To summarize context: I was trying to understand how the ideas of the Erlangen program can be made compatible with the contemporary viewpoint in which classification up to structure-preserving-bijection is the norm.
Say we have an object $M$ with a certain structure. Is there any relationship between
- classifying "sub-objects" of $M$ up to a transformation in the symmetry group (automorphism group) of $M$;
- considering the objects fixed by the subgroups of the symmetry group of $M$?,
I am aware that the general idea of a "sub-object" is not clear as I formulated it; so it's part of my question what is the maximal context in which this idea can be properly formulated.
In any case, we have many examples where this makes sense.
For example, if $M$ is a differentiable manifold, its "symmetry group" would be $Diff(M)$ i.e. the group of diffeomorphisms; the sub-objects would be submanifolds. My intuition says that for this example (and others), 1. and 2. are not equivalent: submanifolds of a manifold are much more complex than the spaces that a subgroup of $Diff(M)$ can fix. But I don't know of any proof in this sense. The same could be said about Riemannian manifolds with metric-invariant diffeomorphisms, algebraic varieties with birational morphisms etc.
Either way, if 1. and 2. are nonequivalent in general - and I would be very surprised if they were equivalent - does this mean that the Erlangen program (in its initial formulation) is limited?
More detailed context:
To my understanding, the underlying idea of the Erlangen program is that the type of geometry we do depends on a subgroup of the symmetries of an ambient space. The correspondence between "geometries" and subgroups is given by looking at the objects fixed by the subgroups; so for example the subgroup of rotations of the isometries of $\mathbb{R}^n$ with the standard metric gives the geometry of the sphere, while restricting even further to the dihedral groups gives the geometries of polygons/polihedra.
My thoughts on this began from the fact that, from a modern viewpoint, it is not immediately clear "up to what level of structure" looking for example at rotations gives us the geometry of the sphere: is it the sphere as a differentiable manifold? As a Riemann surface? As a Riemannian manifold? Since we're dealing with isometries, it seems like we'd somehow end up dealing with the sphere as a Riemannian manifold, however, restricting the metric on the ambient space would only produce a flat metric on the sphere, which is definitely not the usual Riemannian geometry of the sphere.
This got me thinking about the main question.