Closeness to spheres of hypersurfaces with normal curvature bounded below

Abstract

For a Riemannian manifold $M^{n+1}$ and a compact domain $\Omega \subset M^{n+1}$ bounded by a hypersurface $\partial \Omega$ with normal curvature bounded below, estimates are obtained in terms of the distance from $O$ to $\partial \Omega$ for the angle between the geodesic line joining a fixed interior point $O$ in $\Omega$ to a point on $\partial \Omega$ and the outward normal to the surface. Estimates for the width of a spherical shell containing such a hypersurface are also presented.