tangent bundle

Let M be a differentiable manifold. Let the tangent bundle TM of M be(as a set) the disjoint unionMathworldPlanetmath mMTmM of all the tangent spacesPlanetmathPlanetmath to M, i.e., the set of pairs


This naturally has a manifold structureMathworldPlanetmath, given as follows. For M=n, Tn is obviously isomorphicPlanetmathPlanetmathPlanetmath to 2n, and is thus obviously a manifold. By the definition of a differentiable manifold, for any mM, there is a neighborhood U of m and a diffeomorphism φ:nU. Since this map is a diffeomorphism, its derivativePlanetmathPlanetmath is an isomorphismPlanetmathPlanetmathPlanetmath at all points. Thus Tφ:Tn=2nTU is bijectiveMathworldPlanetmathPlanetmath, which endows TU with a natural structure of a differentiable manifold. Since the transition maps for M are differentiableMathworldPlanetmathPlanetmath, they are for TM as well, and TM is a differentiable manifold. In fact, the projection π:TMM forgetting the tangent vector and remembering the point, is a vector bundle. A vector field on M is simply a sectionPlanetmathPlanetmath of this bundle.

The tangent bundle is functorial in the obvious sense: If f:MN is differentiable, we get a map Tf:TMTN, defined by f on the base, and its derivative on the fibers.

Title tangent bundle
Canonical name TangentBundle
Date of creation 2013-03-22 13:58:59
Last modified on 2013-03-22 13:58:59
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 5
Author bwebste (988)
Entry type Definition
Classification msc 58A32
Related topic VectorField
Related topic LieAlgebroids