How to calculate Riemannian Metric by distance function?

  • #1
How to calculate Riemannian Metric by distance function??

Dear Folks:
Here is the problem: in |z|<1, we difine a distance between any two points z1 and z2 by d(z1 , z2) = ln((z1 - b)(z2 - a)/((z1 - a)(z2 - b))) ,where a is the intersection of line z1z2 and the circle which is nearer to z1, and b is the other intersection. How to compute the Remannian Metric ds2.
My classmates calculated it indirectly for he knew the background of the problem?? But is there a direct way??
Many thanks!!
 

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