Index parity of closed geodesics and rigidity of Hopf fibrations

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The Relation of the Morse Index of Closed Geodesics with the Maslov–type Index of Symplectic Paths (2005, background): “From the above theorem, we know that the parity of the Morse index ind(c) of a closed geodesic c on oriented manifold depends only on the associated Poincar´ em apP . Under the conditions p− =0a ndp+ = 0 the formula (1.5) was proved in [ 15 ] by Wilking.”
Clifford algebras and new singular Riemannian foliations in spheres (2014, methodology): “By the work of Grove and Gromoll [5] and Wilking [13], the submersion πC must be in fact a Hopf fibration.”
Nonnegatively and Positively curved manifolds (2007, methodology): “This case was ruled out much later by the author, see [Wilking, 2001].”