I'm trying to prove/disprove the following proposition:
Let $M^2$ be a complete, connected, two-dimensional manifold with positive, constant curvature. If $f:M^2\to\mathbb{R}^3$ an immersion such that $f(M)$ is contained in a sphere $S$, then $f(M)=S$.
One idea is to use Hilbert's theorem on the rigidity of the sphere, but in that case we would need $M$ to be compact, which I don't know is necessarily the case. Another idea is to prove $f(M)$ is an open set of $S$ which, by completeness, would mean $f(M)=S$ (since complete manifolds are not extendable), but I couldn't prove it either.
Any suggestions?