Erlangen Program The Problems Of Nineteenth Century Geometry

Was there one 'geometry' or many? Since Euclid, geometry had meant the geometry of Euclidean space of two dimensions (plane geometry) or of three dimensions (solid geometry). In the first half of the nineteenth century there had been several developments complicating the picture. Mathematical applications required geometry of four or more dimensions; the close scrutiny of the foundations of the traditional Euclidean geometry had revealed the independence of the Parallel Axiom from the others, and non-Euclidean geometry had been born; and in projective geometry new 'points' (at infinity, with complex co-ordinates) had been introduced.

The solution in abstract terms was to use symmetry as an underlying principle, and to state first that different geometries could co-exist, because they dealt with different types of propositions and invariances related to different types of symmetry and transformation. The distinction between affine geometry and projective geometry lies just in the fact that affine-invariant notions such as parallelism are the proper subject matter of the first, while not being principal notions in the second. 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 add required symmetries, you have a more powerful theory but fewer concepts and theorems (which will though be deeper and more general).

2 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 Lie groups. The specific relationships are quite simply described, using technical language.

3 Examples; affine geometry

For example the group of projective geometry in n dimensions is the symmetry group of n-dimensional projective space (the matrix group of size n+1, quotiented by scalar matrices). The affine groupIn mathematics, the affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself. It is a Lie group if K is the real or complex field. There is more than one convenient way to describe th will be the subgroup respecting (mapping to itself, not fixing pointwise) the chosen hyperplane at infinity. This subgroup has a known structure ( semidirect productIn abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. Some equivalent definitions Let G be a group, N a normal subgroup of G and H a subgroup of G''. The following statements are equi of the matrix group of size n with the subgroup of translationEuclidean geometry In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. It can also be interpreted as the addition of a constant vector to every point, or as shiftings). This description then tells us which properties are 'affine'. In Euclidean plane geometry terms, being a parallelogram is: affine transformations always take one parallelogram to another one. Being a circle isn't, since an affine shear will take a circle into an ellipse.

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. It 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.

4 Influence on later work

The long-term effects of the Erlangen programme can be seen all over pure mathematics (see tacit use at congruence (geometry)Euclidean geometry See also: congruence relation In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. More generally, two subsets A and B of Euclidean space R n are c, for example); and the idea of transformations and of synthesis using groups of symmetry is of course now standard too in physicsPhysics (from the Greek, physikos , "natural", and physis , "Nature") is the science of Nature in the broadest sense. Physicists study the behavior and properties of matter in a wide variety of contexts, ranging from the sub-microscopic particles from whi.

When topologyTopology is the study or science of places. It derives its name from the Greek words τοπος meaning place and λογος meaning study, talk. See also earth science, geography, human geography, g is routinely described in terms of properties invariantInvariant may have meanings invariant (computer science), such as a combination of variables not altered in a loop invariant (mathematics), something unaltered by a transformation invariant (music) invariant (physics) conserved by system symmetry. under homeomorphism, one can see the underlying idea in operation. The groups involved will be infinite-dimensional in almost all cases - and not Lie groups - but the philosophy is the same. Of course this mostly speaks to the pedagogical influence of Klein. Books such as those by H.S.M. Coxeter routinely used the Erlangen programme approach to help 'place' geometries. In pedagogic terms, the programme became transformation geometry, a mixed blessing in the sense that it builds on stronger intuitions than the style of Euclid, but is less easily converted into a logical system.

5 Potential critiques of the Erlangen program

Quite often, it appears there are two or more distinct geometries with isomorphic automorphism groups. Just to take a random example, oriented (i.e. reflections not included) elliptic geometry (i.e. the surface of an n-sphere with opposite points identified) and oriented spherical geometry (the same nonEuclidean geometry, but with opposite points not identified) have isomorphic automorphism group, SO(n+1) for even n, but they appear to be distinct. However, it turns out the geometries are very closely related, in a way that can be made precise.

To take another example, elliptic geometries with different radii of curvature have isomorphic automorphism groups, but that does not really count as a critique as all such geometries are isomorphic. General Riemannian geometry falls outside the boundaries of the program.

Now there come some interesting examples:

n-dimensional hyperbolic geometry, n-dimensional de Sitter space and n-1 dimensional inversive geometry all have isomorphic automorphism groups, , the orthochronous Lorentz group, for n>=3. But they are all very distinct geometries, right? Well, here is where some very interesting results come in from physics. It has been shown that physics models in each of the three geometries are " dual" for some models!
n-dimensional anti de Sitter space and n-1 dimensional conformal space with "Lorentzian" signature (contrast that with conformal space with "Euclidean" signature, which is identical to inversive geometry, for 3 dimensions or greater) have isomorphic automorphism groups, but are distinct geometries, right? Once again, there are models in physics with " dualities" between both spaces. See AdS/CFT for more details.

The Erlangen program is therefore still considered fertile, in relation with dualities in physics.

Geometry Group theory

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