Consider the Reimannian metric on $S^{n}$ induced by the Euclidean metric on $\mathbb{R}^{n}$. Use it to define a Riemannian metric on $P^{n}$ and calculate its diameter.
My attempt: I have used the fact that $\mathbb{Z}_{2}$ acts freely, smoothly and properly on $S^{n}$ and hence, we get a Riemannian covering map $\pi: S^{n} \rightarrow S^{n}/ \mathbb{Z}_{2}$. I want to calculate the diameter of $P_{n}$. I was thinking of using geodesics for this purpose. I think we can choose any one point of $P^{n}$ and calculate the maximum distance of points from our fixed point and use that $\pi$ is a local isometry. Am I thinking correctly? I haven't studied curvature yet so is there a way to calculate the diameter without using curvature?