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I need to show a particular map f:M-->N is an isometry (globally). M,N are riemannian manifolds, p is a point on M.

That is, I need to show:

<u,v>_p = <df_p(u),df_p(v)>_f(p) (for all p and for all u,v in T_p(M)?)

I think my problem is that I don't understand what this statement really means. Are u,v elements of the tangent space T_p(M)? Then doesn't that mean I should identify them with partial derivatives? But this doesn't make sense unless I apply them to functions...

I see that in euclidean space we have:

<e_i,e_j>=g_ij where g_ij is the identity matrix of the appropriate size

I don't see how that statement connects to the other one. What is "f" in this context?

Thanks for you help...

That is, I need to show:

<u,v>_p = <df_p(u),df_p(v)>_f(p) (for all p and for all u,v in T_p(M)?)

I think my problem is that I don't understand what this statement really means. Are u,v elements of the tangent space T_p(M)? Then doesn't that mean I should identify them with partial derivatives? But this doesn't make sense unless I apply them to functions...

I see that in euclidean space we have:

<e_i,e_j>=g_ij where g_ij is the identity matrix of the appropriate size

I don't see how that statement connects to the other one. What is "f" in this context?

Thanks for you help...

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