Concept# Synge's theorem

Summary

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.

Official source

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