The n-sphere and the n-dimensional projective plane are locally isometric

[InbredDummy]

Homework Statement

Prove that S^n and P^n(R) are locally isometric.

Homework Equations

So I proved that the antipodal mapping A(p) = -p is an isometry. And I proved that the projection map from S^n to P^n(R) is a local diffeomorphism. I'm just not sure what the Riemannian metric on P^n(R) would be.

The Attempt at a Solution

I think it should be easy once I have the previous two facts, but can anyone help me just put it all together?

Thanks guys

 

[InbredDummy]

EDIT: I got it! I finally got it!

 

[Dick]

EDIT: I got it! I finally got it!

Good! I was going to suggest the metric in the projective plane must be the one that it gets from the sphere, since they didn't give it to you. That would make it trivial. Is it that simple? Or am I just being simple minded?