I A beginner’s thoughts on rotation number -- Resource suggestion request

[Pencilvester]

TL;DR Summary

I’ve minimal exposure to topology, but I’ve been thinking through rotation number for regular smooth curves, and I’d like to know if I’m on the right track as well as what some good resources are to learn more about this.

I have been thinking about rotation number of regular smooth curves in different surfaces. Here is how I’ve been defining these things: a regular smooth curve is a map from $S^1 \rightarrow \mathbb{R}^2$ whose derivative is non-vanishing. If we have a regular smooth curve $\gamma$ as well as a non-vanishing and continuous vector field $\mathbf{V}$ on the same surface as the curve, then $\gamma$’s derivative vector makes an angle with $\mathbf{V}$ at every point on the curve, and this angle changes continuously as we move along $\gamma$. If we take the total change in angle these two vectors make as we traverse the entire curve (with counterclockwise being positive and clockwise being negative) and divide by $2\pi$, we get $\gamma$’s rotation number.
I can see how if we want rotation number to be invariant under homotopy, then $\mathbf{V}$ must be non-vanishing, which is why rotation number cannot be as simply defined for regular smooth curves on $S^2$. But I also see that we can choose to generalize to include non-non-vanishing vector fields— however, we would then need to forbid the curves from crossing the vector fields’ vanishing points.

And here comes the main drive for this post: this is all very speculative on my part, none of it rigorous. I honestly don’t even know for sure if all of this is even correct (i.e. I haven’t proven anything— it just seems right to me). So I was hoping for two things: One, if I’m wrong about any of this, I’d like to know what and how. And two, what are some good resources to learn more about these things?
(I have studied some Reimannian/pseudo-Reimannian geometry as it pertains to general relativity, and I have a pretty crude understanding of basic topology, but I have virtually no knowledge of algebraic topology.)

 

[fresh_42]

TL;DR Summary: I’ve minimal exposure to topology, but I’ve been thinking through rotation number for regular smooth curves, and I’d like to know if I’m on the right track as well as what some good resources are to learn more about this.

(I have studied some Reimannian/pseudo-Reimannian geometry as it pertains to general relativity, and I have a pretty crude understanding of basic topology, but I have virtually no knowledge of algebraic topology.)

Firstly, it's Riemann, not Reimann. Secondly, algebraic topology is exactly what you need here. I don't think that coordinates will be of a lot of help here, e.g. angles. Here is a short introduction
https://scholar.harvard.edu/files/rastern/files/algtoppreview.pdf
and if you want to do it right
https://scholar.harvard.edu/files/rastern/files/algtoppreview.pdf

 

[Pencilvester]

Thanks!

Here is a short introduction
https://scholar.harvard.edu/files/rastern/files/algtoppreview.pdf
and if you want to do it right
https://scholar.harvard.edu/files/rastern/files/algtoppreview.pdf

I will definitely read through this, but did you mean to link to something else for the second one? The URLs appear to be identical.

 

[fresh_42]

Thanks!

I will definitely read through this, but did you mean to link to something else for the second one? The URLs appear to be identical.

Yes, sorry. A buffer mistake. I meant a link to a server of the University of Edinburgh, but it seemed to be still copyright-protected (AP). I hate it, when Western universities do that. I mean, whom can you trust these days?

Here is a source that should be ok:
https://loeh.app.uni-regensburg.de/teaching/topologie1_ws1819/lecture_notes.pdf
At least the author's location and university server coincide so it should be legal.

Edit: This here
http://tomlr.free.fr/Math%E9matique...lgebraic%20Topology%20(Trieste,%20Bruzzo).pdf
is a bit less axiomatic, as far as this is possible, and with more examples.

And if your are interested in a bit more geometry than topology then
https://arxiv.org/pdf/1205.5935
is recommendable.

 

[Pencilvester]

Here is a source that should be ok:
https://loeh.app.uni-regensburg.de/teaching/topologie1_ws1819/lecture_notes.pdf
At least the author's location and university server coincide so it should be legal.

Edit: This here
http://tomlr.free.fr/Math%E9matique...lgebraic%20Topology%20(Trieste,%20Bruzzo).pdf
is a bit less axiomatic, as far as this is possible, and with more examples.

And if your are interested in a bit more geometry than topology then
https://arxiv.org/pdf/1205.5935
is recommendable.

Fantastic, thank you!

 

[fresh_42]

Fantastic, thank you!

Here is another one from Cornell:
https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
with an explicit allowance for non-commercial use. And it has 211 occurrences of the word "sphere"!

 

[dunno1954]

TL;DR Summary: I’ve minimal exposure to topology, but I’ve been thinking through rotation number for regular smooth curves, and I’d like to know if I’m on the right track as well as what some good resources are to learn more about this.

I have been thinking about rotation number of regular smooth curves in different surfaces. Here is how I’ve been defining these things: a regular smooth curve is a map from $S^1 \rightarrow \mathbb{R}^2$ whose derivative is non-vanishing. If we have a regular smooth curve $\gamma$ as well as a non-vanishing and continuous vector field $\mathbf{V}$ on the same surface as the curve, then $\gamma$’s derivative vector makes an angle with $\mathbf{V}$ at every point on the curve, and this angle changes continuously as we move along $\gamma$. If we take the total change in angle these two vectors make as we traverse the entire curve (with counterclockwise being positive and clockwise being negative) and divide by $2\pi$, we get $\gamma$’s rotation number.
I can see how if we want rotation number to be invariant under homotopy, then $\mathbf{V}$ must be non-vanishing, which is why rotation number cannot be as simply defined for regular smooth curves on $S^2$. But I also see that we can choose to generalize to include non-non-vanishing vector fields— however, we would then need to forbid the curves from crossing the vector fields’ vanishing points.

And here comes the main drive for this post: this is all very speculative on my part, none of it rigorous. I honestly don’t even know for sure if all of this is even correct (i.e. I haven’t proven anything— it just seems right to me). So I was hoping for two things: One, if I’m wrong about any of this, I’d like to know what and how. And two, what are some good resources to learn more about these things?
(I have studied some Reimannian/pseudo-Reimannian geometry as it pertains to general relativity, and I have a pretty crude understanding of basic topology, but I have virtually no knowledge of algebraic topology.)

There are works along the similar idea using vector fields.
I list two of them. There are more.

Reinhart, Bruce L.
The winding number on two-manifolds. (English) Zbl 0097.16203
Ann. Inst. Fourier 10, 271-283 (1960).

Chillingworth, D. R. J.
Winding numbers on surfaces. I. (English) Zbl 0221.57001
Math. Ann. 196, 218-249 (1972).

Note that these people use the terminology "winding numbers"
instead of "rotation numbers".

I think their approach is not appropriate for regular homotopy classification of
regular closed curves on surfaces, because their invariant depends on the chosen vector field.

A better approach is as follows: First construct a plane curve from the given closed curve on the surface, and somehow use the rotation number of the plane curve to define the rotation number of the original curve.

(1) When the surface is a 2-sphere, we can also use the stereographic projection w.r.t. the North Pole of the sphere to obtain a closed curve on the euclidean plane. If it happens that the curve on the sphere passes through the North Pole, then modify the curve slightly so that the curve misses the North Pole. The obtained rotation number of the plane curve is well-defined modulo 2. So this defines a mod 2 integer, and this can be used for regular homotopy classification.

(2) When the surface has a complete euclidean/hyperbolic metric, we consider its universal cover (the euclidean plane/the hyperbolic plane).
When the curve is homotopically trivial, we can shrink the curve arbitrarily small so that we can regard it to be a plane curve, and use its rotation number as the rotation number of the original curve.
Assume that the given curve is not null-homotopic. The closed curve on the surface lifts to a non-closed curve on the universal cpver. Immitating the closed curve case, we can define the euclidean rotation number of the lift. Unfortunately, it may not be an integer, and it depends on the choice of the lift.
To remedy this, we connect the two ends of the lift by the shortest geodesic, and subtract its euclidean rotation number form the euclidean rotation number of the lift. It turns out that this difference is a well-defined integer, and it can be used for the regular homotopy classification on the surface.

See the following paper for details.

Rotation numbers of regular closed curves on oriented aspherical surfaces
Yamasaki, Masayuki
https://libir.josai.ac.jp/il/user_contents/02/G0000284repository/pdf/JOS-13447777-1303.pdf

Abstract. Whitney's rotation number classifies regular closed curves on the euclidean plane up to regular homotopy and, when the self-intersections of the curve are transverse double points, there is a combinatorial formula for the rotation number obtained by algebraically counting the self-intersections of the curve. In this paper, I generalize these results to the case of curves on oriented surfaces with a complete euclidean or hyperbolic metric.