7
$\begingroup$

I know that in $1923$ H. Kneser showed that a continuous flow in a Klein bottle without singular points has a periodic trajectory. The original article is this, but does anyone know another old or new proof of this result? (So far I have found the article by Kneser, the article by Nelson G. Markley and some works that use the results of these articles, I am looking for some other idea focused on proving the result mentioned at the beginning) I would really like to read this result, I tried to do it from your original article but the language is too complicated for me. I searched on the internet but found almost nothing about the proof. I asked here but I didn't find any answer even with bounty. I hope to have some help it would help me a lot.

$\endgroup$
2
  • $\begingroup$ I would imagine it to be a contradiction argument. If there is no closed orbit, you argue that the forward-time flow of a point must be clustering in a thin Moebius band, and the same kind of argument as in Poincare-Bendixson tells you the centre of that Moebius band is a closed orbit. You could probably avoid repeating the Poincare-Bendixson argument by covering the Moebius band with an annulus, which is planar. $\endgroup$ May 26 at 20:20
  • $\begingroup$ @RyanBudney Thank you very much for the idea, it is interesting I will try it, if you have time you give it more form as an answer and in case I manage to finish it I will write it too. $\endgroup$
    – Zaragosa
    May 26 at 21:28

2 Answers 2

7
+50
$\begingroup$

The answer below was given to the question as asked originally.


For a more modern, english language proof of Kneser's result, see The Poincaré-Bendixson Theorem for the Klein Bottle, by Nelson G. Markley:

In 1923 Kneser showed that a continuous flow on the Klein bottle without fixed points has a periodic orbit. The purpose of this paper is to prove a stronger version of this theorem. It states that the Klein bottle cannot support a continuous flow with recurrent points which are not periodic.

$\endgroup$
5
  • $\begingroup$ Yes, I did find that article (very interesting, it even puts it as a corollary 5.2) and I also found a thesis based on that article, but I was referring to more current articles (for example after 2000). What I am looking for are new ideas regarding the traditional proofs of this theorem, maybe and just maybe the proof of this theorem has been refined with new tools and in that case I would love to know them. $\endgroup$
    – Zaragosa
    May 26 at 21:06
  • 3
    $\begingroup$ hmm, I'm confused: your question does ask for "another old or new proof of this result" doesn't it? $\endgroup$ May 26 at 21:11
  • $\begingroup$ Sorry, it was my rush to write. Now I'm going to correct that a bit. $\endgroup$
    – Zaragosa
    May 26 at 21:24
  • 3
    $\begingroup$ @Zaragosa Please note that changing a bit a question after an answer is given is considered a bit unfair/ungracious. Also, not giving a complete information on the problem (e.g. a reference) may result into a waste of time for the answerer. $\endgroup$ May 26 at 21:43
  • 3
    $\begingroup$ I know I'm sorry. But I also know that the above answer will get a good upvote, so I don't think it feels a bit unfair. I hope someone can really help me with my question. Greetings. $\endgroup$
    – Zaragosa
    May 26 at 22:45
5
$\begingroup$

It sounds like you are looking for a textbook. I have read bits of Introduction to the geometry of foliations: Part A by Hector and Hirsch. It is well-written, with pictures! They give Kneser's theorem on pages 62-65, after developing the necessary theory.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.