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

        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

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.

Math Pages
Physics Pages
Music Pages
Home Page

Email me, Bill Hammel at

The URL for this document is:
Created: June 11, 1999
Last Updated: May 28, 2000
Last Updated: December 17, 2005