Editing Erlangen program (section)

==Abstract returns from the Erlangen program==
Quite often, it appears there are two or more distinct [[Geometry|geometries]] with [[isomorphic]] [[automorphism group]]s. There arises the question of reading the Erlangen program from the ''abstract'' group, to the geometry.

One example: [[oriented]] (i.e., [[Reflection (mathematics)|reflections]] not included) [[elliptic geometry]] (i.e., the surface of an [[n-sphere|''n''-sphere]] with opposite points identified) and [[oriented]] [[spherical geometry]] (the same [[nonEuclidean geometry|non-Euclidean geometry]], but with opposite points not identified) have [[isomorphic]] [[automorphism group]], [[Special orthogonal 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 geometry|elliptic geometries]] with different [[Radius of curvature (mathematics)|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.

[[Complex numbers|Complex]], [[dual numbers|dual]] and [[split-complex number|double (also known as split-complex) numbers]] appear as homogeneous spaces SL(2,'''R''')/H for the group [[SL2(R)|SL(2,'''R''')]] and its subgroups H=A, N, K.<ref name="raw">{{cite book |last=Kisil |first=Vladimir V. |year=2012 |title=Geometry of Möbius transformations. Elliptic, parabolic and hyperbolic actions of SL(2,R) | location=London |publisher=Imperial College Press|page=xiv+192 |isbn=978-1-84816-858-9 | doi=10.1142/p835}}</ref> The group SL(2,'''R''') acts on these homogeneous spaces by [[linear fractional transformation]]s and a large portion of the respective geometries can be obtained in a uniform way from the Erlangen 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,

:<math>\mathrm{O}(n,1)/\mathrm{C}_2,\ </math>

the [[orthochronous Lorentz group]], for {{nowrap|''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 [[Geometry|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.

In the seminal paper which introduced [[Category theory|categories]], [[Saunders Mac Lane]] and [[Samuel Eilenberg]] stated: "This may be regarded as a continuation of the Klein Erlanger Programm, in the sense that a geometrical space with its group of transformations is generalized to a category with its algebra of mappings."<ref>S. Eilenberg and S. Mac Lane, ''A general theory of natural equivalences'', Trans. Amer. Math. Soc., 58:231–294, 1945. (p. 237); the point is elaborated in Jean-Pierre Marquis (2009), ''From a Geometrical Point of View: A Study of the History of Category Theory'', Springer, {{ISBN|978-1-4020-9383-8}}</ref>

Relations of the Erlangen program with work of [[Charles Ehresmann]] on [[groupoids]] in geometry is considered in the article below by Pradines.<ref>Jean Pradines, ''In [[Ehresmann]]'s footsteps: from group geometries to [[groupoid]] geometries'' (English summary) Geometry and topology of manifolds, 87–157, Banach Center Publ., 76, Polish Acad. Sci., Warsaw, 2007.</ref>

In [[mathematical logic]], the Erlangen program also served as an inspiration for [[Alfred Tarski]] in his analysis of [[Alfred Tarski#Logical notions|logical notions]].<ref>Luca Belotti, ''Tarski on Logical Notions'', Synthese, 404-413, 2003.</ref>