# Tangent bundle

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).[note 1]

In differential geometry, the tangent bundle of a differentiable manifold ${\displaystyle M}$ is a manifold ${\displaystyle TM}$ which assembles all the tangent vectors in M. As a set, it is given by the disjoint union[note 1] of the tangent spaces of M. That is,

${\displaystyle {\begin{array}{lcr}TM=\bigsqcup _{x\in M}T_{x}M=\bigcup _{x\in M}\left\{x\right\}\times T_{x}M=\bigcup _{x\in M}\left\{(x,y)\mid y\in T_{x}M\right\}\\~~~~~~~=\left\{(x,y)\mid x\in M,\,y\in T_{x}M\right\}.\end{array}}}$

where ${\displaystyle T_{x}M}$ denotes the tangent space to ${\displaystyle M}$ at the point ${\displaystyle x}$. So, an element of ${\displaystyle TM}$ can be thought of as a pair ${\displaystyle (x,v)}$, where ${\displaystyle x}$ is a point in ${\displaystyle M}$ and ${\displaystyle v}$ is a tangent vector to ${\displaystyle M}$ at ${\displaystyle x}$. There is a natural projection

${\displaystyle \pi :TM\twoheadrightarrow M}$

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

The tangent bundle comes equipped with a natural topology (described in a section below). With this topology, the tangent bundle to a manifold is the prototypical example of a vector bundle (a fiber bundle whose fibers are vector spaces). A section of ${\displaystyle TM}$ is a vector field on ${\displaystyle M}$, and the dual bundle to ${\displaystyle TM}$ is the cotangent bundle, which is the disjoint union of the cotangent spaces of ${\displaystyle M}$. By definition, a manifold ${\displaystyle M}$ is parallelizable if and only if the tangent bundle is trivial.By definition, a manifold M is framed if and only if the tangent bundle TM is stably trivial, meaning that for some trivial bundle E the Whitney sum TME is trivial. For example, the n-dimensional sphere Sn is framed for all n, but parallelizable only for 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 f : MN is a smooth function, with M and N smooth manifolds, its derivative is a smooth function Df : TMTN.

Other Languages
français: Fibré tangent
한국어: 접다발
հայերեն: Շոշափողակոյտ
Nederlands: Raakbundel