Tangent bundle

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 <math>M</math> is a <math>C^k</math> manifold, and <math>\phi : U \rightarrow \mathbb{R}^n </math>, where <math>U</math> is an open subset of <math>M</math>, and <math>n</math> is the the dimension of the manifold, in the chart <math>\phi(\circ)</math>; furthermore suppose <math> T_{p}M </math> is the tangent space at a point <math> p </math> in <math> M </math>. Then the tangent bundle,
<math> {TM} = \bigcup_{p \in M} T_{p}M </math>
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.