Index parity of closed geodesics and rigidity of Hopf fibrations
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The diameter rigidity theorem of Gromoll and Grove  states that a Riemannian manifold with sectional curvature ≥ 1 and diameter ≥ π/2 is either homeomorphic to a sphere, locally isometric to a rank one symmetric space, or it has the cohomology ring of the Cayley plane Caℙ. The reason that they were only able to recognize the cohomology ring of Caℙ is due to an exceptional case in another theorem [Gromoll and Grove, 1988]: A Riemannian submersion σ:?m→Bb with connected fibers that is defined on the Euclidean sphere ?m is metrically congruent to a Hopf fibration unless possibly (m,b)=(15,8). We will rule out the exceptional cases in both theorems. Our argument relies on a rather unusal application of Morse theory. For that purpose we give a general criterion which allows to decide whether the Morse index of a closed geodesic is even or odd.
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