Tangent bundle
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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 ∈ Tx(M), the tangent space at x. If M is n-dimensional, U is a neighborhood of x, 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
External links
- MathWorld: Tangent Bundle (http://mathworld.wolfram.com/TangentBundle.html)
- PlanetMath: Tangent Bundle (http://planetmath.org/encyclopedia/TangentBundle.html)
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.es:Fibrado tangente