Metric tensor derived from a geodesic

  • #1
56
1
Let we have a 2D manifold. We choose a coordinate system where we can construct all geodesics through any point. Is it enough to derive a metric from geodesic equation? Or do we need to define something else for the manifold?
 

Answers and Replies

  • #2
334
61
Geodesic curves are generated by connection; you can have a connection but do not have metric. Restoring metric compatible with symmetric connection is not a simple problem.
 
  • #3
56
1
Geodesic curves are generated by connection; you can have a connection but do not have metric. Restoring metric compatible with symmetric connection is not a simple problem.
Thanks for reply! Could you tell me what is the complexity? As I see I need to solve partial differential equations. That's it?
 
  • #4
334
61
Yes, you must solve the PDE to restore metric by known connection; these PDE are not obliged to have a suitable solution as well as they are not obliged to have a unique solution
 
  • #5
56
1
There is a question, though. Let we have a curve equation not w.r.t. the affine parameter but a 2D chart. Can I use the geodesic equation with this curve equation to find connection coefficients? If so, will solving further PDE give me a correct metric in this coordinate system?
 
  • #6
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
16,829
6,650
There is a question, though. Let we have a curve equation not w.r.t. the affine parameter but a 2D chart. Can I use the geodesic equation with this curve equation to find connection coefficients? If so, will solving further PDE give me a correct metric in this coordinate system?
In general, no. The geodesic equation does not depend on the anti-symmetric part of the connection coefficients. You can only find the connection coefficients if you also specify the anti-symmetric part (i.e., you need to specify the torsion).
 
  • #7
56
1
In general, no. The geodesic equation does not depend on the anti-symmetric part of the connection coefficients. You can only find the connection coefficients if you also specify the anti-symmetric part (i.e., you need to specify the torsion).
Can my chosen curve/curves family give me information about torsion? Should it have second derivative to have a torsion?
 
  • #8
56
1
Is there really no answer?
 
  • #10
56
1

Related Threads on Metric tensor derived from a geodesic

  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
11
Views
2K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
1
Views
7K
  • Last Post
Replies
1
Views
2K
Replies
3
Views
12K
  • Last Post
Replies
3
Views
3K
Replies
11
Views
11K
  • Last Post
Replies
2
Views
3K
Top