geometry as the study of invariants under certain transformations

An approach to geometry first formulated by Felix Klein in hisErlangen lectures is to describe it as the study of invariants undercertain allowed transformations. This involves taking our space as aset $S$, and considering a subgroup $G$ of the group $Bij(S)$, the setof bijections of $S$. Objects are subsets of $S$, and we consider twoobjects $A,B\subset S$ to be equivalent if there is an $f\in G$ suchthat $f(A)=B$.

A property $P$ of subsets of $S$ is said to be a geometricproperty if it is invariant under the action of the group $G$, whichis to say that $P(S)$ is true (or false) if and only if $P(g(S))$ istrue (or false) for every transformation $g\in G$. For example, theproperty of being a straight line is a geometric property in Euclideangeometry. Note that the question whether or not a certin property isgeometric depends on the choice of group. For instance, in the caseof Euclidean geometry, the property of orthogonality is geometricbecause, 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 ageometric property because, given two orthogonal lines $L_1$ and$L_2$, one can find a transformation $f$ which belongs to the affinegroup such that $f(L_1)$ is not orthogonal to $f(L_2)$.

Invariants can also be numbers. A real-valued function $f$ whosedomain consists of subsets of $S$ is an invariant, or ageometrical quantity if the domain of $X$ is invariant underthe action of $G$ and $f(X)=f(g(X))$ for all subsets $X$ in thedomain of $f$ and all transformations $g\in G$. Familiar examplesfrom Euclidean geometry are the length of line segments, areas oftriangles, and angles. An important feature of the group-theoreticapproach to geometry is that one one can use the techniques ofinvariant theory to systematically find and classify the invariants ofa geometrical system. Using this approach, one can start with thedescription of a geometrical system in terms of a set and a group andrediscover geometric quantities which were originally found by trialand error.

One is not always interested in considering all possible subsets of$S$. For instance, in algebraic geometry, one only cares aboutsubsets which can be defined by sytems of algebraic equations. Toaccommodate this desire, one may revise Klein’s definition byreplacing the set $S$ with a suitable category (such as the categoryof algebraic subsets) to obtain the definition “geometry is the studyof the invariants of a category $C$ under the action of a group $G$which acts upon this category.” Not only is such an approach popularin contemporary algebraic geometry, it is also useful when discussingsuch phenomena as duality transforms which map a point in one space toa line in another space and vice-versa. Such a phenomenon is noteasily accomodated in a set-theoretic framework, but in terms ofcategory theory, the duality transform can be described as acontravariant functor.

Klein’s definition provides an organizing principle for classifyinggeometries. 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 ofthese geometries could become confusing. In the next section, weshall describe several familiar geometric systems from the standpointof 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 arebijections $f:{ℝ}^n\to {ℝ}^n$ that preserve the metric, that is, $d(𝒙,𝒚)=d(f(𝒙),f(𝒚))$ for all $𝒙,𝒚\in {ℝ}^n$. Such maps are calledisometries, and the group is often denoted by $\mathrm{Iso}({ℝ}^n)$. Defining a norm by $|x|=d(𝒙,\mathrm{𝟎})$, for $𝒙\in {ℝ}^n$, we obtain a notion of length or distance.We can also define an inner product $⟨𝒙,𝒚⟩=𝒙\cdot 𝒚$ on ${ℝ}^n$ using the standard dotproduct (this induces the same norm which can now be defined as$|x|=\sqrt{⟨𝒙,𝒙⟩}$).An inner product leads to a definition ofthe angle between two vectors $𝒙,𝒚\in {ℝ}^n$ to be$\mathrm{\angle }𝒙𝒚={\cos }^{-1}\left({\displaystyle \frac{⟨𝒙,𝒚⟩}{|𝒙|\cdot |𝒚|}}\right).$ It is clear that since isometries preserve the metric, they preserve distance and angle. As an example, it can be shownthat the group $\mathrm{Iso}({ℝ}^2)$ consists of translations, reflections, glides, and rotations. Ingeneral, a member $f$ of $\mathrm{Iso}({ℝ}^n)$ has the form $f(𝒙)=𝑼𝒙+𝒄$, where $𝑼$ is an orthogonal $n\times n$ matrix and $𝒄\in {ℝ}^n$.

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 whathappens 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, Euclideangeometry deals with transformations that take objects from one planeto the other, when the planes are parallel to each other. Inaffine geometry, the transformation is between two copies of${ℝ}^2$, but they are no longer required to be parallel to eachother 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, affinegeometry consists of the ordinary Euclidean plane, together with agroup 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 givenstraight line.

Of course, the properties can be generalized to ${ℝ}^n$ and$n-1$ dimensional hyperplanes. A typical tranformation in an affinegeometry is called an affinetransformation (http://planetmath.org/AffineTransformation):$T(𝒙)=𝑨𝒙+𝒃$, where$x\in {ℝ}^n$ and $𝑨$ is an invertible $n\times n$real matrix.

0.1.3 Projective geometry

Projective geometry was motivated by how we see objects in everydaylife. For example, parallel train tracks appear to meet at a point faraway, even though they are always the same distance apart. Inprojective geometry, the primary invariant is that of incidence. Thenotion of parallelism and distance is not present as with Euclideangeometry. There are different ways of approaching projectivegeometry. One way is to add points of infinity to Euclidean space. Forexample, we may form the projective line by adding a point of infinity$\mathrm{\infty }$, called the ideal point, to $ℝ$. We can then createthe projective plane where for each line $l\in {ℝ}^2$, weattach an ideal point, and two ordinary lines have the same idealpoint if and only if they are parallel. The projective plane thenconsists of the regular plane ${ℝ}^2$ along with the idealline, which consists of all ideal points of all ordinary lines. Theidea here is to make central projections from a point sending a lineto another a bijective map.

Another approach is more algebraic, where we form $P(V)$ where V is avector space. When $V={ℝ}^n$, we take the quotient of ${ℝ}^{n+1}-\{0\}$ where $v\sim \lambda \cdot v$ for $v\in {ℝ}^n,\lambda \in ℝ$. The allowed transformations isthe group $PGL({ℝ}^{n+1})$, which is the general linear groupmodulo the subgroup of scalar matrices.

0.1.4 Spherical geometry

Spherical geometry deals with restricting our attention in Euclideanspace to the unit sphere $S^n$. The role of straight lines is taken bygreat circles. Notions of distance and angles can be easily developed,as well as spherical laws of cosines, the law of sines, and sphericaltriangles.