Erlangen program

(Erlanger Programm), a unified view of various geometries, such as Euclidean geometry, affine geometry, and projective geometry, that was formulated by F. Klein in a lecture delivered in 1872 at the University of Erlangen in Germany. The lecture was published in the same year under the title “A Comparative Review of Recent Researches in Geometry.”

In essence, the Erlangen program is concerned with invariants with respect to transformation groups. It is well known, for example, that Euclidean geometry deals with those properties of figures that do not change under motions; figures are defined as equivalent, in the sense of congruent, if they can be made to coincide by a motion. Instead of motions, however, some other set of geometric transformations may be selected, and figures may be declared “equivalent” if they can be obtained from each other by means of transformations in the set. We thereby arrive at a different geometry, which studies those properties of figures that do not change under the transformations in question.

The equivalence introduced for the new geometry must satisfy the following natural conditions: (1) every figure F is equivalent to itself; (2) if F is equivalent to F’, then F’ is equivalent to F; and (3) if F is equivalent to F’ and if F’ is equivalent to F”, then F is equivalent to F”. Accordingly, the following requirements must be imposed on the set of transformations: (1) the set must include an identity transformation, which leaves every figure unchanged; (2) in addition to each transformation Π that carries F into F’, the set must include an “inverse” transformation Π^–1 that carries F’ into F; (3) together with the two transformations Π, and Π₂, which carry Finto F’ and F’ into F’, respectively, the set must include the product Π₂Π₁, of these transformations, which carries F into F” (the expression Π, means that first Π, is performed and then n₂). As a result of requirements (1), (2), and (3), the set being considered is a group of transformations (seeTOPOLOGICAL GROUP). The theory that studies those properties of figures that are preserved under all transformations in a given group is called the geometry of the group.

Different transformation groups yield different geometries. For example, conventional (Euclidean) geometry is based on the group of motions, affine geometry on the group of affine transformations, and projective geometry on the group of projective transformations. Borrowing from the ideas of A. Cayley, Klein showed that Lobachevskii’s non-Euclidean geometry can be derived from the group of projective transformations that carry some circle (or arbitrary conic section) into itself (seeLOBACHEV-SKIAN GEOMETRY). Klein introduced into consideration a broad range of other geometries defined in like manner.

The Erlangen program does not encompass some important branches of geometry, such as Riemannian geometry. Nevertheless, it guided geometric research for many years. J. Schouten and E. Cartan contributed important papers that sought to unite the group-theoretic and differential-geometry approaches to geometry.