UC Riverside

This thesis explores families of metric spaces. It has two parts. First, the Kuratowski embedding is an isometric embedding of a metric space, M, into the space of bound, Borel functions over M, equipped with the sup norm. We extend this map to a family of maps by averaging over metric r--balls. The image of M under this map can be regarded as a deformation. After restricting our metric spaces to Riemannian manifolds, we explore how curvature affects this deformation. Furthermore, we give a complete description of the deformation of the n-sphere. Second, we prove a diffeomorphsim stability theorem. The smallest r so that a metric r--ball covers a metric space M is called the radius of M. The volume of a metric r-ball in the space form of constant curvature k is an upper bound for the volume of any Riemannian manifold with sectional curvature ≥ k and radius ≤ r. We show that when such a manifold has volume almost equal to this upper bound, it is diffeomorphic to a sphere or a real projective space.