I define a subset $M$ of $\mathbb R^n$ to be a "homogeneous Euclidean Manifold" if:

• it is a closed connected smooth submanifold of $\mathbb R^n$,
• for every $p, q$ in $M$, there is an Euclidean isometry $f$ sending $p$ to $q$ fixing $M$ (ie $f(M)=M$ and $f(p)=q$). The problem is to classify the "homogeneous Euclidean Manifolds". I know there are several definitions of symmetric and homogeneous and that these spaces have been more or less classified so it may well be that my question is just a particular case. Thank you