tangent bundle

The tangent bundle is a vector bundle that contains all the tangent spaces of all the points in the manifold.

Definition

$TM=\{(p,v)|p\in M,v\in T_{p}M\}$ Or, if you like arbitrary unions, $TM={⨆}_{p\in M}{T}_{p}M$

Manifold structure

If $M$ is a smooth manifold with $\dim M=n$, then $TM$ has a smooth manifold structure and has $\dim TM=2n$.

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References

@lee2013