from Frankel:
Synge's Theorem
Let [tex]M^{2n}[/tex] be an evendimensional, orientable manifold with positive sectional curvatures, [tex]K(\mathbf{X}\wedge\mathbf{Y}) > 0[/tex]. Then any closed geodesic is unstable, that is, can be shortened by a variation.
Corollary:
A compact, orientable, evendimensional manifold with positive sectional curvatures is simply connected.
to remind you, sectional curvature is the Gaussian curvature determined by two tangent vectors X and Y
