@prefix psr: . @prefix skos: . psr: a skos:ConceptScheme . psr:-V0G085HP-P skos:prefLabel "differential geometry"@en, "géométrie différentielle"@fr ; a skos:Concept ; skos:narrower psr:-RQQ8MB83-B . psr:-RQQ8MB83-B skos:definition """En mathématiques, et plus précisément en géométrie différentielle, le fibré tangent TM associé à une variété différentielle M est la somme disjointe de tous les espaces tangents en tous les points de la variété, soit :

{\\\\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,v)\\\\mid v\\\\in T_{x}M\\ ight\\\\}\\\\\\\\&=\\\\left\\\\{(x,v)\\\\mid x\\\\in M,\\\\,v\\\\in T_{x}M\\ ight\\\\}\\\\end{aligned}}}

où

{\\\\displaystyle T_{x}M}

est l'espace tangent de M en x. Un élément de TM est donc un couple (x, v) constitué d'un point x de M et d'un vecteur v tangent à M en x.
Le fibré tangent peut être muni d'une topologie découlant naturellement de celle de M. Sous cette topologie, il possède une structure de variété différentielle prolongeant celle de M ; c'est un espace fibré de base M, et même un fibré vectoriel.
(Wikipedia, L'Encylopédie Libre, https://fr.wikipedia.org/wiki/Fibr%C3%A9_tangent)"""@fr, """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}

{\\\\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)"""@en ; a skos:Concept ; skos:inScheme psr: ; skos:broader psr:-V0G085HP-P ; skos:prefLabel "fibré tangent"@fr, "tangent bundle"@en ; skos:exactMatch , .