Tangent bundle

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

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

where ${\displaystyle T_{x}M}$ 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 element of the 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 (which is 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 ${\displaystyle M}$ is framed if and only if the tangent bundle ${\displaystyle TM}$ is stably trivial, meaning that for some trivial bundle ${\displaystyle E}$ the Whitney sum ${\displaystyle TM\oplus E}$ 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).
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