### Notes on constructing approximate embeddings

Here are some brief notes on how to construct approximate equivariant isometric embeddings. Refer to On Equivariant Isometric Embeddings for notation and the rest of the proof.

L2(M) decomposes into a direct sum of finite-dimensional eigenspaces of D, and G acts on each eigenspace, so each eigenspace decomposes into irreducible unitary representations. (I am now looking at complex functions, as these will correspond to functions into euclidean space.) Let { fja } be a basis for the vector space of the j-th irreducible representation.

Approximate g by Σi dyi dyi in the Ck topology

Let zi = Σja cija fja approximate yi in the Ck+1 topology, and let Xiabr = Σ Nj-1/2 cija fjb where the sum is over all j such that the j-th representation is equivalent to r.

Proposition 1. X is equivariant
2. zi = Σar Nr1/2 Xiaar
3. F(X) = P(Σi dzi dzi)

It follows from (2) that X is an embedding if z is, which it will be if it approximates y and y is. (3) implies that F(X) approximates g in the Ck topology.