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On the almost negatively curved 3-sphere. (English) Zbl 0651.53032
Proc. 21st. Int. Taniguchi Symp., Katata/Japan, Conf., Kyoto/Japan 1987, Lect. Notes Math. 1339, 78-85 (1988).
[For the entire collection see Zbl 0638.00022.]
This is a nice exposition of a striking fact pointed out in [M. Gromov, J. Differ. Geom. 13, 231-241 (1978; Zbl 0432.53020)]: “The 3- dimensional sphere S 3 carries a metric of almost non positive curvature”. Precisely, for all $$\epsilon >0$$ there exists a Riemannian metric on S 3 with diameter 1 and sectional curvature $$<\epsilon$$. This example has been generalized to all 3-manifolds in [C. Bavard, Compos. Math. 63, 223-236 (1987; Zbl 0642.53047)].
Reviewer: K.Grove

##### MSC:
 53C20 Global Riemannian geometry, including pinching