On equivariant isometric embeddings

Abstract

The representation p can be regarded as a Lie group homomorphism from G into the orthogonal group O(N) which acts on IE N by rotations and reflections; a smooth map X: M ~ I E N is equivariant with respect to p if and only if X(~p) =p(cr) X(p), for all ~r~G, pEM. The main theorem is true in both the C ~ and real analytic categories. We will work in the C ~ category for the time being, and return to the real analytic case in w 4. Moreover, the theorem holds for manifolds with boundary. The main analytic tool used by Nash to prove his isometric embedding theorem is an implicit function theorem based upon the Newton iteration method. The implicit function theorem applies to the equivariant case with virtually no change. In order to apply the implicit function theorem we need to approximate a given G-invariant Riemannian metric on M by a metric induced by an equivariant embedding; we will do this by using the theory of the Laplace operator on compact Riemannian manifolds. According to Gromov and Rokhlin [7], any n-dimensional compact Riemannian manifold can be isometrically embedded in IF, N, where N = (1/2) n(n + 1) + 3 n + 5. No such universal bound is possible in the equivariant case, and in fact, given any positive integer N, it is possible to construct a left invariant