangent bundle construction:
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| :<math>T^2 M = T(TM).\,</math><!-- "\," improves the display of this formula. Do not delete.-->
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| In general, the ''k''-th order tangent bundle <math>T^k M</math> can be defined recursively as <math>T(T^{k-1}M)</math>.
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| A smooth map ''f'' : ''M'' → ''N'' has an induced derivative, for which the tangent bundle is the appropriate domain and range ''Df'' : ''TM'' → ''TN''. 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>.
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| A distinct but related construction are the [[jet bundle]]s on a manifold, which are bundles consisting of [[jet (mathematics)|jets]].
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| ==Canonical vector field on tangent bundle==
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| On every tangent bundle ''TM,'' considered as a manifold itself, one can define a canonical vector field ''V'' : ''TM'' → ''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, <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, while the manifold ''M'' is curved, each tangent space at a point ''m'' <math>T_m M \approx \mathbf{R}^n</math> is flat, so the tangent bundle manifold ''TM'' is locally a product of a curved ''M'' and a flat <math>\mathbf{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"):
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| :<math>T(TM) \approx T(M \times \mathbf{R}^n) \cong TM \times T(\mathbf{R}^n) \cong TM \times (\mathbf{R}^n \times \mathbf{R}^n)</math>
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| and the map <math>TTM \to TM</math> is projection onto the first coordinates:
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| :<math>(TM \to M) \times (\mathbf{R}^n \times \mathbf{R}^n \to \mathbf{R}^n).</math>
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| Splitting the first map via the zero section and the second map by the diagonal yields the canonical vector field.
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| If (''x'', ''v'') are local coordinates for ''TM'', the vector field has the expression
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| :<math> V = \sum_i \left. v^i \frac{\partial}{\partial v^i} \right|_{(x,v)}.</math>
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| 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. Note that this expression for the vector field depends only on ''v,'' not on ''x,'' as only the tangent directions can be naturally identified.
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| Alternatively, consider the scalar multiplication function:
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| :<math>\begin{cases}
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| \mathbf{R} \times TM \to TM \\
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| (t,v) \longmapsto tv
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| \end{cases}</math>
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| The derivative of this function with respect to the variable '''R''' at time ''t'' = 1 is a function ''V'' : ''TM'' → ''TTM'', which is an alternative description of the canonical vector field.
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| 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 León et al.
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| ==Lifts==
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| 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]].
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| The ''vertical lift'' of a function ''f'' : ''M'' → '''R''' is the function ''f<sup>v</sup>'' : ''TM'' → '''R''' defined by <math>f^v=f\circ \pi</math>, where π : ''TM'' → ''M'' is the canonical projection.
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| ==See also==
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| * [[pushforward (differential)]]
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| * [[unit tangent bundle]]
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| * [[cotangent bundle]]
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| * [[frame bundle]]
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| * [[Musical isomorphism]]
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| ==Notes==
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| <references group=note/>
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| ==References==
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| {{citations missing|date=July 2009}}
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| {{Reflist}}<!--added under references heading by script-assisted edit-->
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| * {{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
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| * John M. Lee, ''Introduction to Smooth Manifolds'', (2003) Springer-Verlag, New York. ISBN 0-387-95495-3.
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| * Jurgen Jost, ''Riemannian Geometry and Geometric Analysis'', (2002) Springer-Verlag, Berlin. ISBN 3-540-42627-2
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| * [[Ralph Abraham]] and [[Jerrold E. Marsden]], ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London. ISBN 0-8053-0102-X
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| * 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]
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| ==External links==
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| * {{springer|title=Tangent bundle|id=p/t092110}}
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| * [http://mathworld.wolfram.com/TangentBundle.html Wolfram MathWorld: Tangent Bundle]
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| * [http://planetmath.org/encyclopedia/TangentBundle.html PlanetMath: Tangent Bundle]
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| [[Category:Differential topology]]
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| [[Category:Vector bundles]]
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