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Courbure presque négative en dimension 3. (Almost negative curvature in dimension 3). (French) Zbl 0642.53047
The following striking theorem is proved: Any closed orientable 3- manifold, M, admits a riemannian metric with curve(M)\(\leq 1\), diam(M)\(\leq \epsilon\), and Vol \(M\leq \epsilon\), for any \(\epsilon >0\). The first construction of this kind was given by Gromov in the case \(M=S^ 3\) [cf. P. Buser and D. Gromoll, Gromov’s examples of almost negatively curved metrics on \(S^ 3\) (to appear)]. In these examples the lower bound for the curvature goes to -\(\infty\). This is indeed necessary as was shown recently by Fukaya and Yamaguchi.
Reviewer: K.Grove

MSC:
53C20 Global Riemannian geometry, including pinching
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References:
[1] C. Bavard : thèse de 3ème cycle . Orsay (1984).
[2] A. Besse : Manifolds all of whose geodesics are closed . Springer (1978). · Zbl 0387.53010
[3] P. Buser et D. Gromoll : Gromov’s examples of almost negatively curved metrics on S3, notes .
[4] G. Darboux : Leçons sur la Théorie générale des Surfaces , Vol. 3, Chelsea (réédition) (1972).
[5] H.B. Lawson : Foliations . Bull. A.M.S. 80 (1974) 369-418. · Zbl 0293.57014 · doi:10.1090/S0002-9904-1974-13432-4
[6] B O’Neill : The fundamental equations of a submersion . Michigan Math. J. 13 (1966) 459-469. · Zbl 0145.18602 · doi:10.1307/mmj/1028999604
[7] D. Rolfsen : Knots and links . Publish or Perish (1976). · Zbl 0339.55004
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