geometry as the study of invariants under certain transformations

geometry as the study of invariants under certain transformations

An approach to

geometry

first formulated by Felix Klein in his
Erlangen lectures is to describe it as the study of

invariants

under
certain allowed

transformations

. This involves taking our space as a
set

S

, and considering a

subgroup

G

of the group

B

⁢

i

⁢

j

⁢

(

S

)

, the set
of

bijections

of

S

. Objects are subsets of

S

, and we consider two
objects

A

,

B

⊂

S

to be

equivalent

if there is an

f

∈

G

such
that

f

⁢

(

A

)

=

B

.

A property

P

of subsets of

S

is said to be a

geometric
property

if it is invariant under the action of the group

G

, which
is to say that

P

⁢

(

S

)

is true (or false) if and only if

P

⁢

(

g

⁢

(

S

)

)

is
true (or false) for every transformation

g

∈

G

. For example, the
property of being a

straight line

is a geometric property in

Euclidean
geometry

. Note that the question whether or not a certin property is
geometric depends on the choice of group. For

instance

, in the case
of Euclidean geometry, the property of

orthogonality

is geometric
because, given two lines

L

1

and

L

2

and any transformation

g

which belongs to the

Euclidean group

, the lines

g

⁢

(

L

1

)

and

g

⁢

(

L

2

)

are

orthogonal

if and only if

L

1

and

L

2

are orthogonal.
However, if we consider

affine geometry

, orthogonality is no longer a
geometric property because, given two

orthogonal lines

L

1

and

L

2

, one can find a transformation

f

which belongs to the

affine
group

such that

f

⁢

(

L

1

)

is not orthogonal to

f

⁢

(

L

2

)

.

Invariants can also be numbers. A real-valued function

f

whose
domain consists of subsets of

S

is an invariant, or a

geometrical quantity

if the domain of

X

is invariant under
the action of

G

and

f

⁢

(

X

)

=

f

⁢

(

g

⁢

(

X

)

)

for all subsets

X

in the
domain of

f

and all transformations

g

∈

G

. Familiar examples
from Euclidean geometry are the length of

line segments

,

areas of
triangles

, and angles. An important feature of the group-theoretic
approach to geometry is that one one can use the techniques of
invariant theory to systematically find and classify the invariants of
a geometrical system. Using this approach, one can start with the
description of a geometrical system in terms of a set and a group and
rediscover geometric quantities which were originally found by trial
and error.

One is not always interested in considering all possible subsets of

S

. For instance, in

algebraic geometry

, one only cares about
subsets which can be defined by sytems of

algebraic equations

. To
accommodate this desire, one may revise Klein’s definition by
replacing the set

S

with a suitable

category

(such as the category
of

algebraic

subsets) to obtain the definition “geometry is the study
of the invariants of a category

C

under the action of a group

G

which acts upon this category.” Not only is such an approach popular
in contemporary algebraic geometry, it is also useful when discussing
such phenomena as

duality

transforms which map a point in one space to
a line in another space and vice-versa. Such a phenomenon is not
easily accomodated in a set-theoretic

framework

, but in terms of

category theory

, the duality transform can be described as a

contravariant functor

.

Klein’s definition provides an organizing principle for classifying
geometries. Ever since the discovery of

non-Euclidean geometry

,
geometers have been defined and studied many different geometries.
Without an organizing principle, the discussion and comparison of
these geometries could become confusing. In the next

section

, we
shall describe several familiar geometric systems from the standpoint
of Klein’s definition.

0.1

Basic examples

0.1.1

Euclidean geometry

Euclidean geometry deals with

ℝ

n

as a

vector space

along with a metric

d

. The allowed transformations are
bijections

f

:

ℝ

n

→

ℝ

n

that preserve the metric, that is,

d

⁢

(

𝒙

,

𝒚

)

=

d

⁢

(

f

⁢

(

𝒙

)

,

f

⁢

(

𝒚

)

)

for all

𝒙

,

𝒚

∈

ℝ

n

. Such maps are called

isometries

, and the group is often denoted by

Iso

⁡

(

ℝ

n

)

. Defining a norm by

|

x

|

=

d

⁢

(

𝒙

,

𝟎

)

, for

𝒙

∈

ℝ

n

, we obtain a notion of length or

distance

.
We can also define an

inner product

⟨

𝒙

,

𝒚

⟩

=

𝒙

⋅

𝒚

on

ℝ

n

using the standard

dot
product

(this induces the same norm which can now be defined as

|

x

|

=

⟨

𝒙

,

𝒙

⟩

).
An inner product leads to a definition of
the angle between two vectors

𝒙

,

𝒚

∈

ℝ

n

to be

∠

⁢

𝒙

⁢

𝒚

=

cos

-

1

⁡

(

⟨

𝒙

,

𝒚

⟩

|

𝒙

|

⋅

|

𝒚

|

)

.

It is clear that since isometries preserve the metric, they preserve distance and angle. As an example, it can be shown
that the group

Iso

⁡

(

ℝ

2

)

consists of

translations

,

reflections

, glides, and

rotations

. In
general, a member

f

of

Iso

⁡

(

ℝ

n

)

has the form

f

⁢

(

𝒙

)

=

𝑼

⁢

𝒙

+

𝒄

, where

𝑼

is an orthogonal

n

×

n

matrix and

𝒄

∈

ℝ

n

.

0.1.2

Affine geometry

Unlike Euclidean geometry, we are no longer bound to “

rigid motion

”
transformations in affine geometry. Here, we are interested in what
happens to geometric objects when they undergo a finite series of
“parallel projections”. For example, imagine two

Euclidean planes

(

ℝ

2

) in

ℝ

3

. Loosely speaking, Euclidean
geometry deals with transformations that take objects from one plane
to the other, when the planes are

parallel

to each other. In
affine geometry, the transformation is between two copies of

ℝ

2

, but they are no longer required to be parallel to each
other anymore. Objects from one plane will appear to be “stretched”
in the other. A circle will turn into an

ellipse

, etc…

For

ℝ

2

, in terms of the Kleinian view of geometry, affine
geometry consists of the ordinary Euclidean plane, together with a
group of transformations that

1.

map straight lines to straight lines,

2.

map

parallel lines

to parallel lines, and

3.

preserve ratios of lengths of line segments along a given
straight line.

Of course, the properties can be generalized to

ℝ

n

and

n

-

1

dimensional

hyperplanes

. A typical tranformation in an affine
geometry is called an

affine
transformation

(

http://planetmath.org/AffineTransformation

):

T

⁢

(

𝒙

)

=

𝑨

⁢

𝒙

+

𝒃

, where

x

∈

ℝ

n

and

𝑨

is an

invertible

n

×

n

real matrix

.

0.1.3

Projective geometry

Projective geometry

was motivated by how we see objects in everyday
life. For example, parallel train tracks appear to meet at a point far
away, even though they are always the same distance apart. In
projective geometry, the

primary

invariant is that of incidence. The
notion of parallelism and distance is not present as with Euclidean
geometry. There are different ways of approaching projective
geometry. One way is to add points of

infinity

to Euclidean space. For
example, we may form the

projective line

by adding a point of infinity

∞

, called the ideal point, to

ℝ

. We can then create
the

projective plane

where for each line

l

∈

ℝ

2

, we
attach an ideal point, and two

ordinary lines

have the same ideal
point if and only if they are parallel. The projective plane then
consists of the

regular

plane

ℝ

2

along with the ideal
line, which consists of all ideal points of all ordinary lines. The
idea here is to make central projections from a point sending a line
to another a

bijective

map.

Another approach is more algebraic, where we form

P

⁢

(

V

)

where V is a
vector space. When

V

=

ℝ

n

, we take the

quotient

of

ℝ

n

+

1

-

{

0

}

where

v

∼

λ

⋅

v

for

v

∈

ℝ

n

,

λ

∈

ℝ

. The allowed transformations is
the group

P

⁢

G

⁢

L

⁢

(

ℝ

n

+

1

)

, which is the

general linear group

modulo the subgroup of

scalar matrices

.

0.1.4

Spherical geometry

Spherical geometry

deals with restricting our attention in Euclidean
space to the

unit sphere

S

n

. The role of straight lines is taken by

great circles

. Notions of distance and angles can be easily developed,
as well as spherical laws of cosines, the

law of sines

, and

spherical
triangles

.

Title

geometry as the study of invariants under certain transformations

Canonical name

GeometryAsTheStudyOfInvariantsUnderCertainTransformations

Date of creation

2013-03-22 18:00:29

Last modified on

2013-03-22 18:00:29

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Topic

Classification

msc 51-01

Classification

msc 51-00

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