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  "body": "\n## Statement\n\nLet $(M, g)$ be a compact connected [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ with positive [[sectional curvature|1X42JAA349]].\nThen:\n\n1. If $n$ is even and $M$ is [[orientable|913WEQERBE]], then $M$ is simply connected.\n2. If $n$ is odd, then $M$ is orientable.\n\n## Remarks\n\nThe 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.\nCompactness gives a closed [[geodesic|NSYMQMHPDN]] $\\gamma$ of minimal length in this class.\nA linear-algebra argument shows that [[parallel transport|TVAGN3D6W5]] once around $\\gamma$ fixes a unit vector normal to $\\gamma$.\nPushing $\\gamma$ in the direction of this parallel field decreases length to second order, by the [[second variation formula|ZB91TH6FKV]] and positive curvature, which contradicts minimality.\n\nExample: the real projective plane $\\R P^2$ with the round metric has positive curvature, even dimension and fundamental group of order $2$.\nIt is not orientable, so this does not contradict part 1.\nIts orientability hypothesis is therefore needed.\nIn 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.\n\n## Explanations\n\n### L1 human\n\nPositive curvature makes loops want to shrink.\nSynge's theorem says that for compact spaces of even dimension, every loop can in fact be shrunk to a point, once one excludes the twisted, non-orientable cases.\nIn odd dimensions, it says the space cannot be twisted like a Möbius band.\n\n### L2 human\n\nSynge's theorem is one of the few known topological obstructions to positive sectional curvature.\nFor 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.\nThe 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.\n",
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