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• W2035657091 abstract "One class of problems in riemannian geometry is to determine in what way the conjugate locus structure restricts the topological, differentiable, or metric structure of the manifold. Other than applications of the Morse theory, only a few results of this type are known. For example, if there is one point in a simply connected riemannian manifold possessing no conjugate points, then the manifold is diffeomorphic with euclidean space [12]. E. Hopf [8] has shown that a riemannian structure on the torus T2, with no conjugate pairs along any geodesic, must be flat. The only known results where one allows the existence of conjugate points are in the cases where the first conjugate locus is suitably spherical. For a 2-dimensional compact simply connected riemannian manifold M, Green [7] has shown that, if the first conjugate locus in the tangent space at each point is a sphere of the same radius, then M is isometric to the standard sphere S2. The analogous problem in higher dimensions would be for a compact simply connected riemannian manifold M, for which the first conjugate locus in each Mm is a sphere of the same radius consisting of conjugate points of maximal order. Differentiably such a manifold is a sphere, but the isometry conjecture is unsolved. In this higher dimensional case, assume the first conjugate locus is a sphere of the same radius at each point, but consisting of conjugate points of some constant order less than maximal. In this case one might conjecture the manifolds are isometric to one of the compact irreducible riemannian symmetric spaces of rank 1. However, here the topological question is still open. Allamigeon [3] has shown that, in this case, the manifold has the correct integral cohomology ring. In a somewhat different vein, Klingenberg [11] has shown that, if the conjugate locus structure in the tangent space at one point of a simply connected riemannian manifold is similar to that for one of the compact irreducible rank 1 riemannian symmetric spaces, then the manifold has the integral cohomology ring of the space after which it is modeled. In this paper, we draw some differentiable and topological conclusions from the structure of the conjugate locus at a single point. At first we do not assume the conjugate locus is spherical. If there is one point m in a compact simply connected riemannian manifold M for which each point of the" @default.
• W2035657091 created "2016-06-24" @default.
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• W2035657091 date "1967-07-01" @default.
• W2035657091 modified "2023-09-26" @default.
• W2035657091 title "Conjugate Loci of Constant Order" @default.
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• W2035657091 doi "https://doi.org/10.2307/1970366" @default.
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