Editing Erlangen program (section)

{{Short description|Research program on the symmetries of geometry}}
[[File:Felix_Christian_Klein.jpg | thumb | right | alt=This is an image of Felix Christian Klein, the father of the Erlangen program. | Felix Christian Klein, the father of the Erlangen program.]]
In mathematics, the '''Erlangen program''' is a method of characterizing [[geometry|geometries]] based on [[group theory]] and [[projective geometry]]. It was published by [[Felix Klein]] in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is named after the [[University of Erlangen-Nuremberg|University Erlangen-Nürnberg]], where Klein worked.

By 1872, [[non-Euclidean geometry|non-Euclidean geometries]] had emerged, but without a way to determine their hierarchy and relationships. Klein's method 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 in 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 section]]s, but not about [[circle]]s or [[angle]]s because those notions were not invariant under [[projective transformation]]s (something familiar in [[geometrical perspective]]). The way the multiple languages of geometry then came back together could be explained by the way [[subgroup]]s of a [[symmetry group]] related to each other.

Later, [[Élie Cartan]] generalized Klein's homogeneous model spaces to [[Cartan connection]]s on certain [[principal bundle]]s, which generalized [[Riemannian geometry]].