# Thread: non-euclidean metric on 2-sphere??

1. ## non-euclidean metric on 2-sphere??

Hi,
In the context of calculating the metrics, the 2-sphere is often equipped with the Euclidean metric (identity matrix),
is it always correct?
can we equip the 2-sphere with a NON-Euclidean metric?

Thank you.

2. ## Re: non-euclidean metric on 2-sphere??

I think you have answered your own question, by stating that the matrix should not have to be the identity. Any positive definite matrix will do.

Other than that, there is another metric you can define on the 2-sphere that does not depend on ambient space. You can define the distance between
any two points as the length of the shortest geodesic with these as endpoints.

3. ## Re: non-euclidean metric on 2-sphere??

Thank you for your helpful response , I found that the shortest geodesic between two points X and Y on the 2-sphere is the distance d=arcos(X'.Y),

I dont know how find the matrix form of this distance !! because I wiil need it for computing the pullback metric..;