Geodesic
Conjugate Radius and Sphere Theorem
Canadian mathematical bulletin, Tome 42 (1999) no. 2, pp. 214-220

Voir la notice de l'article provenant de la source Cambridge University Press

Bessa [Be] proved that for given $n$ and ${{i}_{0}}$ , there exists an $\varepsilon (\text{n,}\,{{i}_{0}})\,>\,0$ depending on $n$ , ${{i}_{0}}$ such that if $M$ admits a metric $g$ satisfying $\text{Ri}{{\text{c}}_{(M,g)}}\ge n-1,\text{in}{{\text{j}}_{(M,g)}}\ge {{i}_{0}}\,>\,0$ and $\text{dia}{{\text{m}}_{(M,g)}}\,\ge \,\pi \,-\,\varepsilon $ , then $M$ is diffeomorphic to the standard sphere. In this note, we improve this result by replacing a lower bound on the injectivity radius with a lower bound of the conjugate radius.
DOI : 10.4153/CMB-1999-026-6
Mots-clés : 53C20, 53C21, Ricci curvature, conjugate radius 
Paeng, Seong-Hun; Yun, Jong-Gug. Conjugate Radius and Sphere Theorem. Canadian mathematical bulletin, Tome 42 (1999) no. 2, pp. 214-220. doi: 10.4153/CMB-1999-026-6
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     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1999-026-6/}
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[AC] [AC] Anderson, M. T. and J. Cheeger, Cα-compactness for manifolds with Ricci curvature and injectivity radius bounded below. J. Differential Geom. 35 (1992), 265–281. Google Scholar

[Be] [Be] Bessa, G. P., Differentiable sphere theorems for Ricci curvature. Math. Z. 214 (1993), 245–259. Google Scholar

[BC] [BC] Bishop, R. and Crittenden, B., Geometry of manifolds. Academic Press, New York, 1964. Google Scholar

[BK] [BK] Buser, P. and Karcher, H., Gromov's almost flat manifolds. Astérisque 81 (1981). Google Scholar

[Br] [Br] Brocks, R., Convexité et courbure de Ricci.C. R. Acad. Sci. Paris Sér. I 319 (1994), 73–75. Google Scholar

[Ca] [Ca] Cai, M., A splitting theoremfor manifolds of almost nonnegative Ricci curvature. Ann. Global Anal. Geom. 11 (1993), 373–385. Google Scholar

[CCo1] [CCo1] Cheeger, J. and Colding, T. H., Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. ofMath. 144 (1996), 189–237. Google Scholar

[CCo2] [CCo2] Cheeger, J. and Colding, T. H., On the structure of spaces with Ricci curvature bounded below. Preprint. Google Scholar

[CE] [CE] Cheeger, J. and Ebin, D. G., Comparison theorems in Riemannian geometry. Amsterdam, 1975. Google Scholar

[CH] [CH] Calabi, E. and Hartman, P., On the smoothness of isometries. Duke Math. J. 37 (1970), 741–750. Google Scholar

[Cr] [Cr] Croke, C., An eigenvalue pinching theorem. Invent.Math. 68 (1982), 253–256. Google Scholar

[DSW] [DSW] Dai, X., Shen, Z. and Wei, G., Negative Ricci curvature and isometry group.DukeMath. J. 76 (1994), 59–73. Google Scholar

[DW] [DW] Dai, X. and Wei, G., A comparison estimate of Toponogov type for Ricci curvature.Math. Ann. 303 (1995), 297–306. Google Scholar

[GHL] [GHL] Gallot, S., Hulin, D. and Lafontaine, J., Riemannian geometry. Springer-Verlag, 1987. Google Scholar

[O] [O] Otsu, Y., On manifolds of small excess. Amer. J. Math. 115 (1993), 1229–1280. Google Scholar

[P1] [P1] Paeng, S.-H., Topological entropy for geodesic flows under a Ricci curvature condition. Proc. Amer. Math. Soc. 125 (1997), 1873–1879. Google Scholar

[P2] [P2] Paeng, S.-H., A generalization of Gromov's almost flat manifolds and topological entropy for geodesic flow. Preprint. Google Scholar

[Wei] [Wei] Wei, G., Ricci curvature and Betti number. J. Geom. Anal., to appear. Google Scholar

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