Some notes on a most general definition of "geometry",
first illucidated by Felix Klein that is based on
a set of geometric invariants under a group of tramsformations.
Klein's "Erlanger Program".



According to Felix Klein: A space S is coordinatized by (x_1, x_2, ..., x_n), i.e., there is a bijective map between the points of S and the coordinates. The coordinates are usually local functions on S with values in a field. Relationships between the coordinates of points can be specified. There is an abstract group g acting on S which has a representation as a group of linear substitutions on the coordintes. (could be homogeneous or inhomogeneous)

A "geometry" is defined by the set of relations among the coordinates that are invariant under the group of coordinate substitutions.

Thus, generally, a geometry is defined by a space (set) S, a set R of relations, and the group g of transformations of S that leave R invariant, since the set R is implied by the action of g on S, Thus, most generally the coordinates can be conceptually eliminated.


                      ------------------------
Ultimately an abstract geometry G is defined simply by a set S
and a group g which acts as a transformation group on S.
The relations on S that are left invariant by g are called G-geometric.
                      ------------------------

Example: S is an R^n and g is an R^n which is
         an abelian group, (a group of translations)
         (S, g) is then a geometry with translationally invariant
         relations which may not be rotationally invariant.

Example: S is a Euclidean space and g is IO(n) which is
         the semidirect product of the full group of rotations O(n)
         and an abelian group, (a group of translations) isomorphic
         to R^n.
         (S, g) is a homogeneous space.

Example: Homogeneous spaces
         Let S be a space upon which a Lie Group g acts effectively.
         Let h be an isotropy subgroup of g in its action on S, then

              S is isomorphic to g/h

         This generalizes the second example.

Underlying Klein's notion of geometry are the following modifiable
assumptions:

        1) S will be LOCALLY homeomorphic to an R^n.
           That is, S will be a classical manifold.
           [Essential source of coordinates as n-tuples of field elements]
           [Essential source of dimension as a topological invariant;
            possibility of recursive definition of Brouwer-Uryhson]

        2) Structurally, the R^n is an Abelian ring over (the real)
           field, hence actually a commutative algebra,
           though only the vector space properties are necessary.
           Point-point multiplication is not a part of classical
           geometry and has no meaning in classical intuitive geometry.
           [Essential source of intrinsic geometric holonomy]

        3) The field (of reals) has characteristic zero.
           [Essential source of continuousness of the manifold]
           Note that finite fields have an implicit toroidal
           topology, and that there is a concept of nearness
           which can be refined by a definition of a topology
           that is a proper subset of the power set.
           Alternatively, develop "proximity" independently
           of a topological notion, perhaps more along the line
           a sigma algebra that supports a measure on the Galois
           field.



See also, a review of the historical concepts of Euclidean geometry upon which a modern algebraic understanding of more general geometry is based. Classical Geometry & Physics Redux, which is logically consistent with Klein's group theoretical understandings.

The connections of the essential concept of geometry with groups also extends to Simplicial Homology and to Noncommutative Geometry.



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The URL for this document is:
http://graham.main.nc.us/~bhammel/MATH/geomdef.html
Created: June 11, 1999
Last Updated: May 28, 2000
Last Updated: December 17, 2005