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An estimate on the volume of metric balls. (English) Zbl 0667.53035

A curvature free volume estimate for small geodesic balls in a complete Riemannian manifold due to C. Croke [Ann. Sci. Ec. Norm. Supér., IV. Sér. 13, 419-535 (1980; Zbl 0465.53032)] is improved in some cases by evoking the lower curvature bound.
Reviewer: K.Grove

MSC:

53C20 Global Riemannian geometry, including pinching

Citations:

Zbl 0465.53032
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References:

[1] M. BERGER, Some relations between volume, injectivity radius, and convexity radius in Riemannian manifolds. Differential Geometry and relativity (Cahen, Flato, Eds.), D. Reidel, Dordrecht-Boston, 1976. · Zbl 0342.53038
[2] M. BERGER, Volume et rayon d’injectivite dans les varietes riemanniennes d dimension 3, Osaka J. Math. 14(1977), 191-200. · Zbl 0353.53028
[3] M. BERGER AND J. L. KAZDAN, A Sturm-Liouville inequality with applications t anisoperimetric inequality for volume in terms of injectivity radius, and to Wiedersehen manifolds, in General inequalities 2 (Proceedings of Second Inter-national Conference on General Inequalities, 1978), E. F. Beckenbach (ed.), ISNM47, 367-377, Basel, Birkhauser 1980.
[4] C. B. CROKE, Some isoperimetric inequalities and Eigenvalue estimates, Ann. Sci Ecole Norm. Sup. (4) 13 (1980), 419-535. · Zbl 0465.53032
[5] C. B. CROKE, On the volume of metric balls, Proc. Amer. Math. Soc. 88 (1983), 660-664 · Zbl 0521.53044 · doi:10.2307/2045458
[6] C. B. CROKE, Curvature free volume estimates, Invent, math. 76 (1984), 515-521 · Zbl 0552.53016 · doi:10.1007/BF01388471
[7] J. CHEEGER AND D. EBIN, Comparison Theorems in Riemannian Geometry, Nort Holland, Amsterdam, Oxford, (1975). · Zbl 0309.53035
[8] R. BISHOP AND R. CRITTENDEN, Geometryofmanifolds, Academic Press, New York, 1964 · Zbl 0132.16003
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