Short geodesics and gravitational instantons (Q910734)

The present article is a continuation of the investigations of W. Klingenberg, J. Cheeger and M. Gromov [see \textit{W. Klingenberg}, Ann. Math., II. Ser. 69, 654-666 (1959; Zbl 0133.150); \textit{J. Cheeger}, Am. J. Math. 92, 61-74 (1970; Zbl 0194.529); \textit{M. Gromov}, Comment. Math. Helv. 56, 179-195 (1981; Zbl 0467.53021)]. Let M be a compact, connected n-dimensional Riemannian manifold satisfying the bounds: \(Ric(M)\geq (1-n)k^ 2,\) \(vol(M)\geq v,\) \(diam(M)\leq D.\) The author shows that if \(\gamma\) is a closed loop in M with \([\gamma]^ p\neq 0\) in \(\pi_ 1(M)\) for all \(p\leq N\equiv v_ k(2D)/v,\) then \(\ell (\gamma)\geq Dv/v_ k(2D),\) where \(v_ k(r)\) is the volume of a geodesic ball of radius r in the space form of curvature -k. From this fact he deduces that in the class of compact n-dimensional Riemannian manifolds M with the restrictions \(Ric(M)\geq -k^ 2,\) \(vol(M)\geq v,\) \(diam(M)\leq D,\) there are only finitely many isomorphism classes of \(\pi_ 1(M).\) The author also constructs several examples of compact 4-manifolds with metrics of positive Ricci curvature and in particular an example of a compact simply connected 4-manifold \(M_ 1\) with a family of metrics of positive Ricci curvature, vol\(\geq v\), diam\(\leq D\), with closed geodesics of arbitrarily small length. Besides, he establishes with the help of the examples of 4-manifolds based on the family of gravitational instantons of Gibbons-Hawking that one cannot improve Gromov's theorem (on curvature, diameter and Betti numbers) to positive Ricci curvature in place of sectional curvature.