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 M {\displaystyle M} is a manifold T M {\displaystyle TM} which assembles all the tangent vectors in M {\displaystyle M} . As a set, it is given by the disjoint union of the tangent spaces of M {\displaystyle M} . That is,

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

π : T M ↠ M {\displaystyle \pi :TM\twoheadrightarrow M}

defined by π ( x , v ) = x {\displaystyle \pi (x,v)=x} . This projection maps each element of the tangent space T x M {\displaystyle T_{x}M} to the single point x {\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 T M {\displaystyle TM} is a vector field on M {\displaystyle M} , and the dual bundle to T M {\displaystyle TM} is the cotangent bundle, which is the disjoint union of the cotangent spaces of M {\displaystyle M} . By definition, a manifold M {\displaystyle M} is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold M {\displaystyle M} is framed if and only if the tangent bundle T M {\displaystyle TM} is stably trivial, meaning that for some trivial bundle E {\displaystyle E} the Whitney sum T M ⊕ E {\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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