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    Tangent Bundle

    Basic definition

    In topology, a tangent bundle of a given manifold is a new manifold that consists of the tangent spaces for each point pasted together in a continuous fashion.

    Detailed definition

    Every smooth manifold M has a tangent bundle T M, which consists of the tangent space T M_p at all points p in M. Since a tangent space T M_p is the set of all tangent vectors to M at p, the tangent bundle is the collection of all tangent vectors, along with the information of the point to which they are tangent.

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