Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 283Thus we can conclude that, if the space is complete, then the diametermust be ≤ π because in this case any two points are joined by a segment,which cannot minimize if it has length > π. With some minor modificationsone can now conclude that any complete Riemannian manifold (M, g) withsec ≥ k 2 > 0 must satisfy diam(M, g) ≤ π · k −1 . In particular, M must becompact. Since the universal covering of M satisfies the same curvaturehypothesis, the conclusion must also hold for this space; hence M musthave compact universal covering space and finite fundamental group.In odd dimensions all spaces of constant positive curvature must beorientable, as orientation reversing orthogonal transformation on odd–dimensional spheres have fixed points. This can now be generalized tomanifolds of varying positive curvature. Synge did it in the following way:Suppose M is not simply–connected (or not orientable), and use this tofind a shortest closed geodesic in a free homotopy class of curves (that reversesorientation). Now consider parallel translation around this geodesic.As the tangent field to the geodesic is itself a parallel field, we see thatparallel translation preserves the orthogonal complement to the geodesic.This complement is now odd dimensional (even dimensional), and by assumptionparallel translation preserves (reverses) the orientation; thus itmust have a fixed point. In other words, there must exist a closed parallelfield X perpendicular to the closed geodesic γ. We can now use the abovesecond variation formula¨L(0) =∫ l∫ l{|Ẋ|2 − |X| 2 sec ( ˙γ, X)}dt + g ( ˙γ, A)| l 0 = − |X| 2 sec ( ˙γ, X) dt.00Here the boundary term drops out because the variation closes up at theendpoints, and Ẋ = 0 since we used a parallel field. In case the sectionalcurvature is always positive we then see that the above quantity is negative.But this means that the closed geodesic has nearby closed curves which areshorter. However, this is in contradiction with the fact that the geodesicwas constructed as a length minimizing curve in a free homotopy class.In 1941 Myers generalized the diameter bound to the situation whereone only has a lower bound for the Ricci curvature. The idea is thatRic( ˙γ, ˙γ) = ∑ n−1i=1 sec (E i, ˙γ) for any set of vector–fields E i along γ suchthat ˙γ, E 1 , . . ., E n−1 forms an orthonormal frame. Now assume that thefields are parallel and consider the n − 1 variations coming from the variationalvector–fields sin ( )t · πl Ei . Adding up the contributions from the