Erlangen program

It was published by Felix Klein in 1872 as Vergleichende Betrachtungen über neuere geometrische Forschungen.

Klein's method was fundamentally innovative in three ways: Later, Élie Cartan generalized Klein's homogeneous model spaces to Cartan connections on certain principal bundles, which generalized Riemannian geometry.

Klein also strongly suggested to mathematical physicists that even a moderate cultivation of the projective purview might bring substantial benefits to them.

If you remove required symmetries, you have a more powerful theory but fewer concepts and theorems (which will be deeper and more general).

The affine group will be the subgroup respecting (mapping to itself, not fixing pointwise) the chosen hyperplane at infinity.

The long-term effects of the Erlangen program 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 has become standard in physics.

In pedagogic terms, the program 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.

Such a development enables one to methodically prove the ultraparallel theorem by successive motions.

There arises the question of reading the Erlangen program from the abstract group, to the geometry.

To take another example, elliptic geometries with different radii of curvature have isomorphic automorphism groups.

In the seminal paper which introduced 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.

"[2] Relations of the Erlangen program with work of Charles Ehresmann on groupoids in geometry is considered in the article below by Pradines.