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Homotopy type finiteness theorems for certain precompact families of Riemannian manifolds. (English) Zbl 0647.53035

For fixed \(r>0\) the author studies the class of Riemannian \(n\)-manifolds, \(M\), for which the distance function at any point \(p\in M\) has no critical points [cf. the reviewer and K. Shiohama, Ann. Math., II. Ser. 106, 201-211 (1977; Zbl 0341.53029) and M. Gromov, Comment. Math. Helv. 56, 179-195 (1981; Zbl 0467.53021)] in the ball \(B(p,r)\subset M\). It is proved that any two such manifolds are homotopy equivalent provided their Hausdorff distance [cf. M. Gromov, Structures métriques pour les variétés riemanniennes (Text. Math. 1) Paris 1981; Zbl 0509.53034)] is sufficiently small depending only on \(n\) and \(r\). A similar theorem for general metric spaces is essentially contained in K. Borsuk [Fundam. Math. 41, 168-202 (1954; Zbl 0065.38102)]. The set of homotopy types of compact Riemannian \(n\)-manifolds with bounded embolicvolume \(\text{Vol }M/(\text{inj }M)\) \(n<L\) is shown to be finite.
Reviewer: K.Grove

MSC:

53C20 Global Riemannian geometry, including pinching
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