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A general comparison theorem with applications to volume estimates for submanifolds. (English) Zbl 0416.53027


MSC:

53C22 Geodesics in global differential geometry
53C20 Global Riemannian geometry, including pinching
53C40 Global submanifolds
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References:

[1] M. BERGER , An Extension of Rauch’s Metric Comparison Theorem and some Applications (Illinois J. Math., vol. 6, 1962 , pp. 700-712). Article | MR 26 #719 | Zbl 0113.37003 · Zbl 0113.37003
[2] R. BISHOP , A Relation Between Volume, Mean Curvature and Diameter (Amer. Math. Soc. Not., vol. 10, 1963 , pp. 364).
[3] K. BORSUK , Sur la courbure totale des courbes fermées (Ann. Soc. Polon. Math., vol. 20, 1947 , pp. 251-265). MR 10,60e | Zbl 0037.23602 · Zbl 0037.23602
[4] J. CHEEGER , Finiteness theorems for Riemannian manifolds (Amer. J. Math., vol. 92, 1970 , pp. 61-74). MR 41 #7697 | Zbl 0194.52902 · Zbl 0194.52902
[5] B.-Y. CHEN , On the Total Curvature of Immersed Manifolds, I : An Inequality of Fenchel-Borsuk-Willmore (Amer. J. Math., vol. 93, 1971 , pp. 148-162). MR 43 #3971 | Zbl 0209.52803 · Zbl 0209.52803
[6] W. FENCHEL , Über die Krümmung und Windung geschlossener Raumkurven (Math. Ann., vol. 101, 1929 , pp. 238-252). JFM 55.0394.06 · JFM 55.0394.06
[7] K. GROVE and K. SHIOHAMA , A Generalized Sphere Theorem (Ann. of Math., vol. 106, 1977 , pp. 201-211). MR 58 #18268 | Zbl 0341.53029 · Zbl 0341.53029
[8] C. HEIM , Une borne pour la longueur des géodésiques périodiques d’une variété riemannienne compacte (Thèse, Paris, 1971 ).
[9] W. KLINGENBERG , Contribution to Riemannian Geometry in the Large (Ann. of Math., vol. 69, 1959 , pp. 654-666). MR 21 #4445 | Zbl 0133.15003 · Zbl 0133.15003
[10] T. NAGAYOSHI and Y. TSUKAMOTO , On Positively Curved Riemannian Manifolds with Bounded Volume (Tôhoku Math. J., 2nd series, vol. 25, 1973 , pp. 213-218). Article | MR 49 #11428 | Zbl 0262.53031 · Zbl 0262.53031
[11] H. E. RAUCH , A Contribution to Differential Geometry in the Large (Ann. of Math., vol. 54, 1951 , pp. 38-55). MR 13,159b | Zbl 0043.37202 · Zbl 0043.37202
[12] F. W. WARNER , The Conjugate Locus of a Riemannian Manifold (Amer. J. Math., vol. 87, 1965 , pp. 575-604). MR 34 #8344 | Zbl 0129.36002 · Zbl 0129.36002
[13] F. W. WARNER , Extensions of the Rauch Comparison Theorem to Submanifolds (Trans. Amer. Math. Soc., vol. 122, 1966 , pp. 341-356). MR 34 #759 | Zbl 0139.15601 · Zbl 0139.15601
[14] J. H. C. WHITEHEAD , On the Covering of a Complete Space by the Geodesics through a Point (Ann. of Math., vol. 36, 1935 , pp. 679-704). Zbl 0012.27802 · Zbl 0012.27802
[15] T. J. WILLMORE , Note on Embedded Surfaces (An. Sti. Univ. “Al. I. Cuza”, Iasi, Sect. Ia Mat., 11B, 1965 , pp. 493-496). MR 34 #1940 | Zbl 0171.20001 · Zbl 0171.20001
[16] W. ZILLER , Closed Geodesics on Homogeneous Spaces [Math. Z. (to appear)]. · Zbl 0326.53054
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