Diffeomorphism From Tangent Bundle to Product

[Arkuski]

Show that $TS^1$ is diffeomorphic to $TM×TN$.

($TS^1$ is the tangent bundle of the 1-sphere.)

We can use the theorem stating the following.

If $M$ is a smooth $n$-manifold with or without boundary, and $M$ can be covered by a single smooth chart, then $TM$ is diffeomorphic to $M×ℝ^n.$

Clearly, I must be looking for a single smooth chart on $S^1$, but I am very uncertain on how to go about doing this. Any tips on managing the nuances are greatly appreciated.

 

[fzero]

Show that $TS^1$ is diffeomorphic to $TM×TN$.

($TS^1$ is the tangent bundle of the 1-sphere.)

Have $M$ and $N$ been defined? $TS^1$ has a fairly simple geometric picture, but if we don't know what $M$ and $N$ are, we can't really help you.

We can use the theorem stating the following.

If $M$ is a smooth $n$-manifold with or without boundary, and $M$ can be covered by a single smooth chart, then $TM$ is diffeomorphic to $M×ℝ^n.$

Clearly, I must be looking for a single smooth chart on $S^1$, but I am very uncertain on how to go about doing this. Any tips on managing the nuances are greatly appreciated.

It is impossible to cover $S^1$ with a single chart, since the endpoints of the line would map to the same point.

 

[Arkuski]

Have $M$ and $N$ been defined?

My mistake, I meant $S^1×ℝ$ instead of $M×N$

 

[Mandelbroth]

It is impossible to cover $S^1$ with a single chart, since the endpoints of the line would map to the same point.

This. It's fairly well known that any atlas on $S^1$ must have at least two charts. The reason, as fzero mentions, is a chart must be equipped with a homeomorphism. For us to use a single chart here, if we start at a point, we must necessarily end on that point. Thus, the map from $S^1$ to $\mathbb{R}$ wouldn't even be single-valued, let alone a homeomorphism. Thus, we'd need two charts.

My mistake, I meant $S^1×ℝ$ instead of $M×N$

You mean "instead of $TM\times TN$"?

 

[fzero]

My mistake, I meant $S^1×ℝ$ instead of $M×N$

You mean "instead of $TM\times TN$"?

OK, so let's assume that we have to show that $TS^1$ is diffeomorphic to $S^1\times \mathbb{R}$. Since we know that we need at least two charts to cover $S^1$, we know that we will need at least two charts to cover $TS^1$.

You should start by describing $TS^1$ at a point $p$ of $S^1$. Then pick a covering set of charts for $S^1$ and describe the tangent space on each chart. Ultimately you will want to show that there is a basis for smooth vector fields that is valid at every point of $S^1$.