What do the orbits look like on a torus given as $C\times C$ acted upon by $\mathbb{R}$? I'm trying to better understand how the following group action looks like on a torus. The group $\mathbb{R}$ acts on the torus $C\times C=\{(e^{ix},e^{iy}):x,y\in \mathbb{R}\}$ ($C$ being the unit circle in $\mathbb{C}$) via $$t\cdot (e^{ix},e^{iy})\mapsto (e^{i(x+t)},e^{iy})$$
I am able (I think) to visualize the torus based on the definition of $C\times C$: the first coordinate gives us the ring of unit length in the donut and, as $x$ sweeps through $\mathbb{R}$, the second coordinate gives us the "hollow body" of the donut, again of unit length, as $y$ sweeps through $\mathbb{R}$. So here is where the problems begin for me.


*

*What am I doing wrong to show that this a homomorphism? $$st\cdot (e^{ix},e^{iy})\mapsto (e^{i(x+st)},e^{iy})$$ but $$[s\cdot (e^{ix},e^{iy})][t\cdot (e^{ix},e^{iy})] = (e^{i(x+s)},e^{iy})(e^{i(x+t)},e^{iy}) = (e^{i(2x+s+t)},e^{i2y})$$

*For $t\in \mathbb{R}$, the orbit $\mathbb{R}(t)$ is $\{(e^{i(x+t)},e^{iy}):t\in \mathbb{R}\}$. Is this any different from $C\times C$?
 A: It will be a whole lot easier to study this action by representing the torus as $\mathbf R^2/\mathbf Z^2$. Then your action comes from the action of $\mathbf R$ on $\mathbf R^2$ defined by
$$
t\cdot(x,y)=(x+\tfrac1{2\pi} t,y).
$$
It is clear that this is an action, it is just the action by translation on the right of the additive subgroup $\mathbf R=\mathbf R\times\{0\}$ of $\mathbf R^2$. The orbits are the affine lines parallel to the $x$-axis. In the torus this correpsonds to the circles $C\times\{*\}$, where $*$ is any point of the second factor $C$.
A: Answer to 1.
Thanks to @Dan Rust, I see the mistake in (1). Firstly, let it be clear that the group operation on $\mathbb{R}$ is addition (multiplication cannot give it such structure). Secondly, this line $$[s\cdot (e^{ix},e^{iy})][t\cdot (e^{ix},e^{iy})] = (e^{i(x+s)},e^{iy})(e^{i(x+t)},e^{iy}) = (e^{i(2x+s+t)},e^{i2y})$$ is incorrect: instead of showing composition of permutations I mixed in multiplication of coordinates. It should be $$(s+t)\cdot (e^{ix},e^{iy})\mapsto (e^{i(x+s+t)},e^{iy})$$ and $$s\cdot[t\cdot (e^{ix},e^{iy})] = s\cdot (e^{i(x+t)},e^{iy}) = (e^{i(x+t+s)},e^{iy})$$ which are indeed equal.
Answer to 2.
Thanks to @Johannes Huisman, I see my mistake and use his description of the orbits. The orbits should be written as $$\mathbb{R}(e^{ix},e^{iy}) = \{(e^{i(x+t)},e^{iy}):t\in \mathbb{R}\}$$and not $\mathbb{R}(t) = \{(e^{i(x+t)},e^{iy}):t\in \mathbb{R}\}$. So, for a specific point $(e^{ix},e^{iy})$ on the torus, our orbit will be all points of the form $(e^{i(x+t)},*)$ where $*$ is a fixed point on $C$. As $t$ sweeps through $\mathbb{R}$, we end up with a circle parallel to the plane on which the circle $\{e^{ix}\}$ lies.
