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 , and considering a subgroup
of the group , the set
of bijections
of . Objects are subsets of , and we consider two
objects to be equivalent
if there is an such
that .
A property of subsets of is said to be a geometric
property if it is invariant under the action of the group , which
is to say that is true (or false) if and only if is
true (or false) for every transformation . 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 and and any transformation
which belongs to the Euclidean group, the lines and
are orthogonal if and only if and are orthogonal.
However, if we consider affine geometry
, orthogonality is no longer a
geometric property because, given two orthogonal lines and
, one can find a transformation which belongs to the affine
group such that is not orthogonal to .
Invariants can also be numbers. A real-valued function whose
domain consists of subsets of is an invariant, or a
geometrical quantity if the domain of is invariant under
the action of and for all subsets in the
domain of and all transformations . 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
. 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 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 under the action of a group
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 as a vector space along with a metric . The allowed transformations are
bijections that preserve the metric, that is, for all . Such maps are called
isometries
, and the group is often denoted by . Defining a norm by , for , we obtain a notion of length or distance
.
We can also define an inner product on using the standard dot
product
(this induces the same norm which can now be defined as
).
An inner product leads to a definition of
the angle between two vectors to be
It is clear that since isometries preserve the metric, they preserve distance and angle. As an example, it can be shown
that the group consists of translations
, reflections
, glides, and rotations. In
general, a member of has the form , where is an orthogonal matrix and .
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
() in . 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
, 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 , 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 and
dimensional hyperplanes. A typical tranformation in an affine
geometry is called an affine
transformation (http://planetmath.org/AffineTransformation):
, where
and is an invertible
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 . We can then create
the projective plane
where for each line , 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 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 where V is a
vector space. When , we take the quotient of where for . The allowed transformations is
the group , 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 . 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 |