Tangential bundle

Here the tangential bundle of the circle is illustrated. The first picture shows the tangential spaces on the circle and in the second picture these spaces are combined into a bundle.

Tangential bundle is a term from differential geometry and differential topology . It is about the disjoint union of all tangent spaces . If the tangential bundle has a particularly simple structure, the underlying manifold is called parallelizable .

definition

The tangent bundle of a differentiable manifold is a vector bundle . As a set it is defined as the disjoint union of all tangent spaces of : ${\displaystyle TM}$ ${\displaystyle M}$ ${\displaystyle M}$

${\displaystyle TM:=\bigsqcup _{p\in M}T_{p}M:=\bigcup _{p\in M}\{p\}\times T_{p}M.}$

The vector space structure in the fibers is the structure inherited from the tangent spaces. ${\displaystyle \{p\}\times T_{p}M}$

If M is a -dimensional differentiable manifold and U is an open, contractible neighborhood of , then TU is diffeomorphic to , i.e. locally the tangent bundle TM is diffeomorphic to . ${\displaystyle n}$ ${\displaystyle p\in M}$ ${\displaystyle U\times \mathbb {R} ^{n},}$ ${\displaystyle \mathbb {R} ^{2n}}$

A tangential bundle is given a differentiable structure by the underlying manifold. An atlas of the tangential bundle, in which all maps have the form , is called a local trivialization . The tangential bundle gets the topology and differentiable structure through a local trivialization. ${\displaystyle U\times \mathbb {R} ^{n}}$

A differentiable manifold with a trivial tangential bundle (that is , as a bundle is isomorphic to ) is called parallelizable . ${\displaystyle M}$ ${\displaystyle TM}$ ${\displaystyle M\times \mathbb {R} ^{n}}$

Examples

Parallelizable Manifolds

• ${\displaystyle M=\mathbb {R} ^{n}}$, which is tangent bundle ${\displaystyle TM=\mathbb {R} ^{n}\times \mathbb {R} ^{n}=\mathbb {R} ^{2n}.}$
• Be the 1-sphere . The tangential bundle is the infinitely long cylinder, that is ${\displaystyle S^{1}=\{x\in \mathbb {R} ^{2}:\left\|x\right\|=1\}}$ ${\displaystyle TS^{1}=S^{1}\times \mathbb {R} .}$
• Every finite-dimensional Lie group , because one can choose a basis for the tangent space on the neutral element and then transport it over completely through the group effect in order to obtain a trivialization of . ${\displaystyle G}$ ${\displaystyle T_{e}G}$ ${\displaystyle e}$ ${\displaystyle G}$ ${\displaystyle TG}$
• Every orientable closed manifold. ${\displaystyle 3}$

Non-trivial tangential bundles

• ${\displaystyle TS^{2}}$with , because according to the proposition of the hedgehog there is no constant tangential vector field vanishing anywhere on the sphere. ${\displaystyle S^{2}=\{x\in \mathbb {R} ^{3}:\left\|x\right\|=1\}}$ ${\displaystyle 2}$
• Raoul Bott and John Milnor proved in 1958 as a consequence of the Bott periodicity theorem that and are the only spheres that can be parallelized . ${\displaystyle S^{1},S^{3}}$ ${\displaystyle S^{7}}$

Natural projection

The natural projection is a smooth picture

${\displaystyle \pi \colon TM\to M\,}$

defined by

${\displaystyle (p,v)\mapsto p.}$

There is and . So it applies to everyone . ${\displaystyle p\in M}$ ${\displaystyle v\in T_{p}M}$ ${\displaystyle \;\pi ^{-1}(p)=T_{p}M}$ ${\displaystyle p\in M}$

Cotangent bundle

The cotangent bundle is also defined in the same way as the tangential bundle. Let there be a differentiable manifold and its tangent space at the point , then the dual space of the tangent space, which is called cotangent space , is denoted. The cotangent bundle of is now defined as a disjoint union of the cotangent spaces. That is, it applies ${\displaystyle M}$ ${\displaystyle T_{p}M}$ ${\displaystyle p\in M}$ ${\displaystyle T_{p}^{*}M}$ ${\displaystyle T^{*}M}$ ${\displaystyle M}$

${\displaystyle T^{*}M:=\bigsqcup _{p\in M}T_{p}^{*}M.}$

A differentiable structure can also be defined on the cotangential bundle in a natural way.

Unit tangential bundle

→ Main article : Unit tangent bundle

The unit tangential bundle of a Riemannian manifold with a Riemannian metric consists of all tangential vectors of length 1: ${\displaystyle (M,g)}$ ${\displaystyle g}$

${\displaystyle T^{1}M=\left\{v\in TM\mid g(v,v)=1\right\}.}$

The unit tangent bundle is a fiber bundle , but not a vector space bundle . As the fibers

${\displaystyle T_{p}^{1}M=T^{1}M\cap T_{p}M}$

are diffeomorphic to a sphere , one also speaks of a bundle of spheres .

Vector fields

→ Main article : vector field

A vector field on a differentiable manifold is a mapping that assigns a tangent vector with a base point to each point . In differential topology and differential geometry, one particularly considers smooth vector fields, i.e. those that are smooth mappings from to . ${\displaystyle M}$ ${\displaystyle V\colon M\to TM}$ ${\displaystyle p\in M}$ ${\displaystyle v\in T_{p}M}$ ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle TM}$

literature

• John M. Lee: Introduction to Smooth Manifolds (= Graduate Texts in Mathematics 218). Springer-Verlag, New York NY 2003, ISBN 0-387-95448-1 .
• R. Abraham, Jerrold E. Marsden , T. Ratiu: Manifolds, tensor analysis, and applications (= Applied mathematical sciences 75). 2nd Edition. Springer, New York NY et al. 1988, ISBN 0-387-96790-7 .

Individual evidence

1. ↑ Bott-Milnor: On the parallelizability of the spheres. Bull. Amer. Math. Soc. 64 1958 87-89. ( pdf )