Thread: Synge's Theorem View Single Post
 P: 657 from Frankel: Synge's Theorem Let $$M^{2n}$$ be an even-dimensional, orientable manifold with positive sectional curvatures, $$K(\mathbf{X}\wedge\mathbf{Y}) > 0$$. Then any closed geodesic is unstable, that is, can be shortened by a variation. Corollary: A compact, orientable, even-dimensional 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