# 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.