Loterre: Mathematics: tangent bundle

geometry > differential geometry > tangent bundle

Preferred term

tangent bundle

Definition(s)

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 of the tangent spaces of ${\\displaystyle M}$ . That is,

${\\displaystyle {\egin{aligned}TM&=\igsqcup _{x\\in M}T_{x}M\\\\&=\igcup _{x\\in M}\\left\\{x\ ight\\}\ imes T_{x}M\\\\&=\igcup _{x\\in M}\\left\\{(x,y)\\mid y\\in T_{x}M\ ight\\}\\\\&=\\left\\{(x,y)\\mid x\\in M,\\,y\\in T_{x}M\ ight\\}\\end{aligned}}}$

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

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

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).
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Tangent_bundle)