the tangent bundle of an $m$-dimensional manifold is $2m$

Question

It is often taken as obvious that for $m$-dimensional manifold $M$ in $\mathbb{R}^n$, $TM$ is a $2m$-dimensional manifold in $T\mathbb{R}^n=\mathbb{R}^n\times\mathbb{R}^n$. But still is there an argument for the dimensinality number $2m$?

Answer 1

Whether this claim is obvious or not depends on the definition you've been given of the tangent bundle to a manifold. The "one-line proof" is: dimension is local, $M$ is locally $\mathbb{R}^m$, and you already agree that the tangent bundle to $\mathbb{R}^m$ itself has dimension $2m$.

However, it's possible that your definition of tangent bundle doesn't make it clear how the dimension of the tangent bundle is preserved by homeomorphism. So what you should show (using whatever definition of tangent bundle you have) is: for a point $p \in M$ and a neighborhood $U$ of $p$ with a homeomorphism $\phi: U \stackrel{\sim}{\to} \mathbb{R}^m$, $\phi$ induces a homeomorphism from the tangent bundle to $M$, restricted to $U$, to the tangent bundle of $\mathbb{R}^m$.