On Gromov's large Riemannian manifolds (Q1332628)

The author proves that a Riemannian manifold is hyperspherical if and only if its filling radius is infinite, under suitable geometric hypotheses. The hypotheses are twofold: (i) the Ricci curvature is nonnegative (ii) the injectivity radius is positive. M. Gromov previously proved the result under the hypothesis of nonnegative sectional curvature. There are several further properties of ``largeness'' equivalent to hypersphericity (still assuming the two hypotheses), including the condition of Euclidean volume growth (for the ``biggest'' \(\rho\)-ball). The author's tools include the Anderson-Cheeger compactness theorem [\textit{M. Anderson} and \textit{J. Cheeger}, Geom. Funct. Anal. 1, No. 3, 231-252 (1991; Zbl 0764.53026)] as well as his refinement of it in [\textit{M. Cai}, Ann. Global Anal. Geom. 11, 373-385 (1993)]. A central role is played by Gromov's argument on skeletal retraction [\textit{M. Gromov}, Publ. Math., Inst. Hautes Etud. 53, 213-307 (1983), pp. 258- 260].