Are there Riemannian metrics other than the standard metric induced from the euclidean space on $S^2$ such that the sectional curvature is equal to 1 everywhere? Or is this the unique Riemannian metric of constant curvature (up to scaling)?
[Math] Constant curvature metrics on the sphere
differential-geometryriemannian-geometryspheressurfaces
Best Answer
The standard metric is the only metric of constant sectional curvature $1$ on $S^2$.
More generally, for any positive integer $n$ and any $\lambda \in \mathbb{R}$, there is a unique simply connected Riemannian manifold of dimension $n$ and constant scalar curvature $\lambda$ (which therefore looks like a scaling of either $S^n, \mathbb{R}^n$, or hyperbolic space $H^n$ depending on the sign of $\lambda$).
This is a standard fact in Riemannian geometry, not entirely trivial to prove but not terribly complicated either. Any introductory Riemannian geometry text should have a proof; for instance, it's given as Theorem 4.1 in Chapter 8 of do Carmo and Corollary 10 in section 10.2 of Petersen.