Erlangen program

The Erlangen program referred to the presentation by Felix Klein in his entry into the University of Erlangen scientific manifesto (1872 ). In this he developed the conception of a systematic classification of geometric sub-disciplines, which starts from the idea that the geometry of the properties of figures examined are retained during position changes and therefore a classification by means of the considered possible position changes, ie the approved geometric transformations aiming.

Details of the geometric research program

Felix Klein outlined a geometry beyond Euclidean geometry, namely the hyperbolic geometry by Lobachevsky, for the theory of relativity in physics became important later, and elliptic geometry. These two non-Euclidean geometries were soon important fact in differential geometry.

In each of the thus resulting geometries, the transformations with respect to their corresponding sequential execution form a group, the group of transformations of the geometry. The investigated in the relevant geometry features to stay informed about transformations of the transformation group invariant.

The elementary Euclidean geometry or Kongruenzgeometrie is the geometry of the space of intuition, whose transformation group, the group of motions (ie the translations, rotations or reflections ) which are all lengths and angle- preserving mappings.

  • When you remove from the approved transformations on the length of loyalty and also allows for point dilations, we obtain the äquiforme group of transformations that characterizes the similarity or äquiforme geometry.
  • If one does not also on the angle fidelity, so you get to the transformation group of linear transformations in coordinate representation, ie the collineations that preserve the partial ratio of three points each. They characterize the affine geometry.
  • If one adds finally the space of intuition, yet infinitely distant points at infinity or as intersections of parallels, so let the collineations in this space the cross ratio of four points is invariant and form the group of projective transformations, their associated geometry is projective geometry.

In addition to the mentioned classical geometries, all emerge by restricting the transformation group of projective geometry, you can go to this kind of projective geometry also for elliptic and hyperbolic geometry, these non-Euclidean geometries can be classified thus after the Erlanger program. However, the Erlanger program is not sufficient for a complete classification of all geometries: for example, the theory of general relativity underlying Riemannian geometry not covered by this classification are detected ( Lie groups ).