Homotopy type finiteness theorems for certain precompact families of Riemannian manifolds
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- by Takao Yamaguchi
- Proc. Amer. Math. Soc. 102 (1988), 660-666
- DOI: https://doi.org/10.1090/S0002-9939-1988-0928999-X
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Abstract:
In this paper, we consider a precompact family of Riemannian manifolds with respect to the Hausdorff distance, and prove the homotopy type finiteness of elements in the family. This is an extension in the homotopy type version of the Cheeger and Weinstein finiteness theorems.References
- Marcel Berger, Une borne inférieure pour le volume d’une variété riemannienne en fonction du rayon d’injectivité, Ann. Inst. Fourier (Grenoble) 30 (1980), no. 3, 259–265 (French). MR 597027
- Marcel Berger, Recent trends in Riemannian geometry, Recent trends in mathematics, Reinhardsbrunn 1982 (Reinhardsbrunn, 1982), Teubner-Texte zur Mathematik, vol. 50, Teubner, Leipzig, 1982, pp. 20–37. MR 688824 R. Biship, A relation between volume, mean curvature, and diameter, Notices Amer. Math. Soc. 10 (1963), 364.
- Jeff Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61–74. MR 263092, DOI 10.2307/2373498
- Christopher B. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 4, 419–435. MR 608287
- Kenji Fukaya, A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters, J. Differential Geom. 28 (1988), no. 1, 1–21. MR 950552
- Sylvestre Gallot, Inégalités isopérimétriques, courbure de Ricci et invariants géométriques. I, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 7, 333–336 (French, with English summary). MR 697966
- R. E. Greene and H. Wu, Lipschitz convergence of Riemannian manifolds, Pacific J. Math. 131 (1988), no. 1, 119–141. MR 917868
- Michael Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv. 56 (1981), no. 2, 179–195. MR 630949, DOI 10.1007/BF02566208 —, Volume and bounded cohomology, Inst. Hautes Etudes Sci. Publ. Math. 56 (1983), 223-230.
- Mikhael Gromov, Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts], vol. 1, CEDIC, Paris, 1981 (French). Edited by J. Lafontaine and P. Pansu. MR 682063
- Karsten Grove and Katsuhiro Shiohama, A generalized sphere theorem, Ann. of Math. (2) 106 (1977), no. 2, 201–211. MR 500705, DOI 10.2307/1971164
- Yoe Itokawa, The topology of certain Riemannian manifolds with positive Ricci curvature, J. Differential Geometry 18 (1983), no. 1, 151–155. MR 697986
- Atsushi Katsuda, Gromov’s convergence theorem and its application, Nagoya Math. J. 100 (1985), 11–48. MR 818156, DOI 10.1017/S0027763000000209
- S. B. Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401–404. MR 4518
- Stefan Peters, Cheeger’s finiteness theorem for diffeomorphism classes of Riemannian manifolds, J. Reine Angew. Math. 349 (1984), 77–82. MR 743966, DOI 10.1515/crll.1984.349.77
- G. de Rham, Complexes à automorphismes et homéomorphie différentiable, Ann. Inst. Fourier (Grenoble) 2 (1950), 51–67 (1951) (French). MR 43468
- Takashi Sakai, Comparison and finiteness theorems in Riemannian geometry, Geometry of geodesics and related topics (Tokyo, 1982) Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1984, pp. 125–181. MR 758652, DOI 10.2969/aspm/00310125
- Katsuhiro Shiohama, A sphere theorem for manifolds of positive Ricci curvature, Trans. Amer. Math. Soc. 275 (1983), no. 2, 811–819. MR 682734, DOI 10.1090/S0002-9947-1983-0682734-1
- Alan Weinstein, On the homotopy type of positively-pinched manifolds, Arch. Math. (Basel) 18 (1967), 523–524. MR 220311, DOI 10.1007/BF01899493
- Takao Yamaguchi, Locally geodesically quasiconvex functions on complete Riemannian manifolds, Trans. Amer. Math. Soc. 298 (1986), no. 1, 307–330. MR 857446, DOI 10.1090/S0002-9947-1986-0857446-4
Bibliographic Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 102 (1988), 660-666
- MSC: Primary 53C20
- DOI: https://doi.org/10.1090/S0002-9939-1988-0928999-X
- MathSciNet review: 928999