Erlangen program

An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. This Erlangen Program (Erlanger Programm) — Klein was then at Erlangen — proposed a new kind of solution to the problems of geometry of the time.
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The problems of nineteenth century geometry
Was there one 'geometry' or many? Since Euclid, geometry had meant the geometry of Euclidean space of two dimensions (plane geometry) or of three dimensions (solid geometry). In the first half of the nineteenth century there had been several developments complicating the picture. Mathematical applications required geometry of four or more dimensions; the close scrutiny of the foundations of the traditional Euclidean geometry had revealed the independence of the Parallel Axiom from the others, and nonEuclidean geometry had been born; and in projective geometry new 'points' (points at infinity, points with complex number coordinates) had been introduced.
The solution in abstract terms was to use symmetry as an underlying principle, and to state first that different geometries could coexist, because they dealt with different types of propositions and invariances related to different types of symmetry and transformation. The distinction between affine geometry and projective geometry lies just in the fact that affineinvariant notions such as parallelism are the proper subject matter of the first, while not being principal notions in the second. Then, by abstracting the underlying groups of symmetries from the geometries, the relationships between them can be reestablished at the group level. Since the group of affine geometry is a subgroup of the group of projective geometry, any notion invariant in projective geometry is a priori meaningful in affine geometry; but not the other way round. If you add required symmetries, you have a more powerful theory but fewer concepts and theorems (which will though be deeper and more general).
Homogeneous spaces
In other words, the "traditional spaces" are homogeneous spaces; but not for a uniquely determined group. Changing the group changes the appropriate geometric language.
In today's language, the groups concerned in classical geometry are all very wellknown as Lie groups. The specific relationships are quite simply described, using technical language.
Examples: affine geometry
For example the group of projective geometry in n dimensions is the symmetry group of ndimensional projective space (the matrix group of size n+1, quotiented by scalar matrices). The affine group will be the subgroup respecting (mapping to itself, not fixing pointwise) the chosen hyperplane at infinity. This subgroup has a known structure (semidirect product of the matrix group of size n with the subgroup of translations). This description then tells us which properties are 'affine'. In Euclidean plane geometry terms, being a parallelogram is: affine transformations always take one parallelogram to another one. Being a circle isn't, since an affine shear will take a circle into an ellipse.
To explain accurately the relationship between affine and Euclidean geometry, we now need to pin down the group of Euclidean geometry within the affine group. The Euclidean group is in fact (using the previous description of the affine group) the semidirect product of the orthogonal (rotation and reflection) group with the translations.
Influence on later work
The longterm effects of the Erlangen programme can be seen all over pure mathematics (see tacit use at congruence (geometry), for example); and the idea of transformations and of synthesis using groups of symmetry is of course now standard too in physics.
When topology is routinely described in terms of properties invariant under homeomorphism, one can see the underlying idea in operation. The groups involved will be infinitedimensional in almost all cases  and not Lie groups  but the philosophy is the same. Of course this mostly speaks to the pedagogical influence of Klein. Books such as those by H.S.M. Coxeter routinely used the Erlangen programme approach to help 'place' geometries. In pedagogic terms, the programme became transformation geometry, a mixed blessing in the sense that it builds on stronger intuitions than the style of Euclid, but is less easily converted into a logical system.
For a geometry and its group, an element of the group is sometimes called a motion of the geometry. For example, one can learn about the Poincare halfplane model of hyperbolic geometry through a development based on hyperbolic motions.
Potential critiques of 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 nsphere with opposite points identified) and oriented spherical geometry (the same nonEuclidean 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:
 ndimensional hyperbolic geometry, ndimensional de Sitter space and n1 dimensional inversive geometry all have isomorphic automorphism groups, <math>O(n,1)/\mathbb{Z}_2<math>, the orthochronous Lorentz group, for n>=3. But they are all very distinct geometries, right? Well, here is where some very interesting results come in from physics. It has been shown that physics models in each of the three geometries are "dual" for some models!
 ndimensional anti de Sitter space and n1 dimensional conformal space with "Lorentzian" signature (contrast that with conformal space with "Euclidean" signature, which is identical to inversive geometry, for 3 dimensions or greater) have isomorphic automorphism groups, but are distinct geometries, right? Once again, there are models in physics with "dualities" between both spaces. See AdS/CFT for more details.
The Erlangen program is therefore still considered fertile, in relation with dualities in physics.de:Erlanger Programm