DISCONNECTEDNESS OF SUBLEVEL SETS OF SOME RIEMANNIAN FUNCTIONALS

Let M be a compact manifold of dimension greater than four. Denote by Riem(M) the space of Riemannian structures on M (i.e. of isometry clas ses of Riemannian metrics on M) endowed with the Gromov-Hausdorff metr ic. Let Riem(epsilon)(M)subset of Riem(M) be its subset formed by all Riemannian structures mu such that vol(mu)=1 and inj(mu)greater than o r equal to epsilon; where inj(mu) denotes the injectivity radius of mu . We prove that for all sufficiently small positive epsilon the space Riem(epsilon)(M) is disconnected. Moreover, if epsilon is sufficiently small, then Riem(epsilon)(M) is representable as the union of two non -empty subsets A and B such that the Gromov-Hausdorff distance between any element of A and any element of B is greater than epsilon/9. We a lso prove a more general result with the following informal meaning: T here exist two Riemannian structures of volume one and arbitrarily sma ll injectivity radius on M such that any continuous path (and even any sequence of sufficiently small ''jumps'') in the space of Riemannian structures of volume one on M connecting these Riemannian structures m ust pass through Riemannian structures of injectivity radius ''uncontr ollably'' smaller than the injectivity radii of these two Riemannian s tructures. These results can be generalized for at least some four-dim ensional manifolds. The technique used in this paper can also be used to prove the disconnectedness of many other subsets of the space of Ri emannian structures on M formed by imposing various constraints on cur vatures, volume, diameter, etc.