Synge's theorem

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.

References

do Carmo, Manfredo Perdigão (1992), Riemannian geometry, Mathematics: Theory & Applications, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-3490-2, MR 1138207
Synge, John Lighton (1936), "On the connectivity of spaces of positive curvature", Quarterly Journal of Mathematics (Oxford Series), 7: 316–320, doi:10.1093/qmath/os-7.1.316.

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