Editing Tangent bundle (section)

{{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''<sub>1</sub>}} and {{math|''x''<sub>2</sub>}} of manifold {{math|''M''}} the tangent spaces {{math|''T''<sub>1</sub>}} and {{math|''T''<sub>2</sub>}} have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle {{math|''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 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 ''S<sup>n</sup>'' is framed for all ''n'', but parallelizable only for {{nowrap|1=''n'' = 1, 3, 7}} (by results of Bott-Milnor and Kervaire).