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I just started learning Smooth Manifolds and got stuck on this question:

Show that $T\mathbb S^1$ is diffeomorphic to $\mathbb S^1\times\mathbb R$

I can see that $T\mathbb S^1$ and $\mathbb S^1\times\mathbb R$ are at least isomorphic if I draw a picture with these spaces, but I have trouble to find an exact formula for diffeomorphic map.

aleksandr
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    Do you know that Lie groups are always parallelizable? $S^1$ is a Lie group via multiplication. Alternatively, try to find a global frame for $S^1$. – Ayman Hourieh Jul 06 '14 at 18:22
  • could you say how your book defines the tangent space ? – Rene Schipperus Jul 06 '14 at 18:27
  • @AymanHourieh, Well, I'm just planning to study Lie groups, so the answer to your question is No, but I'll think about it when I get to Lie groups. What do you mean by a global frame? I can describe $\mathbb S^1$ as ${e^{i\phi}:\phi\in[0,2\pi)}$, if it is what you mean. – aleksandr Jul 06 '14 at 18:27
  • @ReneSchipperus, The tangent space is defined as the set of all derivations, linear maps $v:C^\infty(\mathbb S^1)\to\mathbb R$ satisfying the product rule. – aleksandr Jul 06 '14 at 18:32

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One way is have a global description of the tangent space. If $f$ is a function defined on a neighbourhood of point of $S^1$ the we may think of it as a function $$\theta \mapsto f(\cos \theta , \sin \theta)$$

and we can use $\frac{d}{d\theta}$ as a basis for the tangent space. Note that this is global. Just to be clear if the point is given by $\theta_0$ then the vector at this point is $$\frac{d}{d\theta}\Big|_{\theta_0}$$ Now a point of the Tangent space is given by $\left((\cos \theta_0 ,\sin \theta_0), a\frac{d}{d\theta}\Big|_{\theta_0}\right)$

so

$$\left((\cos \theta_0 ,\sin \theta_0), a\frac{d}{d\theta}\Big|_{\theta_0}\right) \mapsto \left((\cos \theta_0 ,\sin \theta_0), a \right)$$ is the desired homeomorphism.

Rene Schipperus
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  • Why is this actually a diffeomorphism? It is clearly a bijection but why is is also smooth? @Rene Schipperus there is no need of charts to show this right? – Mat999 Apr 19 '22 at 14:48