Synge's theorem
/ SING /
Positive curvature does more than bound a manifold's size — depending on whether the dimension is even or odd, and whether the space is twisted or not, it also forces clean answers about orientability and connectedness. Synge's theorem is a sharp parity statement of this kind. It uses the same second-variation idea as Bonnet-Myers, but applied to closed geodesics and twisted by orientation, to conclude that a compact positively curved manifold cannot be both even-dimensional and non-simply-connected if orientable, and is connected-by-orientation in odd dimensions.
Precisely, two clean cases. If M is a compact, orientable, even-dimensional Riemannian manifold with sectional curvature K > 0, then M is simply connected (pi_1(M) is trivial). If M is compact and odd-dimensional with K > 0, then M is orientable. The proof is a gem of the second-variation method. Suppose M is even, orientable, K > 0, but not simply connected; take a shortest closed geodesic gamma in a nontrivial free homotopy class. Parallel transport once around gamma is an orientation-preserving isometry of the normal space (an even-dimensional rotation), which must fix some normal vector; varying gamma in that direction is a second variation whose index form is strictly negative because K > 0, so you can shorten gamma while staying in its homotopy class — contradicting that it was shortest. Hence no such class exists, and pi_1 is trivial.
Synge's theorem is the reason RP^2 (even-dimensional but non-orientable) and the flat Klein bottle behave the way they do, and it cleanly separates what positive curvature can and cannot force. Honest caveats. First, every hypothesis is necessary: RP^{2} = S^2/{±1} has K > 0 and is even-dimensional but non-orientable, with pi_1 = Z/2 — it fails simple connectedness precisely because it is not orientable, so orientability cannot be dropped. Second, in odd dimensions the simple-connectivity conclusion is false (RP^3 has K > 0 and pi_1 = Z/2), which is why the odd case concludes only orientability. Third, K > 0 (sectional) is required — a Ricci bound is not enough for Synge, unlike Bonnet-Myers.
The even-dimensional case in action: any compact orientable surface (n = 2) with K > 0 must be simply connected, hence by classification it is the 2-sphere — there is no positively curved orientable torus or genus-2 surface (consistent with Gauss-Bonnet, which forces total curvature 2pi chi > 0). The necessity of orientability shows in RP^2: K > 0, even-dimensional, but non-orientable, and indeed pi_1 = Z/2 is nontrivial. The odd case: RP^3 = S^3/{±1} carries K > 0 and Synge correctly concludes only that it is orientable (which it is), not that it is simply connected (it is not).
Even + orientable + K>0 forces simply connected (the 2-sphere); RP^2 shows orientability cannot be dropped.
Weinstein's theorem is the companion fixed-point result: an isometry of a compact even-dimensional orientable positively curved manifold that preserves orientation must have a fixed point — the same parity-and-curvature mechanism that drives Synge.