Definitions for tangent bundle
tan·gen·t bun·dle

WikipediaRate this definition:0.0 / 0 votes

1. Tangent bundle

   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 (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 is framed if and only if the tangent bundle TM is stably trivial, meaning that for some trivial bundle 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).

How to pronounce tangent bundle?

1. Alex

   US English

   David

   US English

   Mark

   US English

   Daniel

   British

   Libby

   British

   Mia

   British

   Karen

   Australian

   Hayley

   Australian

   Natasha

   Australian

   Veena

   Indian

   Priya

   Indian

   Neerja

   Indian

   Zira

   US English

   Oliver

   British

   Wendy

   British

   Fred

   US English

   Tessa

   South African

How to say tangent bundle in sign language?

Numerology

1. Chaldean Numerology

   The numerical value of tangent bundle in Chaldean Numerology is: 7

2. Pythagorean Numerology

   The numerical value of tangent bundle in Pythagorean Numerology is: 4

References

1. ^ Wikipedia
   https://en.wikipedia.org/wiki/Tangent_Bundle