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Gromov’s convergence theorem and its application

Published online by Cambridge University Press:  22 January 2016

Atsushi Katsuda
Atsushi Katsuda*
Affiliation:
Department of Mathematics Faculty of Sciences Nagoya University, Nagoya 464, Japan
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One of the basic questions of Riemannian geometry is that “If two Riemannian manifolds are similar with respect to the Riemannian invariants, for example, the curvature, the volume, the first eigenvalue of the Laplacian, then are they topologically similar?”. Initiated by H. Rauch, many works are developed to the above question. Recently M. Gromov showed a remarkable theorem ([7] 8.25, 8.28), which may be useful not only for the above question but also beyond the above. But it seems to the author that his proof is heuristic and it contains some gaps (for these, see § 1), so we give a detailed proof of 8.25 in [7]. This is the first purpose of this paper. Second purpose is to prove a differentiable sphere theorem for manifolds of positive Ricci curvature, using the above theorem as a main tool.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 100 , December 1985 , pp. 11 - 48
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1985

References

[ 1 ] Buser, P. and Karcher, H., Gromov’s almost flat manifolds, Astérisque, 81, Soc. Math. France (1983).Google Scholar
[ 2 ] Cheeger, J., Finiteness theorems for Riemannian manifolds, J. Amer. Math., 92 (1970), 61–74.CrossRefGoogle Scholar
[ 3 ] Cheeger, J. and Ebin, D. G., Comparison Theorems in Riemannian Geometry, North Holland, 1975.Google Scholar
[ 4 ] Cheng, S. Y., Eigenvalue comparison theorems and its geometric applications, J. Math. Z., 143 (1975), 289–297.CrossRefGoogle Scholar
[ 5 ] Croke, C. B., An eigenvalue pinching theorem, Invent. Math., 68 (1982), 253–256.CrossRefGoogle Scholar
[ 6 ] Gromov, M., Almost flat manifolds, J. Differential Geom., 13 (1978), 231–241.CrossRefGoogle Scholar
[ 7 ] Gromov, M., Structures metriques pour les varietes riemanniennes, redige par J. Lafontaine et P. Pansu, Textes math. n°l Cedic/Fernand-Nathan Paris 1981.Google Scholar
[ 8 ] Heintze, E. and Karcher, H., A general comparison theorem with applications to volume estimate for submanifolds, Ann. Sci. Ecole Norm. Sup. 4e serie, t. 11 (1978) 451–470.Google Scholar
[ 9 ] Itokawa, Y., The topology of certain Riemannian manifolds with positive Ricci curvature, J. Differential Geom., 18 (1983), 151–155.CrossRefGoogle Scholar
[10] Kasue, A., Applications of Laplacian and Hessian comparison theorems, Advanced Studies in Pure Math., 3, Geometry of Geodesies and Related Topics, 333–386.Google Scholar
[11] Maeda, M., Volume estimate of submanifolds in compact Riemannian manifolds, J. Math. Soc. Japan, 30 (1978), 533–551.CrossRefGoogle Scholar
[12] Peters, S., Cheeger’s finiteness theorem for diffeomorphism classes of Riemannian manifolds, J. reine angew. Math., 349 (1984), 77–82.Google Scholar
[13] Sakai, T., Comparison and finiteness theorems in Riemannian geometry, Advanced Studies in Pure Math., 3, Geometry of Geodesies and Related Topics, 125–181.Google Scholar
[14] Schwartz, J., Nonlinear Functional Analysis, Gorden and Breach science Pub.Google Scholar
[15] Shiohama, K., A sphere theorem for manifolds of positive Ricci curvature, Trans. Amer. Math. Soc, 275 (1983), 811–819.CrossRefGoogle Scholar
[16] Yamaguchi, T., A differentiate sphere theorem for volume-pinched manifolds, Advanced studies in Pure Math., 3, Geometry of Geodesies and Related Topics, 183–192.Google Scholar