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Tangent bundle

In mathematics, the tangent bundle of a manifold is the union of all the tangent spaces at every point in the manifold.

Definition as directions of curves

Suppose is a
manifold, and , where is an open subset of , and is the dimension of the manifold, in the chart ; furthermore suppose is the tangent space at a point in . Then the tangent bundle,

It is useful, in distinguishing between the tangent space and bundle, to consider their dimensions, n and 2n respectively. That is, the tangent bundle accounts for dimensions in the positions in the manifold as well as directions tangent to it.

Since we can define a projection map, π for each element of the tangent bundle giving the element in the manifold whose tangent space the first element lies, tangent bundles are also fiber bundles.