# Diffeomorphism From Tangent Bundle to Product

• Arkuski
In summary, it is impossible to cover ##S^1## with a single chart, since the endpoints of the line would map to the same point.f

#### 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.

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.

Have ##M## and ##N## been defined?

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

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##"?

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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##.