How to calculate Riemannian Metric by distance function?

[Fangyang Tian]

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!

 

[pat_connell]

The direct way to calculate the Riemannian metric is by using the distance function you have defined. In this case, the metric is defined as: ds2 = (dz1/dz2)^2 * d(z1,z2)^2where dz1 and dz2 are the partial derivatives of z1 and z2 with respect to the other coordinates. For example, if z1 and z2 are two of the coordinates in a 3D space, then dz1/dz2 would be the partial derivative of z1 with respect to z2.Once you have calculated the partial derivatives, you can use them to calculate the metric:ds2 = (dz1/dz2)^2 * d(z1,z2)^2Finally, you can use the metric to calculate the geodesic distance between any two points in the space.