geometrization of n

Note: this entry is being rewritten in response to the discussion ” space has no origin” to make it more useful.

n, the set of n-tuplets of elements of can be made into a geometric space in several ways by imposing various geometric structuresMathworldPlanetmath on it. In this entry, we shall explore four such structures.

0.1 n as an affine space

To make n into an affine spacePlanetmathPlanetmath, we will specify two functions, D:×n×nn and T:n×n×nn as follows:


Intuitively, D(s,p,q) is the point to which q is mapped under dilating space by a factor s about the point p and T(p,q,r) is the point which completesPlanetmathPlanetmathPlanetmathPlanetmath the parallelogramMathworldPlanetmath with vertices p,q,r.

0.2 n as a Euclidean space

In order to make n into a Euclidean space, we specify a distance function d:n×n as follows:


This function is also known as a metric, but one needs to be careful with the term ”metric” because it is sometimes also used to refer to an inner productMathworldPlanetmath.

0.3 n as vector space

In order to make n into a vector spaceMathworldPlanetmath, we specify two maps: the addition map +:n×nn and the scalar productMathworldPlanetmath map ×:×nn. These maps are defined as follows:


0.4 n as an inner product space

In order to make n into an inner product spaceMathworldPlanetmath we specify, in addition to the addition map and the scalar product map, another map, namely the inner product map, :n×n, which is defined as follows:


1 Relations between structures

Some of these geometriesMathworldPlanetmath impose more structure on our space than others. Affine geometry imposes the least structure and inner product space imposes the most. The situation may be summarized in the following diagram, where the arrows mean that we can get from one geometry to the next by imposing more structure:

Affine spaceEuclidean spaceVector SpaceInner product space

2 Symmetry

The four geometric structures described above may be characterized in terms of the symmetry groups which preserve the structures in question. These groups happen to be Lie groups and are as follows:

  1. 1.

    The symmetry group is IGL(n,), the set of all inhomogeneous linear transformations. (Here and in the next item, ”inhomogeneous” means that translationsMathworldPlanetmathPlanetmath are included in the transformation group.)

  2. 2.

    The symmetry group is IO(n,), the set of all inhomogeneous orthogonal transformationsMathworldPlanetmath. These include rotationsMathworldPlanetmath, translations, and reflections.

  3. 3.

    The symmetry group is GL(n,), the set of all homogeneousPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath linear transformations.

  4. 4.

    The symmetry group is O(n,), the set of all homogeneous orthogonal transformations.

These different groups act differently on the same underlying space and hence we have different invariantsMathworldPlanetmath. These invariants may be described in algebraic and in geometric terms.

3 Generalizations

Although we presented considered n, most of these constructions also work more generally. In this sectionMathworldPlanetmathPlanetmathPlanetmath, we shall discuss under what conditions we can use the same constructions to impose geometric structure on 𝕂n when 𝕂 is a algebraic system.

4 Abuses of language

Below is the old entry, which will be erased as soon as it has been supplanted by the new entry.

Affine space of dimensionPlanetmathPlanetmathPlanetmath n over a field k is kn, the set of of elements of k. The reason that the finite dimensional vector space kn has such a special name “affine space” is because it is acted upon by the set of affine transformations, which are transforms of the form


and hence is the natural setting for affine geometry. (For more information on affine geometry, please see section 1.1.2 of the entry on geometry.) It is worth mentioning that the name “affine space” is used primarily in geometry and in commutative algebra. (In algebraMathworldPlanetmathPlanetmath, one does not concern oneself with affine transforms and simply uses the term “affine space” as synonym for “vector space”).

Title geometrization of n
Canonical name GeometrizationOfmathbbRn
Date of creation 2013-03-22 12:03:50
Last modified on 2013-03-22 12:03:50
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 29
Author rspuzio (6075)
Entry type Topic
Classification msc 14-00
Related topic AffineTransformation