Synge's theorem

Remarks

The idea of the proof: if the conclusion fails, there is a nontrivial free homotopy class of loops along which parallel transport has the required orientation behaviour. Compactness gives a closed geodesic $\gamma$ of minimal length in this class. A linear-algebra argument shows that parallel transport once around $\gamma$ fixes a unit vector normal to $\gamma$. Pushing $\gamma$ in the direction of this parallel field decreases length to second order, by the second variation formula and positive curvature, which contradicts minimality.

Example: the real projective plane $\R P^2$ with the round metric has positive curvature, even dimension and fundamental group of order $2$. It is not orientable, so this does not contradict part 1. Its orientability hypothesis is therefore needed. In odd dimensions, $\R P^3$ is orientable, as part 2 says, but it is not simply connected, so part 1 has no odd-dimensional analogue.

Going deeper

For a student

Synge's theorem is one of the few known topological obstructions to positive sectional curvature. For instance, it shows that $\R P^2 \times \R P^2$, whose orientable double cover is not simply connected, has no metric of positive curvature. The proof is the model for later arguments that combine second variation with the holonomy of a closed geodesic, such as Weinstein's fixed point theorem.