Tangent bundle - Definition and Overview

In mathematics, the tangent bundle of a manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure.

The tangent bundle of manifold M is usually denoted by T(M) or just TM. Any element of T(M) is a pair (x,v) where v ∈ T_x(M), the tangent space at x. If M is n-dimensional and φ : U → Rⁿ is a coordinate chart then the preimage V of U in T(M) admits a map to ψ : V → Rⁿ × Rⁿ defined by ψ(x, v) = (φ(x), dφ(v)). This map is taken to be a chart (by definition) and it defines structure of smooth 2n-dimensional manifold on T(M).

See also: Cotangent bundle

External links

• MathWorld: Tangent Bundle (http://mathworld.wolfram.com/TangentBundle.html)
• PlanetMath: Tangent Bundle (http://planetmath.org/encyclopedia/TangentBundle.html)