Felix Klein suggested in his Erlangen program a way to classify geometries based on group theory. According to Wikipedia, we have the following definition:
A Klein geometry is a pair (G, H) where G is a Lie group and H is a closed Lie subgroup of G such that the (left) coset space G/H is connected.
Regarding this program I have several questions:
- Did someone actually carry it out systematically? It seems that Klein himself didn't do so (please correct me if I am wrong here).
- On the Wikipedia website mentioned above, they give several examples for Klein geometries, e.g. Euclidean, spherical, conformal, projective, affine, and hyperbolic geometry, together with their groups and invariants. What other important Klein geometries are missing?
- I have no serious knowledge about Lie theory so maybe the following question is silly or trivial or ridiculous -- but still: In how far is the classification of closed Lie subgroups of the projective group finished?
Please note that I'm not interested in books that take an even more general approach like the one by Sharpe (Differential Geometry: Cartan's Generalization of Klein's Erlangen Program). This seems to be well above my head and complexifies things to a degree that I cannot follow anymore.