Lines with irrational slope in a torus

differential-geometrysmooth-manifolds

I'm trying understand one point about Example 15.9 in the Introduction to Manfolds by Tu (p. 167). I'll reproduce the example here. See this post, which addressed a separate question I had.

Example 15.9 (Lines with irrational slope in a torus). Let $G$ be the torus $\mathbb{R}^2/\mathbb{Z}^2$ and $L$ a line through the origin in $\mathbb{R}^2$. The torus can also be represented by the unit square with the opposite edges identified. The image $H$ of $L$ under the projection $\pi:\mathbb{R}^2\longrightarrow\mathbb{R}^2/\mathbb{Z}^2$ is a closed curve if and only if the line $L$ goes through another lattice point, say $(m,n)\in\mathbb{Z}^2$. This is the case if and only if the slope of $L$ is $n/m$, a rational number or $\infty$; then $H$ is the image of finitely many line segments on the unit square. It is a closed curve diffeomorphic to a circle and is a regular submanifold of $\mathbb{R}^2/\mathbb{Z}^2$ (Figure 15.1).

If the slope of L is irrational, then its image H on the torus will never close up. In
this case the restriction to L of the projection map, $f := \pi\big|_L : L \rightarrow \mathbb{R}^2/\mathbb{Z}^2$, is a one-to-one
immersion. We give H the topology and manifold structure induced from f.

  1. I don't understand why $f$ is an immersion. How would I prove this?
  2. Why is the subspace topology of H in $\mathbb{R}^2/\mathbb{Z}^2$ a strict subset of the topology on $H$ induced from $ f : L \xrightarrow{\sim} H$? For the induced topology, I understand that a basic open set of H is the image under $f$ of an open set in $L$. I also understand that basic open sets in the subspace topology will be the intersection of H with the image of an open ball in $\mathbb{R^2}$.

Best Answer

  1. I don't understand why f is an immersion. How would I prove this?

Hint Here's a brief outline:

  1. The inclusion $\iota :L \hookrightarrow \Bbb R^2$ is an immersion.
  2. The map $\pi: \Bbb R^2 \to \Bbb R^2 / \Bbb Z^2$ is a covering map. In particular, it is a local diffeomorphism and hence an immersion.
  3. The composition of immersions is again an immersion.
  1. Why is the subspace topology of H in $\mathbb{R}^2/\mathbb{Z}^2$ a strict subset of the topology on $H$ induced from $ f : L \xrightarrow{\sim} H$? For the induced topology, I understand that a basic open set of H is the image under $f$ of an open set in $L$. I also understand that basic open sets in the subspace topology will be the intersection of H with the image of an open ball in $\mathbb{R^2}$.

See the answer to this question mentioned by Lee Mosher. In particular, if we pick an open ball $B \subset \Bbb R^2$ that intersects $L$, then $\pi(B \cap L)$ is by definition open in the subset topology, but any neighborhood (in the subspace topology) of any point in $\pi(B \cap L)$ contains the image of points in $L$ that are not in $\pi(B \cap L)$, hence $\pi(B \cap L)$ is not open in the subspace topology.

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