  A compact Riemannian manifold may be immersed into Euclidean space by using
high frequency Laplace eigenfunctions. We study the geometry of the manifold
viewed as a metric space endowed with the distance function from the ambient
Euclidean space. As an application we give a new proof of a result of
Burq-Lebeau and others on upper bounds for the sup-norms of random linear
combinations of high frequency eigenfunctions.
