how to prove that quotient group $\mathbb{R}/\mathbb{Z}$ and the circle are diffeomorphism?

Question

Are quotient group $\mathbb{R}/\mathbb{Z}$ and the circle are diffeomorphism? How to prove? Hope someone give me some advise or some reference documents. Thank you.

Answer 1

I know some information for the problem and I hope it helps you by considering another point of view. We know that the set of all complex numbers of absolute value $1$ makes a multiplicative group denoted by $T$. We can easily verify that for ant fix $y\in\mathbb R$, the map $f_y:\mathbb R\to T,~~ f_y(x)=\exp(iyx)$ is a homomorphism. So we can have $\mathbb R/\mathbb Z\cong T$. It is just to verify that the latter isomorphism and its inverse are both differentiable. Sorry if this is due to Group theory. Above function can be regarded as $$g:\mathbb R\to T=S^1\\g(x)=\exp(2\pi xi)$$ also.