# Homeomorphism of torus and injective curve

I have an exercise in topology and I am struggling with it:

a) Proof that a homeomorphism $$S^1 \times S^1\to \mathbb T ^2$$ exists.

b) For which $$p,q \in \mathbb R$$ is the curve $$\gamma(p,q):=\mathbb{R}\to S^1 \times S^1 , t\mapsto (\exp(2* \pi*t*p),\exp(2*\pi*t*q)$$ injective?

c) Draw a sketch of the curve $$\gamma(2,3)$$ for $$r=1$$ and $$R=2$$.

I think the answer for a) is that $$f((a,b),(c,d))=((c+R)*a,(c+R)*b,d*r)$$ is the requested homeomorphism with the inverse $$f^{-1}(a,b,c)= (\frac{a}{\sqrt{a^2+b^2}}, \frac{y}{\sqrt{a^2+b^2}},\frac{\sqrt{a^2+b^2}-R}{r},\frac{c}{r})$$.

My Problem is b) and c). In b) I think that I have to choose $$p$$ and $$q$$ so that the $$\exp$$ is not periodic, is this right? If it is right how can I find this $$p$$ and $$q$$ and if it is nit true, what I have to do? And at the moment im completley confused which effect the parameter have for the sketch in c).

Thank you :)

Definition: $$\mathbb T^2$$ is the surface of revolution generated by revolving the circle $$\{(x,0,z):(x-R)^2+z^2=r^2\}$$ around the $$z$$-axis.

• Your question is not complete unless you tell us what definition of $\mathbb T^2$ you are using. This is important not just to understand your equations, which seem somewhat random, because also because one of the most common definitions, $\mathbb T^2$ is defined to be $S^1 \times S^1$. – Lee Mosher Nov 4 '19 at 22:44
• I added the definition. – alpaka123 Nov 4 '19 at 23:03

For b. As for $$q=0$$ or $$p=0$$ is clear that the map is not injective, we may assume $$pq\neq 0$$. Note that if $$\gamma$$ is not inyective there should exist $$t_1,t_2$$ such that (I added an $$i$$ that I think you forgot).
$$(e^{2\pi it_1 p},e^{2\pi it_1 q})=(e^{2\pi it_2 p},e^{2\pi it_2 q})\quad \Rightarrow \quad e^{2\pi i(t_1-t_2) p}=1=e^{2\pi i(t_1-t_2) q}$$
Therefore $$(t_1-t_2) p=n\in \mathbb Z$$ and $$(t_1-t_2) q=m\in \mathbb Z$$. Therefore $$p/q=(n/m)$$ is rational. Which proves that $$\gamma$$ is NOT injective if $$p/q$$ is rational. I leave to you the proof of the reverse implication but the final result is $$\gamma$$ is injective if and only if $$p/q$$ is irrational.
That gives a fairly good idea for c. As (2/3) is rational $$\gamma$$ is not injective. Moreover you have to look for the smallest $$t$$ such that both $$2t$$ and $$3t$$ are integers. 2,3 are co-prime then the smallest such $$t$$ is 1 and the image $$\gamma(2,3)(\mathbb R)$$ is equal to $$\gamma(2,3)([0,1])$$. Lastly noting than $$e^{2\pi i 2t},e^{2\pi i 3t}$$ covers the circle two and three times for $$0\leq t\leq 1$$ respectively. you see the plot is a line that winds around one direction two times and around the other three times.