On the variety of manifolds without conjugate points
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- by Robert Gulliver PDF
- Trans. Amer. Math. Soc. 210 (1975), 185-201 Request permission
Abstract:
The longest geodesic segment in a convex ball of a riemannian manifold, where the convexity is ensured by an upper bound on sectional curvatures, is the diameter. This and related results are demonstrated and applied to show that there exist manifolds with sectional curvatures of both signs but with-out conjugate points.References
- Richard L. Bishop and Richard J. Crittenden, Geometry of manifolds, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. MR 0169148
- R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49. MR 251664, DOI 10.1090/S0002-9947-1969-0251664-4
- Armand Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 111–122. MR 146301, DOI 10.1016/0040-9383(63)90026-0
- Patrick Eberlein, Geodesic flow in certain manifolds without conjugate points, Trans. Amer. Math. Soc. 167 (1972), 151–170. MR 295387, DOI 10.1090/S0002-9947-1972-0295387-4
- Patrick Eberlein, When is a geodesic flow of Anosov type? I,II, J. Differential Geometry 8 (1973), 437–463; ibid. 8 (1973), 565–577. MR 380891
- Eberhard Hopf, Statistik der Lösungen geodätischer Probleme vom unstabilen Typus. II, Math. Ann. 117 (1940), 590–608 (German). MR 2722, DOI 10.1007/BF01450032
- Wilhelm Klingenberg, Riemannian manifolds with geodesic flow of Anosov type, Ann. of Math. (2) 99 (1974), 1–13. MR 377980, DOI 10.2307/1971011
- Richard K. Miller and George R. Sell, Existence, uniqueness and continuity of solutions of integral equations, Ann. Mat. Pura Appl. (4) 80 (1968), 135–152. MR 247395, DOI 10.1007/BF02413625
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 210 (1975), 185-201
- MSC: Primary 53C20; Secondary 58F15
- DOI: https://doi.org/10.1090/S0002-9947-1975-0383294-0
- MathSciNet review: 0383294