Is there exist a shortest closed curve in the homology class of a simple closed curve

  • Thread starter hyperttt
  • Start date
  • #1
3
0
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?
 

Answers and Replies

  • #2
CompuChip
Science Advisor
Homework Helper
4,302
47


"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.
 
  • #3
3
0


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.
 
  • #4
3
0


can anyone help?
 
  • #5
709
0


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.
 
  • #6
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
19


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?
 

Related Threads on Is there exist a shortest closed curve in the homology class of a simple closed curve

  • Last Post
Replies
0
Views
3K
Replies
0
Views
2K
Replies
4
Views
1K
Replies
7
Views
2K
Replies
2
Views
4K
Replies
16
Views
4K
  • Last Post
Replies
8
Views
3K
Replies
15
Views
4K
Top