International Congress of Mathematicians

316 Anton PetruninTheorem A. Given any ö > 0, there exists a positive constant io = io($) >0 such that the injectivity radius of any simply connected compact 3-dimensionalRiemannian manifold with Ricc> 5, K < 1, is bounded from below by io-Moreover they made a conjecture that this result should be also true for higherdimensions. Later on some new examples of manifolds with positively pinched curvaturewere found by Alloff and Wallach, Eschenburg and Bazaikin ([AW], [E], [B])which disprove this conjecture in general, but since then closely related conjecturesappeared on almost each list of open problems in Riemannian geometry. Thetheorem which we proved can be formulated as follows:Theorem B. Given any m and any ö > 0, there exists a positive constant io =io(m,ö) > 0 such that the injectivity radius of any simply connected compact m-dimensional Riemannian manifold with finite second homotopy group and Ricc> ö,K < 1, is bounded from below by io(m,ö).Theorem B generalizes the Burago-Toponogov Theorem A to arbitrary dimensionsand is also in even dimensions interesting, since there is no Synge theoremfor positive Ricci curvature. For sectional curvature pinching a similar result wasobtained independently by Fang and Rong [FR].Now I will turn to one proof of this statement which is described in the appendixof [PT] (The main part of paper contains an other proof). This proof makesuse of a generalized notion of Riemannian manifold, which was also described byGromov in the end of section 8 + of [G3], and employs a "tangential" version ofGromov-Hausdorff convergence. Here I will just give an informal analogy whichdescribes this notion. The formal aspects and all further details can be found in[PT].One may think about a manifold as a set of charts and glueing mappings. Fora Riemannian manifold, denoting the disjoint union of all charts with the pulledback metrics by (U,g), the set of all glueing maps defines an isometric pseudo-groupaction by a pseudogroup G on (U,g). Here is the definition of a pseudogroup action:Definition. A pseudogroup action (or pseudogroup of transformations) on a manifoldM is given by a set G of pairs of the form p = (D p ,p), where D p is an opensubset of M and p is a homeomorphism D p —t M, so that the following propertieshold:(1) p,q £ G implies po q = (q^1 (D p n q(D p j),po q) £ G;(2) p £ G implies p^1= (p(D p ),p^1)£ G;(3) (M, id) £ G;(4) if P is a homeomorphism from an open set D c M into M and D =(j a D a , where D a are open sets in M, then the property (D,p) £ G is equivalent to(D a ,p\i) a ) £ G for any a.We call the pseudo-group action natural if in addition the following is true:(i)' If (D,p) £ G and p can be extended as a continuous map to a boundarypoint x £ dD, then there is an element (D',p r ) £ G such that x £ D', D c D' andP'\D =P-To form a manifold this action must be in addition properly discontinuous