Seeking the Shortest Closed Curve in a Homology Class

In summary: 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.
  • #1
hyperttt
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?
 
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  • #2


"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


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


can anyone help?
 
  • #5


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


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?
 

1. What is the significance of seeking the shortest closed curve in a homology class?

Seeking the shortest closed curve in a homology class is important in topology and geometry because it helps us understand the properties of surfaces and spaces. It also has practical applications in fields such as computer graphics and robotics.

2. How is the shortest closed curve in a homology class determined?

The shortest closed curve in a homology class can be determined by using a mathematical concept called the geodesic. This is the shortest path between two points on a curved surface or space, and it can be generalized to find the shortest closed curve in a homology class.

3. Can the shortest closed curve in a homology class be found for any surface or space?

It is not always possible to find the shortest closed curve in a homology class for every surface or space. In some cases, it may not exist, and in others, it may be too complex to calculate. However, for simpler surfaces and spaces, it is possible to find the shortest closed curve using mathematical algorithms.

4. How does the shortest closed curve in a homology class relate to the concept of curvature?

The shortest closed curve in a homology class is closely related to the concept of curvature, which measures the amount of bending or deviation from a straight line on a surface or space. The geodesic, which is used to find the shortest closed curve, is also a measure of curvature.

5. What are some real-world applications of finding the shortest closed curve in a homology class?

As mentioned before, finding the shortest closed curve in a homology class has practical applications in fields such as computer graphics and robotics. It can also be useful in navigation and map-making, as well as in understanding the properties of materials and structures in engineering and physics.

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