Is π(L) a Submanifold of the Torus?

In summary, Fix a,b \in \mathbb{R} with a \neq 0. Let L = \{(x,y) \in \mathbb{R}^2:ax+by = 0\} and let \pi:\mathbb{R}^2 \rightarrow \mathbb{T}^2 be the canonical projection map. If \frac{b}{a} \notin \mathbb{Q}, then \pi(L) (with the subspace topology) is not a submanifold of \mathbb{T}^2. The conversation discusses different methods for proving that \pi(L) is not locally Euclidean, with the conclusion that a brute force method using the Hurw
  • #1
jgens
Gold Member
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I am trying to prove the following result: Fix [itex]a,b \in \mathbb{R}[/itex] with [itex]a \neq 0[/itex]. Let [itex]L = \{(x,y) \in \mathbb{R}^2:ax+by = 0\}[/itex] and let [itex]\pi:\mathbb{R}^2 \rightarrow \mathbb{T}^2[/itex] be the canonical projection map. If [itex]\frac{b}{a} \notin \mathbb{Q}[/itex], then [itex]\pi(L)[/itex] (with the subspace topology) is not a submanifold of [itex]\mathbb{T}^2[/itex].

I am having difficulty however showing that [itex]\pi(L)[/itex] is not locally Euclidean. From drawing a few pictures, I think every neighborhood of [itex]\pi(0)[/itex] is disconnected (which would be enough to complete the proof), but I am having difficulty showing this. Any help?
 
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  • #2
Look at p.158 of the book of John Lee.
 
  • #3
quasar987 said:
Look at p.158 of the book of John Lee.

Thanks! I (finally) figured out a brute force method using the Hurwitz Theorem that works, but Lee's solution is much cleaner.
 

FAQ: Is π(L) a Submanifold of the Torus?

What is "Irrational Winding of the Torus"?

The "Irrational Winding of the Torus" is a mathematical concept that refers to the idea of wrapping a string or curve around the surface of a torus in a way that cannot be expressed as a ratio of two integers.

How is the "Irrational Winding of the Torus" different from regular winding?

Regular winding involves wrapping a string or curve around the surface of a torus in a way that can be expressed as a ratio of two integers. However, with irrational winding, this ratio is not possible, resulting in a more complex and non-repeating pattern.

What are some real-life examples of "Irrational Winding of the Torus"?

One example is the winding of a string around a doughnut-shaped torus. Another example is the motion of a planet around the sun, where the orbit is not a perfect circle but instead follows a more complex, irrational path.

Why is the concept of "Irrational Winding of the Torus" important in mathematics?

The concept of irrational winding helps us to better understand and visualize objects and phenomena that have complex and non-repeating patterns. It also has applications in fields such as physics and engineering where irrational winding can be used to model and predict various systems and processes.

How does the concept of "Irrational Winding of the Torus" relate to other mathematical concepts?

The concept of irrational winding is closely related to other mathematical concepts such as irrational numbers and fractals. It also has connections to topology and geometry, as the torus is a 3-dimensional geometric object.

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