The Hierarchy of Geometries Johar M. Ashfaque
1
Dimension One: Lines
We bring together some very simple and basic facts about the classical one-dimensional geometries.
1.1
The Euclidean Line
The Euclidean line is modeled by the real numbers R with transformation group Ismtr(R) consisting of parallel shifts (x 7→ x + v, v ∈ R) and of reflections with respect to any point. The subset of all parallel shifts is a subgroup of Isom(R), the motion group Ismtr+ (R) of the Euclidean line. To specify an element of Ismtr+ (R), it suffices to indicate any point of the line and its image. The distance between two points x, y ∈ R defined in the standard way by d(x, y) = |x − y| is an invariant of Imstr(R). Given a point x and its image y there are two elements of Imstr(R) that take x to y: one is a parallel shift and the other a reflection in the midpoint of the segment joining x and y.
1.2
The Hyperbolic Line
The hyperbolic line can be modeled by the open interval (−1, 1) with the distance function d(x, y) =
1 | log(h1, −1, x, yi)| 2
where h1, −1, x, yi is the cross ratio of the points 1, −1, x and y h1, −1, x, yi =
|x − 1| |y + 1| · . |x + 1| |y − 1|
Isometries of the hyperbolic line include shifts (the one-dimensional analogs of parallel translations) given by the formula x+v Tv : [−1, 1] → [−1, 1], x 7→ xv + 1 where v is a real number of absolute value less than 1.The composition of any two shifts is a shift and this operation has a physical interpretation in the theory of relativity.
1.3
The Circle
The geometry of the circle S1 is the one-dimensional analog of spherical geometry. Its transformation group, O(2), consists of rotations and symmetries. O(2) contains SO(2) the group of all rotations as a subgroup. To specify an element r ∈ SO(2) it suffices to indicate a single point x ∈ S1 and its image r(x). If one defines the distance between two points on the circle in the natural way (as angle between the radii passing through them), one does not obtain a metric space (the triangle inequality does not hold). However, this distance function supplies the circle with a local metric space structure. 1
There is a natural morphism between the Euclidean line and the circle, namely the exponential covering map given by R 3 ψ 7→ eiψ ∈ S1 . This covering plays a key role in elementary topology and its visualization.
1.4
The Elliptic Line
The elliptic line is the one-dimensional analog of Riemann’s elliptic plane. It can be modeled as the circle with diametrically opposite points identified. Its transformation group Õ(2) consists of rotations and symmetries (any symmetry has two fixed points whose diameters form a ninety degree angle). However, the group Õ(2) is actually isomorphic to O(2) (rotations by ψ corresponding to rotations by 2ψ and symmetries acting in the same way) and it is easy to construct an isomorphism of geometries between the elliptic line and the circle. Thus the elliptic line is just another model of the geometry of the circle.
1.5
The Affine Line
For the points of the affine line Aff1 , we can take the real numbers but its trasnformation group, the affine group Aff(1) is much bigger than Ismtr(R). The group Aff(1) consists of all the transformations of the form x 7→ ax + b where a, b ∈ R and a 6= 0.
1.6
The Projective Line
The projective line RP 1 is obtained from the real line R by adding the point at infinity so that topologically it is the circle. Its transformation group consists of those bijections of R ∪ ∞ that preserve the cross ratio of any four points a, b, c and d [(a − c)/(b − c)] : [(a − d)/(b − d)]. Note that RP 1 can also be defined as the set of lines passing through the origin of the plane R2 . Its transformation group is the group of non-singular two-by-two matrices considered up to multiplication of the columns of the matrix by non-zero constants. These two definitions of the projective line are equivalent.
2
Dimension Two: Planes
We summarize some properties of the classical two-dimensional geometries. We will rely on the not so standard terminology concerning the transformation groups. We say that a transformation group possesses two degrees of freedom if any of its elements is determined by two points and their images.
2.1
The Euclidean Plane
We denote the Euclidean plane by R2 . The plane is non-compact and orientable. Its transformation group Ismtr(R2 ) contains the subgroup of motions Ismtr+ (R2 ) which possesses two degrees of freedom in the sense that any motion is determined by two points and their images. The basic invariant of this geometry is the distance between two points.
2.2
The Sphere
We denote the sphere by S2 . It is compact and orientable. Its transformation group denoted by Ismtr(S2 ) = O(3) contains the subgroup Ismtr+ (S2 ) = SO(3) of orientation-preserving isometries which has two degrees of freedom. The basic invariant of this geometry is the distance between two points. Note that this distance is not the distance induced from the Euclidean space by the standard embedding of the sphere. It is the angular distance, also known as geodesic distance, between points. 2
2.3
The Hyperbolic Plane
We denote the hyperbolic plane by H2 . It is non-compact and orientable. Its transformation group denoted Ismtr(H2 ) contains the subgroup Imstr+ (H2 ) of orientation-preserving isometries which possesses two degrees of freedom. The basic invariant is the distance between two points.
2.4
The Elliptic Plane
We denote the elliptic plane by Ell2 . It is compact and non-orientable. Its transformation group denoted Ismtr(Ell2 ) possesses two degrees of freedom. The basic invariant of this geometry is the distance between two points.
2.5
The Affine Plane
We denote the affine plane by Aff2 . It is non-compact and orientable. Its transformation group denoted by Aff(2) contains the subgroup Aff+ (2) of orientation-preserving affine transformations which has three degrees of freedom. The basic invariant of this geometry is the ratio of three collinear points.
2.6
The Projective Plane
We denote the projective plane by RP 2 . It is compact and non-orientable. Its transformation group denoted by P roj(2) has four degrees of freedom. The basic invarint of this geometry is the cross ration of four collinear points. R2 S2 H2 Ell2 Aff2 RP 2
Compactness No Yes No Yes No Yes
Orientability Yes Yes Yes No Yes No
Degrees of Freedom 2 2 2 2 3 4
3
Invariant Distance Distance Distance Distance Ratio Cross Ratio