, 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
, to consider their dimensions, n
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.