Abstract Returns From The Erlangen Program
Quite often, it appears there are two or more distinct geometries with isomorphic automorphism groups. There arises the question of reading the Erlangen program from the abstract group, to the geometry.
One example: oriented (i.e., reflections not included) elliptic geometry (i.e., the surface of an n-sphere with opposite points identified) and oriented spherical geometry (the same non-Euclidean geometry, but with opposite points not identified) have isomorphic automorphism group, SO(n+1) for even n. These may appear to be distinct. It turns out, however, that the geometries are very closely related, in a way that can be made precise.
To take another example, elliptic geometries with different radii of curvature have isomorphic automorphism groups. That does not really count as a critique as all such geometries are isomorphic. General Riemannian geometry falls outside the boundaries of the program.
Some further notable examples have come up in physics.
Firstly, n-dimensional hyperbolic geometry, n-dimensional de Sitter space and (n−1)-dimensional inversive geometry all have isomorphic automorphism groups,
the orthochronous Lorentz group, for n ≥ 3. But these are apparently distinct geometries. Here some interesting results enter, from the physics. It has been shown that physics models in each of the three geometries are "dual" for some models.
Again, n-dimensional anti de Sitter space and (n−1)-dimensional conformal space with "Lorentzian" signature (in contrast with conformal space with "Euclidean" signature, which is identical to inversive geometry, for three dimensions or greater) have isomorphic automorphism groups, but are distinct geometries. Once again, there are models in physics with "dualities" between both spaces. See AdS/CFT for more details.
The covering group of SU(2,2) is isomorphic to the covering group of SO(4,2), which is the symmetry group of a 4D conformal Minkowski space and a 5D anti de Sitter space and a complex four-dimensional twistor space.
The Erlangen program can therefore still be considered fertile, in relation with dualities in physics.
Relations of the Erlangen program with work of C. Ehresmann on groupoids in geometry is considered in the article below by Pradines.
Read more about this topic: Erlangen Program
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