An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. This Erlanger Program (Klein at the time was at Erlangen) proposed a new kind of solution to the problems of geometry of the time.

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 non-Euclidean geometry had been born; and in projective geometry new 'points' (at infinity, with complex co-ordinates) had been introduced.

The solution in abstract terms was to use symmetry as an underlying principle, and to state first that different geometries could co-exist, 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 affine-invariant 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 groupss of symmetries from the geometries, the relationships between them can be re-established 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).

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 well-known as Lie groups. The specific relationships are quite simply described in technical language.

For example the group of projective geometry in n dimensions is the symmetry group of n-dimensional 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 translationss). This description then tells use 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. It is in fact (using the previous description of the affine group) the semi-direct product of the orthogonal (rotation and reflection) group with the translations.

Influence on later work

The long-term 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 infinite-dimensional 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.

Potential critique of the Erlanger program

Quite often, it appears there are two or more distinct geometries with isomorphic automorphism groups. Just to take a random 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 nonEuclidean geometry, but with opposite points not identified) have isomorphic automorphism group, SO(n+1) for even n, but they appear to be distinct. However, it turns out both geometries are very closely related in a precise manner.

To take yet another example, elliptic geometries with different radii of curvature have isomorphic automorphism groups, but that does not really count as a critique as all such geometries are isomorphic.

Now here comes some interesting examples:

So, maybe the Erlanger program can teach us some things about dualities in physics!