- #1
Pencilvester
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- 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.)
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.)