Are any two geodesics the same after applying an isometry?

In summary, the conversation discusses the possibility of using an isometry to push a known geodesic forward in a Riemannian manifold to obtain all geodesics. This is generally true for homogeneous and isotropic manifolds, but may not hold for others. Examples of manifolds with geodesics of different lengths are provided to support this argument. However, it is noted that this may not hold for manifolds where some geodesics are closed and others are not.
  • #1
nonequilibrium
1,439
2
Hello, I was wondering the following.

Suppose you start with a Riemannian manifold M. Say you know one geodesic.
Pushing this geodesic forward through an isometry M -> M gives again a geodesic.
Can this procedure give you all geodesics?

Thinking of the plane or the sphere it seems obviously true.
 
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  • #2
This is certainly true for homogeneous and isotropic manifolds. If these conditions fail, then this property might not be recoverable.

Edit: Find a manifold with geodesics of different lengths and since the length of a geodesic should be invariant under isometries this provides an ample number of examples. Do let me know if some part of this argument does not check out though since I have not thought it through very thoroughly.
 
  • #3
Sure on a manifold where some geodesics are closed(?) and some are not this can't be true
 

1. Are all geodesics preserved after applying an isometry?

No, not all geodesics are preserved after applying an isometry. Only geodesics that have the same starting and ending points will remain the same.

2. Can an isometry change the shape of a geodesic?

Yes, an isometry can change the shape of a geodesic. This is because an isometry is a transformation that preserves distances and angles, so it can alter the curvature of a geodesic.

3. Do all isometries have the same effect on geodesics?

No, not all isometries have the same effect on geodesics. Different types of isometries, such as translations, reflections, and rotations, will have different effects on the shape and orientation of a geodesic.

4. Can an isometry create a new geodesic that did not exist before?

No, an isometry cannot create a new geodesic that did not exist before. Isometries only transform existing geodesics, they cannot create new ones.

5. How does an isometry affect the length of a geodesic?

An isometry does not change the length of a geodesic. As mentioned before, isometries preserve distances, so the length of a geodesic will remain the same after applying an isometry.

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