A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter

Let $$ be the space of closed n-dimensional Riemannian manifolds (M, g) with $$ and $$ . In this paper we consider sequences $$ in $$ converging in the Gromov-Hausdorff topology to a compact metric space Y. We show, on the one hand, that the limit space of this sequence has at most codimension one if there is a positive number r such that the quotient $$ can be uniformly bounded from below by a positive constant C(n, r, Y) for all points $$ . On the other hand, we show that if the limit space has at most codimension one then for all positive r there is a positive constant C(n, r, Y) bounding the quotient $$ uniformly from below for all $$ . As a conclusion, we derive a uniform lower bound on the volume and a bound on the essential supremum of the sectional curvature for the closure of the space consisting of all manifolds in $$ with $$ .