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Collapsing and soul theorem in three dimensions. (English) Zbl 0901.53025

Séminaire de théorie spectrale et géométrie. Année 1996-1997. St. Martin D’Hères: Univ. de Grenoble I, Institut Fourier, Sémin. Théor. Spectrale Géom., Chambéry-Grenoble. 15, 163-166 (1997).
The present article is devoted to the study of the topology of the isometry classes of closed \(n\)-dimensional Riemannian manifolds \(M ^n\) with a lower sectional curvature bound collapsing to an Alexandrov space of lower dimension \(X\). The author notes that “some extremal cases” have been studied before: for \(\dim(X)= n\), see G. Perelman [‘A.D. Alexandrov’s spaces with curvatures bounded from below. II’, preprint] and [St. Petersbg. Math. J. 5, 205-213 (1994; Zbl 0815.53072)]; for \(\dim(X) = 0\), see K. Fukaya and T. Yamaguchi [Ann. Math., II. Ser. 136, 253-333 (1992; Zbl 0770.53028) ]. The author had previously considered (with T. Shioya) the cases when \(n = 3\) and \(\dim(X) = 1,2\) [‘Collapsing three-manifolds under a lower curvature bound’, in preparation] and with K. Fukaya the case when \(n=3\) and \(\dim(X) = 0\) [loc. cit.].
In this article the author considers the cases where \( 1 \leq \dim(X) \leq n-1\), giving an extension of the soul theorem due to J. Cheeger and D. Gromoll [ Ann. Math., II. Ser. 96, 413-443 (1972; Zbl 0246.53049)].
Other soul theorems have been proved; for rigidity theorems see M. Strake [Manuscr. Math. 61, 315-325 (1988; Zbl 0653.53023)]; for the metric structure of open manifolds of nonnegative curvature see V. B. Marenich [Ukr. Geom. Sb. 26, 79-96 (1988; Zbl 0532.53030); Sov. Math., Dokl. 39, 404-407 (1989; Zbl 0699.53050)]. Extensions of the soul theorem: by means of a splitting theorem [J. Yim, Ann. Global Anal. Geom. 6, 191-206 (1988; Zbl 0668.53025)] and by studying the asymptotic behaviour of the set of rays [S. J. Mendonça, Comment. Math. Helv. 72, 331-348 (1997)]; another soul theorem: D.-T. Zhou [Int. Math. Res. Not. 1994, 209-214 (1994; Zbl 0815.53049)].
The author has previously published along these lines: see [Ann. Math., II. Ser. 133, 317-357 (1991; Zbl 0737.53041) and Sémin. Congr. 1, 601-642 (1996; Zbl 0885.53041)].
For the entire collection see [Zbl 0882.00016].

MSC:

53C20 Global Riemannian geometry, including pinching
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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