An extension procedure for manifolds-with-boundary.
Abstract. Manifolds-with-boundary tend to be "negatively curved" objects since geodesics can bifurcate along the boundary. Nevertheless, there is a way to devise an extension procedure to locally isometrically embed any manifold-with-boundary into an Alexandrov space of curvature bounded below. This leads to upper estimates on volume, diameter, and dimension, as well as to precompactness and finiteness theorems. Then, time permitting, we will outline how this procedure, under certain curvature and injectivity radius bounds, can be applied to yield a fiber bundle structure for any manifold-with-boundary which is sufficiently Gromov-Hausdorff close to a given closed manifold.