  We study the a.s. convergence of a sequence of random embeddings of a fixed
manifold into Euclidean spaces of increasing dimensions. We show that the limit
is deterministic. As a consequence, we show that many intrinsic functionals of
the embedded manifolds also converge to deterministic limits. Particularly
interesting examples of these functionals are given by the Lipschitz-Killing
curvatures, for which we also prove unbiasedness, using the Gaussian kinematic
formula.
