  In my talk I will discuss the following results which were obtained in joint
work with Wilderich Tuschmann. 1. For any given numbers $m$, $C$ and $D$, the
class of $m$-dimensional simply connected closed smooth manifolds with finite
second homotopy groups which admit a Riemannian metric with sectional curvature
$\vert K \vert\le C$ and diameter $\le D$ contains only finitely many
diffeomorphism types. 2. Given any $m$ and any $\delta>0$, there exists a
positive constant $i_0=i_0(m,\delta)>0$ such that the injectivity radius of any
simply connected compact $m$-dimensional Riemannian manifold with finite second
homotopy group and Ricci curvature $Ric\ge\delta$, $K\le 1$, is bounded from
below by $i_0(m,\delta)$. I also intend to discuss Riemannian megafolds, a
generalized notion of Riemannian manifolds, and their use and usefulness in the
proof of these results.
