projective geometry

projective geometry

1

Subspace geometries

Given a

vector space

V

,

dim

⁡

V

>

0

, the

projective geometry

of

V

is the set of all

subspaces

of

V

ordered by

set inclusion

. It is typically denoted

P

⁢

G

⁢

(

V

)

or

P

⁢

(

V

)

. The vector space may be over a field or a division ring.

The

partially ordered set

(poset) of all subspaces of a vector space is a

geometric lattice

. So next to a

boolean lattice

, it is one of the best lattices one could expect. However, an alternative to viewing projective geometry

P

⁢

G

⁢

(

V

)

as a

lattice

is one of viewing

P

⁢

G

⁢

(

V

)

as a

geometry

. For this we assign the to

P

⁢

G

⁢

(

V

)

.

Points

Points in projective geometry are the 1-dimensional subspaces. These are often denoted

P

⁢

G

⁢

(

V

)

0

and are the atoms of the lattice.

P

⁢

G

⁢

(

V

)

0

is often referred to as the

projective space

of

V

and denoted

k

⁢

P

n

where

V

=

k

n

+

1

, especially in topological settings. Common examples include

ℝ

⁢

P

n

,

ℂ

⁢

P

n

.
(More on projective spaces (

http://planetmath.org/ProjectiveSpace

).)

Lines

As in the usual

Euclidean geometry

two distinct points should determine a unique line in the geometry. As two

linearly independent

vectors span a plane in

V

, in to make two points in the geometry

P

⁢

G

⁢

(

V

)

determine a line, we must define a line in projective space to a 2-dimensional subspace.

Remark 1

.

There is a conflict of terminology at this stage. Often it is useful to define a projective space as the set of all lines through the origin in a vector space

V

. But if we elect to view a projective space as a projective geometry then the lines in this definition correspond to points in the geometry, and lines are now the planes of the vector space containing the origin. Sometimes for clarity the phrase “projective point” and “

projective line

” can be used to resolve ambiguity.

Planes

A

projective plane

for a vector space

V

is a 3-dimensional subspace.
The study of projective planes extends beyond the consideration of the poset of vector spaces however, and is the beginning of many interesting combinatorial problems. See the following

section

on non-Desarguesian planes for further details.

Hyperplanes

Hyperplanes are subspaces of

V

, sometimes called

codimension

1 subspaces. If

V

is finite dimensional then points and hyperplanes are in a

1-1 correspondence

. This correspondence leads to many situations where an exchange of a point with a hyperplane is considered. The simplest of these exchanges occurs through the notion of a

perpendicular

subspace, i.e.: one may say a point is perpendicular to an entire hyperplane, and a hyperplane is perpendicular to just one point.

When

dim

⁡

V

=

2

then every point is a hyperplane which leads to many degenerate properties causing

dim

⁡

V

≠

2

conditions in many

theorems

of projective geometry.

2

Dual Geometries

Given a finite dimensional vector space

V

, the

dual space

V

*

of all

linear functionals

is

isomorphic

as a vector space but it is also possible to

P

⁢

G

⁢

(

V

)

to

P

⁢

G

⁢

(

V

*

)

in a dual manner.

For every subspace

W

of

V

define

W

∙

=

{

f

∈

V

*

:

(

W

)

⁢

f

=

0

}

.

That is,

W

∙

is the set of all

functionals

which contain

W

in their kernel (

nullspace

). It follows

dim

⁡

W

∙

=

dim

⁡

V

-

dim

⁡

W

and the map
from

P

⁢

G

⁢

(

V

)

to

P

⁢

G

⁢

(

V

*

)

determined by

W

↦

W

∙

is

order-reversing

. It is also evident that under the natural isomorphism of

V

≅

V

*

*

we can further take

W

=

W

∙

∙

.

3

Morphisms of Projective Geometry

Morphisms

from one projective geometry to another are defined as order-preserving maps, also called

projectivities

. In some

context

order-reversing maps may also be included which leads to the study of

dualities

and

polarities

.

Given an order-reversing map

P

⁢

G

⁢

(

V

)

→

P

⁢

G

⁢

(

W

)

, the map

P

⁢

G

⁢

(

V

)

→

P

⁢

G

⁢

(

W

*

)

determines a canonical order-preserving map so that one can indeed consider simply the order-preserving maps between projective geometries.

A

collineation

is a

function

which maps any three

collinear points

(i.e.: three 1 dimensional subspaces which all lie in a single 2-dimensional subspace) to three collinear points. These determine a unique order-preserving map between the two projective geometries. Thus the morphisms of projective geometry are often identified with collineations. This is preferable when authoring theorems in the of geometry.

Remark 2

.

Some authors prefer collineations to any projectivity

P

⁢

G

⁢

(

V

)

→

P

⁢

G

⁢

(

V

)

. Although there is no uniformity in these definitions, each takes

collinear

triples to collinear triples thus preserving the geometries under consideration.

4

Notations for projective geometries

When the

dimension

d

of

V

is finite we may write

P

⁢

G

⁢

(

d

-

1

,

k

)

where

k

is the field (or division ring) of the vector space. Notice that the

d

-

1

indicates the dimension of the geometry, not the dimension of the vector space, though one can be attained from the other.

When

k

is real, or complex,

P

⁢

G

⁢

(

1

,

k

)

is often denoted

ℝ

⁢

P

1

and

ℂ

⁢

P

1

instead. Once again the

1

denotes the dimensions of the geometry, and in this case also the manifold, not the vector space from which it is derived.

When

|

k

|

=

q

we may further write

P

⁢

G

⁢

(

d

,

q

)

.

5

P

⁢

G

⁢

(

1

,

k

)

The 1 dimensional vector spaces have points and hyperplanes, but every point is a hyperplane. Every

permutation

of points is a collineation. So the projective line is exceptional in many ways. Including in the

fundamental theorem of projective geometry

.

5.1

Abstract Projective Geometry

Projective geometry in general can be axiomatized as achieved by Hilbert.
The axioms precisely characterize the subspace lattice of a finite dimensional
vector space, but the

converse

is not generally true. Indeed, already for 1-dimensional geometries, so called

projective lines

, i.e.: a set of points, it is clear that not all such geometries can be captured as the subspaces of a vector space. For example, there is no vector space with exactly 2 one dimensional subspaces. Such geometries however are of little interest as a geometry to themselves (though they are pivotal as sub-geometries) for they have no .

When an abstract geometry is infinite dimensional or 2-dimensional, it is possible that it is not isomorphic to the geometry of subspaces of any vector space. These geometries are termed “non-Desarguesian” as they do not carry with them a version of Desargues’ theorem. These geometries are a rich area of study, especially so called

projective planes

– geometries of dimension 2. Despite not having the

structure

of a subspace geometry, so far every non-Desarguesian projective plane has still had order

p

k

for some prime

p

.
This has lead to the following unsolved problem:

Are their any projective planes of order not a power of a prime?

References

1

Gruenberg, K. W. and Weir, A.J.

Linear Geometry 2nd Ed.

(English)
[B] Graduate Texts in Mathematics. 49. New York - Heidelberg - Berlin: Springer-Verlag. (1977), pp. x-198.

2

Kantor, W. M.

Lectures notes on

Classical Groups

.

3

Taylor, Donald E.

The geometry of the classical groups

Sigma Series in Pure Mathematics. 9.
Heldermann Verlag, Berlin, xii+229, (1992), ISBN 3-88538-009-9.

Title

projective geometry

Canonical name

ProjectiveGeometry

Date of creation

2013-03-22 15:51:25

Last modified on

2013-03-22 15:51:25

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

26

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 51A10

Classification

msc 51A30

Classification

msc 51A05

Classification

msc 51D25

Related topic

PolaritiesAndForms

Related topic

SesquilinearFormsOverGeneralFields

Related topic

Perspectivity

Defines

projective geometry

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