A remark on equivariant Riemannian isometric embeddings preserving symmetries

This remark pertains to isometric embeddings endowed with certain geometric properties. We study two embeddings problems for the universal cover M of an n-dimensional Riemannian torus (ⁿ,g). The first concerns the existence of an isometric embedding of M into a bounded subset of some Euclidean space D₁, and the second one seeks an isometric embdding of M that is equivariant with respect to the deck transformation group of covering map. By using a known trick in a novel way, our idea yields results with D₁ = N+2n and D₂ = N+n, where N is the Nash dimension of ⁿ. However, we doubt whether these bounds are optimal.