Erlangen program
In mathematics, the Erlangen program is a method of characterizing
By 1872, 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 geometrical perspective). The way the multiple languages of geometry then came back together could be explained by the way subgroups of a symmetry grouprelated to each other.
- Klein made much more explicit the idea that each geometrical language had its own, appropriate concepts, thus for example projective geometry rightly talked about
Later, Élie Cartan generalized Klein's homogeneous model spaces to Cartan connections on certain principal bundles, which generalized Riemannian geometry.
The problems of nineteenth century geometry
Since
With every geometry, Klein associated an underlying group of symmetries. The hierarchy of geometries is thus mathematically represented as a hierarchy of these groups, and hierarchy of their invariants. For example, lengths, angles and areas are preserved with respect to the Euclidean group of symmetries, while only the incidence structure and the cross-ratio are preserved under the most general projective transformations. A concept of parallelism, which is preserved in affine geometry, is not meaningful in projective geometry. Then, by abstracting the underlying groups of symmetries from the geometries, the relationships between them can be re-established at the group level. Since the group of affine geometry is a subgroup of the group of projective geometry, any notion invariant in projective geometry is a priori meaningful in affine geometry; but not the other way round. If you remove required symmetries, you have a more powerful theory but fewer concepts and theorems (which will be deeper and more general).
Homogeneous spaces
In other words, the "traditional spaces" are homogeneous spaces; but not for a uniquely determined group. Changing the group changes the appropriate geometric language.
In today's language, the groups concerned in classical geometry are all very well known as
Examples
For example, the group of
To explain accurately the relationship between affine and Euclidean geometry, we now need to pin down the group of Euclidean geometry within the affine group. The Euclidean group is in fact (using the previous description of the affine group) the semi-direct product of the orthogonal (rotation and reflection) group with the translations. (See Klein geometry for more details.)
Influence on later work
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.
When
In his book Structuralism (1970) Jean Piaget says, "In the eyes of contemporary structuralist mathematicians, like Bourbaki, the Erlangen program amounts to only a partial victory for structuralism, since they want to subordinate all mathematics, not just geometry, to the idea of structure."
For a geometry and its group, an element of the group is sometimes called a motion of the geometry. For example, one can learn about the Poincaré half-plane model of hyperbolic geometry through a development based on hyperbolic motions. Such a development enables one to methodically prove the ultraparallel theorem by successive motions.
Abstract returns from the Erlangen program
Quite often, it appears there are two or more distinct
One example:
To take another example,
Some further notable examples have come up in physics.
Firstly, n-dimensional hyperbolic geometry, n-dimensional de Sitter space and (n−1)-dimensional inversive geometry all have isomorphic automorphism groups,
the
Again, n-dimensional
The covering group of SU(2,2) is isomorphic to the covering group of SO(4,2), which is the symmetry group of a 4D conformal Minkowski space and a 5D anti-de Sitter space and a complex four-dimensional twistor space.
The Erlangen program can therefore still be considered fertile, in relation with dualities in physics.
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 Program, 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
In mathematical logic, the Erlangen program also served as an inspiration for Alfred Tarski in his analysis of logical notions.[4]
References
- ISBN 978-1-84816-858-9.
- ISBN 978-1-4020-9383-8
- Ehresmann's footsteps: from group geometries to groupoidgeometries (English summary) Geometry and topology of manifolds, 87–157, Banach Center Publ., 76, Polish Acad. Sci., Warsaw, 2007.
- ^ Luca Belotti, Tarski on Logical Notions, Synthese, 404-413, 2003.
- Klein, Felix (1872) "A comparative review of recent researches in geometry". Complete English Translation is here https://arxiv.org/abs/0807.3161.
- Sharpe, Richard W. (1997) Differential geometry: Cartan's generalization of Klein's Erlangen program Vol. 166. Springer.
- ISBN 0-486-63433-7.
- Covers the work of Lie, Klein and Cartan. On p. 139 Guggenheimer sums up the field by noting, "A Klein geometry is the theory of geometric invariants of a transitive transformation group (Erlangen program, 1872)".
- Thomas Hawkins (1984) "The Erlanger Program of Felix Klein: Reflections on Its Place In the History of Mathematics", Historia Mathematica 11:442–70.
- "Erlangen program", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Lizhen Ji and Athanase Papadopoulos (editors) (2015) Sophus Lie and Felix Klein: The Erlangen program and its impact in mathematics and physics, IRMA Lectures in Mathematics and Theoretical Physics 23, European Mathematical Society Publishing House, Zürich.
- Felix Klein (1872) "Vergleichende Betrachtungen über neuere geometrische Forschungen" ('A comparative review of recent researches in geometry'), Mathematische Annalen, 43 (1893) pp. 63–100 (Also: Gesammelte Abh. Vol. 1, Springer, 1921, pp. 460–497).
- An English translation by Mellen Haskellappeared in Bull. N. Y. Math. Soc 2 (1892–1893): 215–249.
- The original German text of the Erlangen program can be viewed at the University of Michigan online collection at [1], and also at [2] in HTML format.
- A central information page on the Erlangen program maintained by John Baez is at [3].
- ISBN 0-486-43481-8
- (translation of Elementarmathematik vom höheren Standpunkte aus, Teil II: Geometrie, pub. 1924 by Springer). Has a section on the Erlangen program.