Suppose $S$ is a compact surface where in every point $P$ the principal curvatures $C_{1}(P)$ and $C_{2}(P)$ fulfill $C_{1}(P)=C_{2}(P)$.
Is $S$ necessarily a sphere? My idea is that the compacity hypothesis implies that the image of the Gauss curvature is bounded, and that the equality of principal curvatures implies the isotropy of its distribution. Is that enough to conclude?
Edit: if the map $P\mapsto K(P)$ is analytic, Liouville theorem may allow us to deduce it is constant on $S$.