Erlangen Program

Erlangen Program

An influential research program 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 solution to the problem how to classify and characterize geometries on the basis of projective geometry and group theory.

At that time, a family of new non-Euclidean geometries had already emerged, without adequate clarifications of their mutual hierarchy and relationships. Klein's suggestion was fundamentally innovative in three ways:

• projective geometry was emphasized as the unifying frame for all other geometries considered by him. In particular, Euclidean geometry was more restrictive than affine geometry, which is turn is more restrictive than projective geometry.

• Klein proposed that group theory, a branch of mathematics that uses algebraic methods to abstract the idea of symmetry, was the most useful way of organizing geometrical knowledge; at the time it had already been introduced into the theory of equations in the form of Galois theory.

• Klein made much more explicit the idea that each geometrical language had its own, appropriate concepts, thus for example projective geometry rightly talked about conic sections, but not about circles or angles because those notions were not invariant under projective transformations (something familiar in geometrical perspective). The way the multiple languages of geometry then came back together could be explained by the way subgroups of a symmetry group related to each other.

Ultimately Élie Cartan generalized Klein's homogeneous model spaces to (Cartan) connections on certain principal bundles. Simultaneously, this view generalizes classical Riemannian geometry. For an introduction to this perspective, see the text by Sharpe.