This is an exercise from a Riemannian textbook. It claims that if a riemannian manifolld is diffeomorphism to $S^2$, then for every $x\in M$, the intersection of its conjugate locus and cut locus is not empty, that is, $K(x)\cap C(x)\neq\varnothing$. The hint says that one way to solve this is to use Jordan curve theorem which states that a curve $\gamma$ on $S^2$ homeomorphism to $S^1$ splits the sphere into exactly two connected open sets, with $\gamma$ as their common boundary.
I have made some progress, but didn't fully solve it: If the cut locus $C(x)$ is a regular curve, then assume $K(x)\cap C(x)=\varnothing$ I can always find a section of $\exp_x(C(x))$ that is homeomorphism to $S^1$, then ues Jordan curve theorem to derive a contradiction. But I don't konw if this is true.
Any help will be appreciated.