On equivariant isometric embeddings of Riemannian manifolds with symmetries

Let [Formula: see text] be a [Formula: see text]-smooth, [Formula: see text]-dimensional Riemannian manifold that is diffeomorphic to [Formula: see text] and admits a properly discontinuous, cocompact, isometric, fixed-point-free action by a group. This work proves the existence of a [Formula: see text] equivariant isometric embedding of [Formula: see text] into some Euclidean space [Formula: see text] with [Formula: see text] which is the same as the optimal dimension bound in Matthias Günther’s results.