Thread: Synge's Theorem
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lethe
#8
Nov19-03, 04:36 PM
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P: 657
from Frankel:
Synge's Theorem

Let [tex]M^{2n}[/tex] be an even-dimensional, 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, 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