tangent bundle (original) (raw)

Let M be a differentiable manifold. Let the tangent bundle T⁢M of M be(as a set) the disjoint unionMathworldPlanetmath ∐m∈MTm⁢M of all the tangent spacesPlanetmathPlanetmath to M, i.e., the set of pairs

This naturally has a manifold structureMathworldPlanetmath, given as follows. For M=ℝn, T⁢ℝn is obviously isomorphicPlanetmathPlanetmathPlanetmath to ℝ2⁢n, and is thus obviously a manifold. By the definition of a differentiable manifold, for any m∈M, there is a neighborhood U of m and a diffeomorphism φ:ℝn→U. Since this map is a diffeomorphism, its derivativePlanetmathPlanetmath is an isomorphismPlanetmathPlanetmathPlanetmath at all points. Thus T⁢φ:T⁢ℝn=ℝ2⁢n→T⁢U is bijectiveMathworldPlanetmathPlanetmath, which endows T⁢U with a natural structure of a differentiable manifold. Since the transition maps for M are differentiableMathworldPlanetmathPlanetmath, they are for T⁢M as well, and T⁢M is a differentiable manifold. In fact, the projection π:T⁢M→M 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:M→N is differentiable, we get a map T⁢f:T⁢M→T⁢N, defined by f on the base, and its derivative on the fibers.