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Let $T^2\subset\mathbb{C}^2$ denote the (usual) torus. Let $a\in\mathbb{R}$ be an irrational number and define a map $f(t)=(e^{2\pi it}, e^{2\pi i at})$. Prove that:

(a) $f$ is injective

(b) $f$ is an immersion, but not an embedding

(c) $f(\mathbb{R})$ is dense in $T^2$

(d) $f(\mathbb{R})$ is not an embedded submanifold of $T^2$

I've already shown (a) and (b), but (c) has me stumped. (I haven't tried my hand at (d) yet.)

For ease I've just been working with a square (with edges identified). I took an open set (wlog, an open ball) in the square, but it's not clear what will happen if the image of $f$ doesn't hit this ball. It seems like we want to somehow get that the image "wraps" back onto itself, which would imply that $a$ is rational, but... I don't see how to get there.

Any help is appreciated!

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I think you want to show directly that the image does hit any ball. Translate the ball down along the direction (1,a) such that its center sits at, say, the horizontal edge of the square. What can you say about the intersection of the image and the horizontal edge? –  Henning Makholm Nov 4 '11 at 21:48
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How did you prove that $f$ is not an embedding? –  Mariano Suárez-Alvarez Nov 5 '11 at 2:43
    
@Mariano: I showed that $f(0)$ is a limit point of $f(\mathbb{Z})$. Since $\mathbb{Z}$ has no limit in $\mathbb{R}$, $f$ can't be homeomorphic onto its image. –  Bey Nov 5 '11 at 14:55
    
@Henning: What do mean translate the ball along (1,a)? –  Bey Nov 6 '11 at 19:18
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@Bey, if the original ball has center at $(x,y)$ in an $1\times1$ square, consider the one centered at $(x-\frac1a y \bmod 1, 0)$ instead. If the moved ball is hit, then the original ball will be hit $\frac 1a y$ time later. So it is enough to show that every ball centered on the lower edge is hit. –  Henning Makholm Nov 6 '11 at 23:01

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up vote 3 down vote accepted

I think you want to show directly that the image does hit any ball. For any ball with center $(x,y)$ in the unit-square-with-edges-identified, translate the ball down along the direction $(1,a)$ such that its center sits at, say, the vertical edge of the square, with center $(x,y)-x(1,a)=(0,y-ax)$. If the translated ball is hit, the original ball will be hit $x$ time later.

Thus, it is enough to show that every ball centered on the vertical edge is hit. What can you say about the intersection of the image and the vertical edge?

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