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    Shiohama, Katsuhiro 2000. Vol. 1, Issue. , p. 865.
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On the number of diffeomorphism classes in a certain class of Riemannian manifolds

  • Takao Yamaguchi (a1)
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The study of finiteness for Riemannian manifolds, which has been done originally by J. Cheeger [5] and A. Weinstein [13], is to investigate what bounds on the sizes of geometrical quantities imply finiteness of topological types, —e.g. homotopy types, homeomorphism or diffeomorphism classes-— of manifolds admitting metrics which satisfy the bounds. For a Riemannian manifold M we denote by RM and KM respectively the curvature tensor and the sectional curvature, by Vol (M) the volume, and by diam(M) the diameter.

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Copyright
COPYRIGHT: © Editorial Board of Nagoya Mathematical Journal 1985
References
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[ 1 ] Bishop, R. and Crittenden, R., Geometry of manifolds, Academic Press, New-York, 1964.
[ 2 ] Buser, P. and Karcher, H., Gromov’s almost flat manifolds, Astérisque, 1981.
[ 3 ] Cheeger, J., Comparison and finiteness theorems for Riemannian manifolds, Ph. D. Thesis, Princeton Univ., 1967.
[ 4 ] Cheeger, J., Pinching theorems for a certain class of Riemannian manifolds, Amer. J. Math., 91 (1969), 807–834.
[ 5 ] Cheeger, J., Finiteness theorems for Riemannian manifolds, Amer. J. Math., 92 (1970), 61–74.
[ 6 ] Cheeger, J. and Ebin, D., Comparison theorems in Riemannian geometry, North-Holland, 1975.
[ 7 ] Greene, R., Complete metrics of bounded curvature on noncompact manifolds, Arch. Math., 31 (1978), 89–95.
[ 8 ] Gromov, M., Almost flat manifolds, J. Differential Geom., 13 (1978), 231–241.
[ 9 ] Gromov, M., Structures métriques pour les variétés riemanniennes, rédigé par Lafontaine, J. et Pansu, P., Cedic-Fernand Nathan, Paris, 1981.
[10] Heintz, E. and Karcher, H., A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. Ecole Norm. Sup., 11 (1978), 451–470.
[11] Maeda, M., Volume estimate of submanifolds in compact Riemannian manifolds, J. Math. Soc. Japan, 30 (1978), 533–551.
[12] Shikata, Y., On a distance function on the set of differentiable structures, Osaka J. Math., 3 (1966), 65–79.
[13] Weinstein, A., On the homotopy type of positively-pinched manifolds, Arch. Math., 18 (1967), 523–524.
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Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
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