Tangent bundle

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 : Rⁿ → U is a coordinate chart then the preimage V of U in T(M) admits a map to ψ : Rⁿ × Rⁿ → V 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
• Geodesic
• Lie derivative
• Differential form

External links

• MathWorld: Tangent Bundle
• PlanetMath: Tangent Bundle

References

• Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-4267-2
• Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X
• Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0.