The Nash embedding theorem states that every Riemannian
manifold can be isometrically embedded into some Euclidean space with
dimension bound. Isometric means preserving the length of every
path. Nash's proof involves sophisticated perturbations of the
initial embedding, so not much is known about the geometry of the
resulted embedding.
     In this talk, using the eigenfunctions of the Laplacian
operator, we construct canonical isometric embeddings of compact
Riemannian manifolds into Euclidean spaces, and study the geometry of
embedded images. They turn out to have large mean curvature
(intuitively, very bumpy), but the extent of oscillation is about the
same at every point. More can be said about global quantities like
the center of mass. This is a joint work with Xiaowei Wang.