In mathematics, specifically Riemannian geometry, Synge's theorem is a classical result relating the curvature of a Riemannian manifold to its topology. It is named for John Lighton Synge, who proved it in 1936.Theorem and sketch of proof
Let M be a closed Riemannian manifold with positive sectional curvature. The theorem asserts:
If M is even-dimensional and orientable, then M is simply connected.
If M is odd-dimensional, then it is orientable.
In particular, a closed manifold of even dimension can support a positively curved Riemannian metric only if its fundamental group has one or two elements.
The proof of Synge's theorem can be summarized as follows. Given a geodesic S1 → M with an orthogonal and parallel vector field along the geodesic (i.e. a parallel section of the normal bundle to the geodesic), then Synge's earlier computation of the second variation formula for arclength shows immediately that the geode
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