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. Let *M* be a compact 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.