Let $M$ be a compact 2-manifold with positive sectional curvature. Show that any two closed geodesics of $M$ must intersect.
The familiar example is $S^n$, where the closed geodesics are great circles, where any 2 geodesics must intersect. I'm wondering if there is a technic that transforms any 2-manifold with positive sectional curvature continuously into $S^n$, while in the mean time preserving geodesics? Specifically, let $c$ be the closed geodesic, $M_t$ be the transformation of $M$ with parameter $t\in [0, \infty)$, and let $c_t$ be the image of $c$ in $M_t$. Is is possible to find $M_t$ such that $c_t$ are geodesics, and $\lim_{t\rightarrow \infty} M_t \approx S^2$? If such transformation exists then the problem is trivial.
