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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Created: June 11, 1999

Last Updated: May 28, 2000

Last Updated: December 17, 2005