geometrization of ℝ^n

geometrization of

ℝ

n

Note: this entry is being rewritten in response to the discussion ”

http://planetmath.org/?op=getmsg&id=6329

euclidean space

has no origin” to make it more useful.

ℝ

n

, the set of

n

-tuplets of elements of

ℝ

can be made into a geometric space in several ways by imposing various geometric

structures

on it. In this entry, we shall explore four such structures.

0.1

ℝ

n

as an affine space

To make

ℝ

n

into an

affine space

, we will specify two

functions

,

D

:

ℝ

×

ℝ

n

×

ℝ

n

→

ℝ

n

and

T

:

ℝ

n

×

ℝ

n

×

ℝ

n

→

ℝ

n

as follows:

D

⁢

(

s

,

(

a

1

,

…

,

a

n

)

,

(

b

1

,

…

,

b

n

)

)

=

(

(

1

-

s

)

⁢

a

1

+

s

⁢

b

1

,

…

,

(

1

-

s

)

⁢

a

n

+

s

⁢

b

n

)

T

⁢

(

(

a

1

,

…

,

a

n

)

,

(

b

1

,

…

,

b

n

)

,

(

c

1

,

…

,

c

n

)

)

=

(

b

1

+

c

1

-

a

1

,

…

,

b

n

+

c

n

-

a

n

)

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

completes

the

parallelogram

with vertices

p

,

q

,

r

.

0.2

ℝ

n

as a Euclidean space

In order to make

ℝ

n

into a Euclidean space, we specify a distance function

d

:

ℝ

n

×

ℝ

n

→

ℝ

as follows:

d

⁢

(

(

a

1

,

…

,

a

n

)

,

(

b

1

,

…

,

b

n

)

)

=

(

a

1

-

b

1

)

2

+

⋯

+

(

a

n

-

b

n

)

2

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 product

.

0.3

ℝ

n

as vector space

In order to make

ℝ

n

into a

vector space

, we specify two maps: the

addition

map

+

:

ℝ

n

×

ℝ

n

→

ℝ

n

and the

scalar product

map

×

:

ℝ

×

ℝ

n

→

ℝ

n

. These maps are defined as follows:

(

a

1

,

…

,

a

n

)

+

(

b

1

,

…

,

b

n

)

=

(

a

1

+

b

1

,

…

,

a

n

+

b

n

)

c

×

(

a

1

,

…

,

a

n

)

=

(

c

⁢

a

1

,

…

,

c

⁢

a

n

)

0.4

ℝ

n

as an inner product space

In order to make

ℝ

n

into an

inner product space

we specify, in addition to the addition map and the scalar product map, another map, namely the inner product map,

⋅

:

ℝ

n

×

ℝ

n

→

ℝ

, which is defined as follows:

(

a

1

,

…

,

a

n

)

⋅

(

b

1

,

…

,

b

n

)

=

a

1

⁢

b

1

+

…

+

a

n

⁢

b

n

1

Relations between structures

Some of these

geometries

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 space

↙

↘

Euclidean space

Vector Space

↘

↙

Inner 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.

The symmetry group is

I

⁢

G

⁢

L

⁢

(

n

,

ℝ

)

, the set of all
inhomogeneous

linear transformations

. (Here and in the next item,
”inhomogeneous” means that

translations

are included in the

transformation group

.)

2.

The symmetry group is

I

⁢

O

⁢

(

n

,

ℝ

)

, the set of all
inhomogeneous

orthogonal transformations

. These include

rotations

,
translations, and

reflections

.

3.

The symmetry group is

G

⁢

L

⁢

(

n

,

ℝ

)

, the set of all

homogeneous

linear transformations.

4.

The symmetry group is

O

⁢

(

n

,

ℝ

)

, the set of all
homogeneous orthogonal transformations.

These different groups act differently on the same underlying space and hence we have different

invariants

. These invariants may be described in

algebraic

and in geometric terms.

3

Generalizations

Although we presented considered

ℝ

n

, most of these constructions also work more generally. In this

section

, we shall discuss under what conditions we can use the same constructions to impose geometric structure on

𝕂

n

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

dimension

n

over a field

k

is

k

n

, the set of

http://planetmath.org/node/6617

n

-tuplets of elements of

k

. The reason that the finite dimensional vector space

k

n

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

(

x

1

⋮

x

n

)

↦

(

m

11

⋯

m

1

⁢

n

⋮

⋱

⋮

m

n

⁢

1

⋯

m

n

⁢

n

)

⁢

(

x

1

⋮

x

n

)

+

(

v

1

⋮

v

n

)

and hence is the natural setting for affine geometry. (For more information on affine geometry, please see section 1.1.2 of the

http://planetmath.org/node/3824

topic entry on geometry.) It is worth mentioning that the name “affine space” is used primarily in geometry and in commutative algebra. (In

algebra

, one does not concern oneself with affine transforms and simply uses the term “affine space” as synonym for “vector space”).

Title

geometrization of

ℝ

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

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