Let $(M^m,g)$ be a compact smooth Riemannian manifold of dimension $m$. From the celebrated Nash Embedding Theorem, we know there exists a (smooth) isometric embedding $M\hookrightarrow\mathbb R^n$ on the Euclidean space of dimension $n\leq m(3m+11)/2$.

My (quite loose) question is whether the

amount of symmetryof $(M,g)$ should influence this estimate on the maximal possible codimension for which it embeds isometrically in Euclidean space. In other words, is there any intuition if highly symmetric (vs. generic) metrics on $M$ impose smaller (vs. larger) bounds on the codimension of the isometric embedding?

By *amount of symmetry* I mean any possible ways to measure how symmetric $(M,g)$ is, perhaps the most usual being the *symmetry rank* (rank of the isometry group); *symmetry degree* (dimension of the isometry group) and *cohomogeneity* (dimension of the orbit space $M/Iso(M)$).

For example, assume there is an isometric action of the circle $S^1$ on $(M,g)$. Should we (expect to) be able to embed $(M,g)$ isometrically into an Euclidean space with lower (higher?) codimension than if $M$ was endowed with a generic metric?

About a possible converse question: if we *know* a certain metric on $M$ allows it to be embedded with "low codimension", should this indicate anything about how symmetric this metric is? Maybe taking arbitrarily small perturbations of the embedded submanifold one could destroy the symmetries while keeping it embedded (so that there is no relation between those things)?