Seeking the Shortest Closed Curve in a Homology Class

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Discussion Overview

The discussion revolves around the question of whether there exists a shortest closed curve within the homology class of a given simple closed curve on a compact Riemannian surface. Participants explore the relationship between homology and homotopy classes and the implications for the existence of geodesics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about the existence of a shortest closed curve in the homology class of a simple closed curve on a compact Riemannian surface.
  • Another participant references a theorem stating that every homotopy class of closed curves contains a shortest geodesic, suggesting a potential avenue for exploration.
  • A different participant clarifies that the original question pertains specifically to homology classes, not homotopy classes, indicating a distinction that needs to be addressed.
  • One suggestion involves covering the curve with open balls and deforming it to create a piecewise smooth geodesic, which may locally minimize length but raises questions about global minimization.
  • This participant also proposes that starting with a length-minimizing curve in the homotopy class could lead to discovering an even shorter curve through geodesic approximation.
  • Another participant questions whether knowledge of the homotopy class can provide insights into the homology class, suggesting a potential connection between the two concepts.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between homology and homotopy classes, with no consensus reached on whether a shortest closed curve exists in the homology class. The discussion remains unresolved.

Contextual Notes

The discussion highlights the complexity of the relationship between homology and homotopy classes, as well as the challenges in proving the existence of a shortest closed curve. Specific assumptions and definitions related to Riemannian metrics and curve properties are not fully explored.

hyperttt
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I am seeking the answer to the following question: Given a simple closed curve on a compact Riemannian surface(a compact surface with a Riemannian metric), whether there exists, in the homology class of this simple closed curve, a (single) closed curve which has the shortest length measured with respect to the given Riemannian metric? In other words, whether there exists a shortest path which is composed by a single closed curve in the homology class of a given simple closed curve? I think this might be interesting, can you help me?
 
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"Let M be a compact Riemannian manifold. [...] Likewise, every homotopy class of closed curves in M contains a curve which is shortest and geodesic."

That's theorem 1.5.1 from this book. Also the proof is given there (and luckily for you, within the previewable pages). Actually I suspect (I didn't look at it nor try it) that it's not very hard so maybe you want to try it yourself first.
 


Well, The question I posted is still unsolved.
first, thank CompuChip. But what I am asking is the existence of the geodesic in the "homology" class of a given simple closed curve, rather than the existence of the geodesic in the "homotopy" class of a simple closed curve. Can you help? Thanks a lot for your prospective help.
 


can anyone help?
 


Not sure of the proof but I would try covering the curve with open balls then deforming it to a geodesic segment in each ball. This would give you a piece wise smooth geodesic in the same homotopy class. It will locally minimize length but maybe not globally. You can then try to show that by smoothing out the kinks you get a geodesic that is even shorter.

This argument should work but as is begs the question because there still may be even a shorter curve that is not a geodesic.

So maybe you start out with a length minimizing curve in the same homotopy class. If it is not a geodesic apply this procedure of geodesic approximation to get even a shorter curve.

Details to be worked out.
 


hyperttt said:
Well, The question I posted is still unsolved.
first, thank CompuChip. But what I am asking is the existence of the geodesic in the "homology" class of a given simple closed curve, rather than the existence of the geodesic in the "homotopy" class of a simple closed curve. Can you help? Thanks a lot for your prospective help.
Surely knowing something about the homotopy class tells you something about the homology class?
 

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