Tangent bundle

Tangent bundle is a term used in differential geometry and differential topology. It is the disjoint union of all tangent spaces. If the tangent bundle, a particularly simple structure, then it is called the underlying manifold parallelizable.

Definition

The tangent bundle of a differentiable manifold is a vector bundle. When quantity is defined as the disjoint union of all tangent spaces of:

The vector space structure in the fibers is inherited from the tangent structure.

If M is a -dimensional differentiable manifold and U be an open, contractible around, then TU is diffeomorphic to the tangent bundle TM is called locally is diffeomorphic to.

A tangent is replaced by the underlying manifold again a differentiable structure. This is called an atlas of the tangent bundle, in which all cards have the form, a local trivialization. The topology and differentiable structure gets the tangent bundle by a local trivialization.

A differentiable manifold with trivial tangent bundle ( ie is isomorphic to a bundle ) is called parallelizable.

Examples

Parallelizable manifolds

  • , The tangent bundle is
  • Be the 1- sphere. The tangent bundle is the infinitely long cylinder, ie
  • Every finite-dimensional Lie group, because you can choose a basis for the tangent space at the neutral element and then transported by the group effect over all, to obtain a trivialization of.
  • Any orientable closed manifold.

Nontrivial tangent bundle

  • With, because after the hairy ball theorem, there is on the sphere no nowhere vanishing, continuous tangent vector field.
  • Raoul Bott and John Milnor proved in 1958 as a consequence of the Bott - Periodizitätssatz that and the only parallelizable spheres are.

Natural projection

The natural projection is a smooth map

Defined by

Here, and. It applies to everyone.

Cotangent bundle

Similar to the tangent bundle is also defined the cotangent bundle. Let be a differentiable manifold and its tangent space at the point, so is referred to as the dual space of the tangent space, which is called cotangent space. The cotangent bundle of is now defined as a disjoint union of Kotangentialräume. That is, it is

Can again define naturally a differentiable structure on the cotangent bundle.

Unit tangent bundle

The unit tangent bundle of a Riemannian manifold with Riemannian metric consists of all tangent vectors of length 1:

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

Diffeomorphic to be a sphere, one also speaks of a sphere bundle.

Vector fields

A vector field on a differentiable manifold is a mapping which assigns to each point a tangent vector with foot point. In differential topology and differential geometry is considered especially smooth vector fields, ie, those which are smooth pictures from to.

Atlas (topology) Topological space Smooth function
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