International Congress of Mathematicians

330 Xiaochun RongCorollary 2.4 (Generalized Margulis Lemma). ([CCR2]) Let M n be a closedmanifold of nonpositive sectional curvature. If the Ricci curvature is negative atsome point, then for every metric with \sec\ < 1, there is a point with injectivityradius > ö(n) > 0.Another consequence is a geometric obstruction for a nontrivial F-structure.Corollary 2.5 (Nonexistence of F-structure). ([CCR2]) If a closed manifoldM admits a metric of nonpositive sectional curvature such that the Ricci curvatureis negative at some point, then M does not admit a nontrivial F-structure.3. Collapsed manifolds with bounded sectionalcurvature and diameterIn this section, we discuss the class of collapsed manifolds of bounded sectionalcurvature whose diameters are also bounded. By the Gromov's compactness theorem,any sequence of such collapsed manifolds contains a convergent subsequence;see [GLP]. Hence, without loss of the generality, we only need to consider convergentcollapsing sequences.(3.1) Let M GH > X denote a sequence of closed manifolds converging to a compactmetric space X such that |secMf | < 1 and dim(X) < n.Main Problem 3.2. For i large, investigate relations between geometry and topologyof M and that of X. The following are some specific problems and questions.(3.2.1) Find topological obstructions for the existence of M as in (3.1).(3.2.2) To what extent is the topology of the M in (3.1) stable when i is sufficientlylarge?(3.2.3) Under what additional conditions is it true that {M} as in (3.1) containsa subsequence of constant diffeomorphism type? If all M are diffeomorphic, thento what extent do the metrics converge?Note that by the Cheeger-Gromov convergence theorem, the above problemsare well understood in the noncollapsed situation dim(X) = n.Theorem 3.3 (Convergence). ([Ch], [GLP]) Let Aff ^-y X be as in (3.1)except dim(X) = n. Then for i large, M is diffeomorphic to some fixed M nwhich is homeomorphic to X and there are diffeomorphisms, fi : M n —t M, suchthat the pulled back metrics, f*(gi), converge to a metric, g^, in the C 1 '®-topology(0