| We investigate manifolds with small excess, curvature bounded from below, and diameter bounded from above. In particular, we prove the Double Soul conjecture of K. Grove and P. Petersen in dimension 3. In order to prove this we give some new ideas and results on collapsing of Riemannian manifolds. A. D. Alexandrov's spaces enter here as limit spaces, in the (pointed) Gromov-Hausdorff topology, of collapsing sequences of Riemannian manifolds whose curvatures are uniformly bounded from below. |
