Are $\mathbb{R}P^3$ and $T^1S^2$ isometric?

Question

It is well-known that 3-dimensional real projective space $\mathbb{R}P^3$ is diffeomorphic to $T^1S^2$, the unit tangent bundle of the 2-sphere. However, I could not find any reference to whether these spaces are also isometric as Riemannian manifolds, where $\mathbb{R}P^3$ is given its canonical metric of constant curvature 1 (coming from its double cover by $S^3$), and $T^1 S^2$ is given the Sasaki metric. I feel like this should be true, but I am not sure. Can anyone help? Any reference would be appreciated.