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 a Euclidean isometry $f$ of $\mathbb R^n$ sending $p$ to $q$ fixing $M$ (i.e., $f(M)=M$ 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.



Classification of "homogeneous" submanifolds of ℝⁿ
Such $M$ are called extrinsically homogeneous submanifolds of euclidean space.


Reference: for example the book by Berndt-Console-Olmos, Submanifolds and holonomy.


The set of all euclidean isometries that fix $M$ is a closed subgroup $G$ of the group of euclidean motions $E(n)$, and $M$ is an orbit of $G$. So the question is to classify all (compact connected) orbits of all closed subgroups of $E(n)$.


Since $M$ is assumed to be bounded, it has a euclidean center of mass, which one can assume to be the origin. That center of mass is a common fixed point of $G$. So $G$ is a closed subgroup of the orthogonal group, and the orbit $M$ is contained in a sphere around the origin. These are examples of isoparametric submanifolds of the sphere (Edit: This is not correct, see the comment by Claudio Gorodsky).


Classification is probably asking too much. For example, all compact homogeneous spaces occur as such orbits in the sphere (Mostow).


