  Motivated by questions of manifold learning, we study a sequence of random
manifolds, generated by embedding a fixed, compact manifold $M$ into Euclidean
spheres of increasing dimension via a sequence of Gaussian mappings. One of the
fundamental smoothness parameters of manifold learning theorems is the reach,
or critical radius, of $M$. Roughly speaking, the reach is a measure of a
manifold's departure from convexity, which incorporates both local curvature
and global topology.
  This paper develops limit theory for the reach of a family of random,
Gaussian-embedded, manifolds, establishing both almost sure convergence for the
global reach, and a fluctuation theory for both it and its local version. The
global reach converges to a constant well known both in the reproducing kernel
Hilbert space theory of Gaussian processes, as well as in their extremal
theory.
