Abh. Math. Sem. Univ. Hamburg 60 (1990), 61-69
2-dimensional Minkowski Planes and Desarguesian Derived Altine Planes By...

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Steinke G.F.

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Abh. Math. Sem. Univ. Hamburg 60 (1990), 61-69

2-dimensional Minkowski Planes and Desarguesian Derived Altine Planes By G.E STEINKE

1 Introduction and Results A Minkowski plane J / / = (P, ~ , {[[+, [I-}) consists of a set P of points, a set ~ff of circles (considered as subsets of P) and two equivalence relations [[+ and [[_ on P (parallelism) such that for every point p the derived incidence structure dp = (Ap, dfp) is an affine plane (derived affine plane at p) where Ap consists of all the points not parallel to p (i.e. of all points that are neither (+)-parallel nor (-)-parallel to p), and ~ p consists of all circles through p (without the point p) and all parallel classes (i.e. [[+-equivalence classes or [[_-equivalence classes) not passing through p. In particular there is a unique circle through three pairwise nonparallel points. The classical model of a Minkowski plane is obtained as the geometry of plane sections of a nondegenerated quadric of index 2 in a 3-dimensional projective space over a commutative field; in this case dp is a desarguesian plane. These Minkowski planes can be characterized by the theorem of MIQUEL and, therefore, are also called miquelian Minkowski planes. In [2] it was shown that a finite Minkowski plane of odd order with at least one desarguesian derived affine plane is miquelian; this result is due to the remarkable theorem of SEGRE that every oval in a finite desarguesian affine plane of odd order is indeed a quadric. For finite Minkowski planes of even order HEISE [5] proved that they are all miquelian. In particular all finite Minkowski planes of order _ 8 are miquelian. Generally, a Minkowski plane is miquelian if and only if every derived affine plane is desarguesian and if every translation in any derived affine plane that fixes all (+)-parallel classes is induced from an automorphism of the Minkowski plane (see [4], the same result is true for (-)-parallel classes). There is an analogon to the result on finite Minkowski planes if one imposes topological conditions. To begin with, a topological Minkowski plane is a Minkowski plane Jg = (P,o~ff, {[[+, [[_}) where P and off carry topologies such that the geometric operations of joining three pairwise nonparallel points by a circle, of forming touching circles, and intersection of parallel classes and circles are continuous operations in their domain of definition (compare [7]). It follows from [6] w and [8] that in the case that P is locally

62

G.E Steinke

compact, connected, and of finite topological dimension, the dimension of P can only be either 2 or 4. For short such planes are called 2-dimensional Minkowski planes (dim P = 2) or 4-dimensional Minkowski planes (dim P = 4), respectively. Similarly to finite Minkowski planes of odd order a 4dimensional Minkowski plane with at least one desarguesian derived affine plane is classical (see L6WEN [6] 2.5). This is a consequence of the beautiful result of BUCHANAN [1] which in analogy to the mentioned theorem of SEGRE proves that a closed oval in the complex desarguesian projective plane is already a conic. For abbreviation we call a point p of a Minkowski plane J/{ desarguesian, if the derived affine plane at p is desarguesian, and we denote the set of all desarguesian points of J / by ~ . In this note we consider 2-dimensional Minkowski planes exclusively and ask whether such a plane is classical if it has sufficiently many desarguesian points. The answer lies in between the general result of HARTMANN [4] in which, in particular, every point has to be desarguesian and the above mentioned result of L6WEN [6] for 4-dimensional Minkowski planes in which one single desarguesian point suffices. We first state the worst possible case: In [3] HARTMANN constructed a family of real Minkowski planes which actually are topological 2-dimensional Minkowski planes, although this last fact was not mentioned. This family contains a subfamily of Minkowski planes such that precisely the derived affine planes at points parallel to one special point are all desarguesian. HARTMANN'S examples depend on two real parameters rl, r2 > 0 (the classical real Minkowski plane is obtained only for rl = r2 = 1). The subfamily mentioned consists of those models with rl = r2 :fi 1. We give a short description of these Minkowski planes: Fix r > 0 and let f r ( X ) ~" XIX[ - r - 1 for x E IR, x :fi 0. The point space of the Minkowski plane Jl(r) is P = (N U {~}) x (R U {oo}) ; circles of J/g(r) are of the form

{(x,y) ~ Z l y

-=mx+t}U{(o%c~)}

(m, t E ~ . , m ~ O )

or of the form {(x,y) E ~2 l Y = a f r ( x - b ) + c ,

x ~ b}U{(b, oo),(oo, c)}

(a,b,c E ~., a -~ 0).

Two points (x, y), (u, v) E P are (+)-parallel if and only if x = u and they are (-)-parallel if and only if y = v. Desarguesian points of J/g(r), r :fi 1, are precisely the infinite points (~,y) and (x, oo), x,y E R U {oo}; Jr is the classical Minkowski plane, in this case all points are desarguesian. With this notation we prove: Theorem. Let J~ = (P, ~'~, { ][+, ]]_}) be a 2-dimensional Minkowski plane. a) I f the set ~ of desarguesian points of J/t contains one (+)-parallel class or one (-)-parallel class and at least one further point, then ~ is isomorphic to one of the Minkowski planes J#(r) of HARTMANN.

2-dimensional Minkowski Planes

63

b) I f ~ contains one (+)-parallel class or one (-)-parallel class and at least three pairwise nonparallel points, then sg is isomorphic to the classical real Minkowski plane.

2

Comparison of Derived Attine Planes at two Nonparallel Desarguesian Points

Let Jr = (P,J~c,{l[+, [I-}) be a 2-dimensional Minkowski plane. We choose three pairwise nonparallel points p0, pl, q such that Po, pl E 9 . For i = 0, 1 we denote the derived affine plane of J// at pi by d i . We coordinatize d i such that Pl-i becomes the point (0,0), q becomes the point (1, 1) and such that the (-)-parallel class through Pl-i and the (+)-parallel class through Pl-i become the X- and Y-axis, respectively. We compare both coordinate systems on those points that belong to both derived affine planes (i.e. points that are not parallel to P0 and to pl ). As usual, we identify the Y -axis with IR ; since the derived affine planes are desarguesian the ternary operation on the Y-axis given by the coordinatization comes from the usual addition and multiplication in IR. We express the coordinates of ~ 1 in the coordinates of ~ 0 . To do so, let f : ~,. \ {0}

) ~. \ {0}

be a bijection such that the point (1,y) @ (1,0) in the coordinates of d 0 has the coordinates (1,f(y)) in a l l . From the topological properties of s// it follows that f is a homeomorphism of ~ \ {0} that can be extended to a homeomorphism F of the one-point compactification lRtA {~} -~ S 1 (1-sphere) of IR by F ( ~ ) = 0 and F(0) = 0o. By the choice of q one has F(1) = f(1) = 1. Since in both affine planes the line of slope 1 through the origin is represented by the same circle through the three points P0,Pl, q, we obtain the coordinate transformation qo : ( ~ \ {0}) x ( ~ \ {0})

,(~\{O})x(~\{O}):(x,y),

,(f(x),f(y))

which transforms the coordinates of ~ 1 into the coordinates of d 0 . This transformation r extends to : P ---* P : (x,y),

, (F(x),F(y))

on the whole Minkowski plane J//. We look at the multiplication in the coordinate fields. Since the point q has the coordinates (1, 1) in both affine planes and because parallels to the X-axis or Y -axis are the same in both derived affine planes the multiplication of two points ~ p0, pl on the respective Y-axis yields essentially the same point, i.e. the resulting points in ~r and d l are (-)-parallel and their coordinates are correlated by the mapping f . Therefore we obtain a functional equation for f , namely (1)

f(xy)

=

f(x)f(y)

for all

x,y c ~ \ {0}.

64

G.E Steinke

The continuous solutions of (1) are well known. These are

fr(x)

(2)

=

x" Ixl - ' - a

for

x ~ 0, where

r E I/.

From F(0) = oo we infer that r > 0. Circles of ~ passing through the point P0 are lines in ~r in the coordinates of d 0 extended to the Minkowski plane J / / t h e y can be described as

{(x,y)~e

[y=rnx+t}

(re, t E N . , m ~ O )

where we use the convention m. oo = oo. m = oo and oo + t = t + oo = oo for m, t E R , m 7~ 0. Similarly, circles of J / passing through the point pl can be expressed in the coordinates of d l and, using ~o, even in the coordinates of ~r We obtain {(x,y) E P l y = F - ~ ( m F ( x ) + t ) }

(m, t E F,~, m-J=O).

Therefore, if we know the parameter r of F in (2), we know all circles of J/g through the point Po = (o0, oo) or through the point Pl = (0, 0). In the sequel we fix three points po, pl,p2 E ~ such that pl,p2 are not parallel to Po and look on the parameters rl and r2 of the coordinate transformation functions corresponding to the derived affine planes at po, pl and Po, P2 respectively. By considering certain functional equations we obtain rl = r2, and, if all three points are pairwise nonparallel, we even obtain /'1 ~

3

r2 ~

1.

Three Pairwise Nonparallel Desarguesian Points

Suppose that @ contains three pairwise nonparallel desarguesian points po, pl,p2. We may assume that in the desarguesian coordinates of the derived affine plane ~ 0 at P0 the points Pl,P2 have the coordinates (0,0) and (a, b) respectively, where a, b @ 0, oo. According to 2 there is a coordinate transformation corresponding to the derived affine planes at P0 and Pl with parameter r > 0 such that circles passing through Pl = (0, 0) can be described by {(x,y) E P l y = F - l ( m F ( x ) + t ) } (re, t E N . , m(=O) where F denotes the extension of fr to IR U {oo}. With the help of the translation (x,y), , ( x - a , y - b ) we similarly can describe circles passing through the point p2 = (a, b) by {(x,y) E e l Y = G - l ( m G ( x - - a ) + t ) + b }

(m, t E R , m ~ O)

where G denotes the extension of fs, s > 0 suitable, to R U {oo}.

2-dimensional Minkowski Planes

65

In either description we determine the unique circle K through the pairwise nonparallel points (0,0), (a,b) and ( - a , oo). With the multiplicative law (1) we obtain: K

=

{(x,y) E P I Y = (b/F-l(2)) "F-I(F(x/a) + 1)}

K

=

{(x,y) E P l Y = (b/G-l( 1 - G(2))). G - l ( G ( ( x / a ) - 1) + G(2)) + b } .

and

Replacing x/a by u we therefore have (3) F-I(F(u) + 1 ) / F - l ( 2 )

=

G(2))/G-I(1 - G(2)) + 1

6-1(6(u

- 1) +

for all

uERU{oo}.

For u = 2 equation (3) gives (note that F(1) = G(1) = 1): F - l ( F ( 2 ) + 1)/F-1(2) = G-l(1 + G(2))/G-I(1 - G(2)) + 1. Setting u = 1/2 in (3) and using (1) we similarly find: F - I ( F ( 2 ) + 1 ) / F - l ( 2 ) = G-1(G(4) - 1)/G-l(1 - G(2)) + 2. This results in the equation G-1(1 + 6(2)) = G-~(G(4) - 1)) + 6-1(1 - G(2)) into which only one of the two unknown functions enters. Since G(4) - 1 = (6(2) - 1)(G(2) + 1) the last equation, after division by G-1(6(4) - 1) and multiplication by 2, yields

1/G-1(G(1/2)- 1) + 1

(4)

1/G-1(G(1/2) + 1) - 1.

=

Let h+ : (0, +00) h_ : (0, + ~ )

) (0, +oo)

:

t~

) (2 t - 1) 1/t-l- 1

) (0, +oo)

:

tt

) (2 t + l )

1/t-l;

then h+, h_ are differentiable functions. Moreover, acccording to (4), these functions coincide on s (the parameter of G): (4')

h+(s)

= 1/6--1(6(1/2)

-- 1) qt_ l = 1 / 6 - - 1 ( 6 ( 1 / 2 )

+ 1) -- 1 = h _ ( s ) .

It is easy to see that the derivate o f h+ is strictly positive and that the derivative o f h_ is strictly negative; hence h+ is strictly increasing on (0, +oo) and h_ is strictly decreasing on (0, +oo). Therefore there is only one solution o f the equation h+(t) = h_(t), namely t = s according to (4'). But h+(1) = 2 = h_(1), thus s = 1. For u = oo in (3) we finally obtain 1/F-~(2) = 2/6-10

- 6(2)) + 1 = 2.

F r o m this and the definition o f F it follows 1/2 = F-1(2) = 2 -1/r ; hence r = 1. In particular F = G.

66 4

G.E Steinke Three Desarguesian Points of which Precisely Two Are (-)-Parallel

Suppose that ~ contains three desarguesian points pi (i = 0, 1, 2) such that pl and P2 are (-)-parallel and p0 is not parallel to Pl (and P2). We may assume that in the desarguesian coordinates of the derived affine plane d o at Po the points pl,p2 have the coordinates (0,0) and (a,0), a ~ 0, 0o, respectively. Again, there are parameters r, s > 0 such that circles passing through the point pt = (0, 0) have the form {(x,y) E P l y = f - l ( m F ( x ) + t ) }

(m, t E N . , m--riO)

where F denotes the extension of fr to ~ U {co}, and circles passing through the point P2 = (a, 0) have the form {(x,y) e P l y = G - l ( m G (

x-a)+t)}

(m, t e lR, m r

where G denotes the extension of fs to I1 u {co}. Without loss of generality we may assume that r < s (otherwise we interchange the role of PI and p2). Contrary to the case in the previous chapter there is no circle through Pl and p2 because these two points are parallel; therefore we cannot compare different descriptions of the same circle in order to prove F = G. Instead, we consider the circles through the points ( - a , co), (co, 1), (0, 0) and ( - a , o0), (0o, 1), (a, 0) respectively. From the condition that these two circles have only the points ( - a , co) and (co, 1) in common we shall infer the desired result F=G. The circles K~ passing through ( - a , co), (co, l) and Pl = (0,0) and Kz passing through ( - a , co), (co, 1) and p2 = (a,0) are described by K1

=

{(x,y) e P [Y = F - l ( F ( x / a ) + t)}

K2

=

{(x,y) e P i Y = G - I ( G ( ( x / a ) - 1)/G(2) + 1)}.

Replacing x / a by u we then have the condition that the equation (5)

F-I(F(u) + 1) = G-I(G((u - 1)/2) + 1)

has the only solutions u = 0o and u = - 1 . Let H = F o G-l . Then H(u) = u. [u[t-1 for u 6 R , and H ( ~ ) = oo ; here t = r/s, thus 0 < t < 1. With this notation and the coordinate transformation u ~ G(u) we obtain from (5) that the equation

H - I ( H ( z ) + 1) = 1 + G((G-I(z) - 1)/2) has the only solutions z = - 1 and z = 0. We abbreviate the left side by L(z) and the right side by R(z) and consider the restrictions of these two mappings on the interval I = [0, 1). They are continuous in 1 with image c R and they are continuously differentiabte in I' = (0, 1). An easy conputation shows L(O) -- R(O) = 1

2-dimensional Minkowski Planes L(z) = (z t + l) l/t,

L(1) = H -l (2) E IR,

R(z) = l + 2S(z - 1 ~ s - l ) -s

67 (zEI')

lim R(z) = +oo z---~ 1

L'(z) = (z t

+ 1)-l+l/tzt-l,

R'(z) = 2s(1 -- z-l/s) -s-1

(z E I')

R'(O) = 2 ~ > 1

lira U ( z ) = lim z t-1 = { 1 z-~0 z--,0 +oo

for

t= 1 0< t< 1

If 0 < t < 1, then L(0) = R(0) and L'(0) > R'(0) ; hence L(z) > R(z) for all z > 0 sufficiently close to 0. But L(z) < R(z) for z < 1 sufficiently close to 1. By continuity of L and R there is a point z0 E I t such that L(zo) = R(zo) contrary to the condition that the equation L(z) = R(z) has only the solutions z=-I andz=0. Therefore t = 1, thus r = s and F = G. 5

Proof of the Theorem

Suppose that ~ contains a parallel class H of desarguesian points and a further desarguesian point P0 r 17. Without loss of generality we may assume that 11 is the (-)-parallel class of the point p~ I1+ P0 (otherwise we interchange the role of II+ und 1[_). We choose one point pl ~ l-I \ {p~} and coordinatize the derived affine plane at p0 as in 2 such that P0 becomes the point (~, ~ ) and Pl becomes the point (0,0). Then the parallel class 11 consists of all points (a,0), a E R U {~}. Let F be the extension of fr to R U {~} which corresponds to the coordinate transformation of the derived affine planes at p0 and pl (compare 2). From 4 we know that up to translation the same F corresponds to the coordinate transformation of the derived affine planes at P0 and p E I I \ {p~}. Therefore circles not passing through the point poo have the form {(x,y) E P [ y = F - l ( m F ( x - a ) + t ) }

(rn, t E R ,

[email protected]).

It remains to determine the form of the circles through p~ = (~,0) or equivalently the lines of the derived affine plane at p~. After the coordinate transformation (x, y), , (x, F(y)) the point p~ becomes (o% oo) and circles not passing through (0% oo) are of the form K(a,t,m)

=

{(x,y) E ( ~ U { o o } ) • (a,t, m E R ,

mr

We fix m, t E ~-., m 5~ 0, and choose z ~ R \ {0, - 1 , + 1}. Then we consider the two points q0 = (0, t) and qz = ( z , - ( m / r ) F ( z - 1) + t - re~r) on the circle K = K ( 1 , t - - ( m / r ) , - - m / r ) and the third point PI/~ = ( 1 / z , ~ ) not on K. Since

68

G.F. Steinke

z 4: 0, ~ , __+1 these three points are pairwise nonparallel, hence there exists a unique circle Lz through these points; moreover, because of z :~ 0, the circle passes not through ( ~ , ~ ) and therefore Lz is of the form K(a',t',m') for suitable a', t', m' E R , m' ~ O. An easy computation shows that a'

=

1/z

t'

=

t - - ( m / r ) . ( F ( z - - 1) + 1)/(F(z 2 - 1) + 1)

m'

=

--(m/r) . r ( z ) . ( V ( z - - 1) + 1)/(F(z 2 - 1) + 1).

By definition Lz intersects K in the two points q0 and qz where qz converges to qo for z --* 0; equally Pl/z converges to poo for z ~ 0. Since the point space P is compact and connected, the Minkowski plane J / is K 1-coherent (see [7] 3.16) which means in our situation that the circles Lz converge to the unique circle L through poo = lim pl/z which is tangential to K at the z---*0

l

point q0 = lira qz (i.e. L and K have only the point qo in common); for a z---*0

precise definition of Kl-coherence see [7] 2.1. We calculate the limit circle L pointwise; it is the graph {(x, h(x)) I x 6 R U {oo}} of a suitable mapping h : I1 u {oo} --. R u {oo}. Since L passes through p~ = (oo,oo) and q0 = (0, t), we have h(oo) = oo and h(0) = t. Let Hz(x) = m'F(x - a') + t', i.e. Lz = {(x, Hz(x)) I I1 U {oe}}. Using the formulae for a', t', m' we rewrite Hz (x) for x 7~ 0, oo as follows H~ (x) =

m

F(z-1)+l

r

z2

F(xz-1)+l

F(z 2 -- 1) + 1

z

xz

x + t.

Since F is continuously differentiable in - 1 , de l'Hospital's rule may be applied to obtain ~i~(F(u - 1) + 1)/u = u--,01imF'(u -- 1) = F'(--1) = --r.

Therefore h(x) = lim Hz(x) = m . x + t ; z-"*O

hence in the given coordinate system lines of the derived affine plane at poo are just the usual euclidean lines. So we have the following description of the circles of J / : they are of the form {(x,y) E P l y = m x + t }

(m, tEP,~, m4=O)

or of the form {(x,y) E P l y = m F ( x - a ) + b }

(a,b, m E R ,

m~O)

where F(x)

=

x" ]xl -r-1

F(0)

=

oo

F(oo)

=

0.

for

x E I 1 \ {0},

2-dimensional Minkowski Planes

69

This shows that J / / i s isomorphic to the Minkowski plane ~/(r) of HARTMANN described in the introduction. If ~ contains a further desarguesian point p E I I not (+)-paraUel to p0, then ~ contains three pairwise nonparallel points (viz. p, P0 and a point p' E I-I). As shown in 3 then r = 1, and M/ is isomorphic to J//(1) which is the classical real Minkowski plane. References

[1] T. BUCHANAN,Ovale und Kegelschnitte in der komplexen projektive Ebene. Math. Phys. Sem. Ber. 26 (1979), 244-260. [2] Y. CHEN, G. KAERLEIN,Eine Bemerkung fiber endliche Laguerre- und MinkowskiEbenen, Geom. Dedicata 2 (1973), 193-194. [3] E. HARTMANN,Beispiele nicht einbettbarer reeller Minkowskiebenen, Geom. Dedicata 10 (1973), 155-159. [4] E. HARTMANN,Minkowski-Ebenen mit Transitivitiitseigenschaften, Result. Math. 5 (1982), 136-148. [5] W. HEISE,Minkowski-Ebenen gerader Ordnung, J. Geom. 5 (1974), 83. [6] R. L6WEN, Projectivities and the Geometric Structure of Topological Planes, In: Geometry - - von Staudt's Point of View (ed. by P. Plaumann and K. Strambach), Bad Windsheim 1981, pp. 339-372. [7] A. SCHENKEL,Topologische Minkowski-Ebenen. Dissertation, Erlangen-Niirnberg 1980. [8] G.F. STEINKE,Topological Properties of Locally Compact Connected Minkowski Planes and their Derived Affine Planes, Geom. Dedicata 32 (1989), 341-351.

Eingegangen am 05.05.1989 in revidierter Fassung am 12.06.1989

Author's address: Giinter E Steinke, Mathematisches Seminar der Universitiit Kiel, Ludewig-Meyn-Str. 4, D-2300 Kiel 1, Federal Republic of Germany

2-dimensional Minkowski Planes and Desarguesian Derived Altine Planes By G.E STEINKE

1 Introduction and Results A Minkowski plane J / / = (P, ~ , {[[+, [I-}) consists of a set P of points, a set ~ff of circles (considered as subsets of P) and two equivalence relations [[+ and [[_ on P (parallelism) such that for every point p the derived incidence structure dp = (Ap, dfp) is an affine plane (derived affine plane at p) where Ap consists of all the points not parallel to p (i.e. of all points that are neither (+)-parallel nor (-)-parallel to p), and ~ p consists of all circles through p (without the point p) and all parallel classes (i.e. [[+-equivalence classes or [[_-equivalence classes) not passing through p. In particular there is a unique circle through three pairwise nonparallel points. The classical model of a Minkowski plane is obtained as the geometry of plane sections of a nondegenerated quadric of index 2 in a 3-dimensional projective space over a commutative field; in this case dp is a desarguesian plane. These Minkowski planes can be characterized by the theorem of MIQUEL and, therefore, are also called miquelian Minkowski planes. In [2] it was shown that a finite Minkowski plane of odd order with at least one desarguesian derived affine plane is miquelian; this result is due to the remarkable theorem of SEGRE that every oval in a finite desarguesian affine plane of odd order is indeed a quadric. For finite Minkowski planes of even order HEISE [5] proved that they are all miquelian. In particular all finite Minkowski planes of order _ 8 are miquelian. Generally, a Minkowski plane is miquelian if and only if every derived affine plane is desarguesian and if every translation in any derived affine plane that fixes all (+)-parallel classes is induced from an automorphism of the Minkowski plane (see [4], the same result is true for (-)-parallel classes). There is an analogon to the result on finite Minkowski planes if one imposes topological conditions. To begin with, a topological Minkowski plane is a Minkowski plane Jg = (P,o~ff, {[[+, [[_}) where P and off carry topologies such that the geometric operations of joining three pairwise nonparallel points by a circle, of forming touching circles, and intersection of parallel classes and circles are continuous operations in their domain of definition (compare [7]). It follows from [6] w and [8] that in the case that P is locally

62

G.E Steinke

compact, connected, and of finite topological dimension, the dimension of P can only be either 2 or 4. For short such planes are called 2-dimensional Minkowski planes (dim P = 2) or 4-dimensional Minkowski planes (dim P = 4), respectively. Similarly to finite Minkowski planes of odd order a 4dimensional Minkowski plane with at least one desarguesian derived affine plane is classical (see L6WEN [6] 2.5). This is a consequence of the beautiful result of BUCHANAN [1] which in analogy to the mentioned theorem of SEGRE proves that a closed oval in the complex desarguesian projective plane is already a conic. For abbreviation we call a point p of a Minkowski plane J/{ desarguesian, if the derived affine plane at p is desarguesian, and we denote the set of all desarguesian points of J / by ~ . In this note we consider 2-dimensional Minkowski planes exclusively and ask whether such a plane is classical if it has sufficiently many desarguesian points. The answer lies in between the general result of HARTMANN [4] in which, in particular, every point has to be desarguesian and the above mentioned result of L6WEN [6] for 4-dimensional Minkowski planes in which one single desarguesian point suffices. We first state the worst possible case: In [3] HARTMANN constructed a family of real Minkowski planes which actually are topological 2-dimensional Minkowski planes, although this last fact was not mentioned. This family contains a subfamily of Minkowski planes such that precisely the derived affine planes at points parallel to one special point are all desarguesian. HARTMANN'S examples depend on two real parameters rl, r2 > 0 (the classical real Minkowski plane is obtained only for rl = r2 = 1). The subfamily mentioned consists of those models with rl = r2 :fi 1. We give a short description of these Minkowski planes: Fix r > 0 and let f r ( X ) ~" XIX[ - r - 1 for x E IR, x :fi 0. The point space of the Minkowski plane Jl(r) is P = (N U {~}) x (R U {oo}) ; circles of J/g(r) are of the form

{(x,y) ~ Z l y

-=mx+t}U{(o%c~)}

(m, t E ~ . , m ~ O )

or of the form {(x,y) E ~2 l Y = a f r ( x - b ) + c ,

x ~ b}U{(b, oo),(oo, c)}

(a,b,c E ~., a -~ 0).

Two points (x, y), (u, v) E P are (+)-parallel if and only if x = u and they are (-)-parallel if and only if y = v. Desarguesian points of J/g(r), r :fi 1, are precisely the infinite points (~,y) and (x, oo), x,y E R U {oo}; Jr is the classical Minkowski plane, in this case all points are desarguesian. With this notation we prove: Theorem. Let J~ = (P, ~'~, { ][+, ]]_}) be a 2-dimensional Minkowski plane. a) I f the set ~ of desarguesian points of J/t contains one (+)-parallel class or one (-)-parallel class and at least one further point, then ~ is isomorphic to one of the Minkowski planes J#(r) of HARTMANN.

2-dimensional Minkowski Planes

63

b) I f ~ contains one (+)-parallel class or one (-)-parallel class and at least three pairwise nonparallel points, then sg is isomorphic to the classical real Minkowski plane.

2

Comparison of Derived Attine Planes at two Nonparallel Desarguesian Points

Let Jr = (P,J~c,{l[+, [I-}) be a 2-dimensional Minkowski plane. We choose three pairwise nonparallel points p0, pl, q such that Po, pl E 9 . For i = 0, 1 we denote the derived affine plane of J// at pi by d i . We coordinatize d i such that Pl-i becomes the point (0,0), q becomes the point (1, 1) and such that the (-)-parallel class through Pl-i and the (+)-parallel class through Pl-i become the X- and Y-axis, respectively. We compare both coordinate systems on those points that belong to both derived affine planes (i.e. points that are not parallel to P0 and to pl ). As usual, we identify the Y -axis with IR ; since the derived affine planes are desarguesian the ternary operation on the Y-axis given by the coordinatization comes from the usual addition and multiplication in IR. We express the coordinates of ~ 1 in the coordinates of ~ 0 . To do so, let f : ~,. \ {0}

) ~. \ {0}

be a bijection such that the point (1,y) @ (1,0) in the coordinates of d 0 has the coordinates (1,f(y)) in a l l . From the topological properties of s// it follows that f is a homeomorphism of ~ \ {0} that can be extended to a homeomorphism F of the one-point compactification lRtA {~} -~ S 1 (1-sphere) of IR by F ( ~ ) = 0 and F(0) = 0o. By the choice of q one has F(1) = f(1) = 1. Since in both affine planes the line of slope 1 through the origin is represented by the same circle through the three points P0,Pl, q, we obtain the coordinate transformation qo : ( ~ \ {0}) x ( ~ \ {0})

,(~\{O})x(~\{O}):(x,y),

,(f(x),f(y))

which transforms the coordinates of ~ 1 into the coordinates of d 0 . This transformation r extends to : P ---* P : (x,y),

, (F(x),F(y))

on the whole Minkowski plane J//. We look at the multiplication in the coordinate fields. Since the point q has the coordinates (1, 1) in both affine planes and because parallels to the X-axis or Y -axis are the same in both derived affine planes the multiplication of two points ~ p0, pl on the respective Y-axis yields essentially the same point, i.e. the resulting points in ~r and d l are (-)-parallel and their coordinates are correlated by the mapping f . Therefore we obtain a functional equation for f , namely (1)

f(xy)

=

f(x)f(y)

for all

x,y c ~ \ {0}.

64

G.E Steinke

The continuous solutions of (1) are well known. These are

fr(x)

(2)

=

x" Ixl - ' - a

for

x ~ 0, where

r E I/.

From F(0) = oo we infer that r > 0. Circles of ~ passing through the point P0 are lines in ~r in the coordinates of d 0 extended to the Minkowski plane J / / t h e y can be described as

{(x,y)~e

[y=rnx+t}

(re, t E N . , m ~ O )

where we use the convention m. oo = oo. m = oo and oo + t = t + oo = oo for m, t E R , m 7~ 0. Similarly, circles of J / passing through the point pl can be expressed in the coordinates of d l and, using ~o, even in the coordinates of ~r We obtain {(x,y) E P l y = F - ~ ( m F ( x ) + t ) }

(m, t E F,~, m-J=O).

Therefore, if we know the parameter r of F in (2), we know all circles of J/g through the point Po = (o0, oo) or through the point Pl = (0, 0). In the sequel we fix three points po, pl,p2 E ~ such that pl,p2 are not parallel to Po and look on the parameters rl and r2 of the coordinate transformation functions corresponding to the derived affine planes at po, pl and Po, P2 respectively. By considering certain functional equations we obtain rl = r2, and, if all three points are pairwise nonparallel, we even obtain /'1 ~

3

r2 ~

1.

Three Pairwise Nonparallel Desarguesian Points

Suppose that @ contains three pairwise nonparallel desarguesian points po, pl,p2. We may assume that in the desarguesian coordinates of the derived affine plane ~ 0 at P0 the points Pl,P2 have the coordinates (0,0) and (a, b) respectively, where a, b @ 0, oo. According to 2 there is a coordinate transformation corresponding to the derived affine planes at P0 and Pl with parameter r > 0 such that circles passing through Pl = (0, 0) can be described by {(x,y) E P l y = F - l ( m F ( x ) + t ) } (re, t E N . , m(=O) where F denotes the extension of fr to IR U {oo}. With the help of the translation (x,y), , ( x - a , y - b ) we similarly can describe circles passing through the point p2 = (a, b) by {(x,y) E e l Y = G - l ( m G ( x - - a ) + t ) + b }

(m, t E R , m ~ O)

where G denotes the extension of fs, s > 0 suitable, to R U {oo}.

2-dimensional Minkowski Planes

65

In either description we determine the unique circle K through the pairwise nonparallel points (0,0), (a,b) and ( - a , oo). With the multiplicative law (1) we obtain: K

=

{(x,y) E P I Y = (b/F-l(2)) "F-I(F(x/a) + 1)}

K

=

{(x,y) E P l Y = (b/G-l( 1 - G(2))). G - l ( G ( ( x / a ) - 1) + G(2)) + b } .

and

Replacing x/a by u we therefore have (3) F-I(F(u) + 1 ) / F - l ( 2 )

=

G(2))/G-I(1 - G(2)) + 1

6-1(6(u

- 1) +

for all

uERU{oo}.

For u = 2 equation (3) gives (note that F(1) = G(1) = 1): F - l ( F ( 2 ) + 1)/F-1(2) = G-l(1 + G(2))/G-I(1 - G(2)) + 1. Setting u = 1/2 in (3) and using (1) we similarly find: F - I ( F ( 2 ) + 1 ) / F - l ( 2 ) = G-1(G(4) - 1)/G-l(1 - G(2)) + 2. This results in the equation G-1(1 + 6(2)) = G-~(G(4) - 1)) + 6-1(1 - G(2)) into which only one of the two unknown functions enters. Since G(4) - 1 = (6(2) - 1)(G(2) + 1) the last equation, after division by G-1(6(4) - 1) and multiplication by 2, yields

1/G-1(G(1/2)- 1) + 1

(4)

1/G-1(G(1/2) + 1) - 1.

=

Let h+ : (0, +00) h_ : (0, + ~ )

) (0, +oo)

:

t~

) (2 t - 1) 1/t-l- 1

) (0, +oo)

:

tt

) (2 t + l )

1/t-l;

then h+, h_ are differentiable functions. Moreover, acccording to (4), these functions coincide on s (the parameter of G): (4')

h+(s)

= 1/6--1(6(1/2)

-- 1) qt_ l = 1 / 6 - - 1 ( 6 ( 1 / 2 )

+ 1) -- 1 = h _ ( s ) .

It is easy to see that the derivate o f h+ is strictly positive and that the derivative o f h_ is strictly negative; hence h+ is strictly increasing on (0, +oo) and h_ is strictly decreasing on (0, +oo). Therefore there is only one solution o f the equation h+(t) = h_(t), namely t = s according to (4'). But h+(1) = 2 = h_(1), thus s = 1. For u = oo in (3) we finally obtain 1/F-~(2) = 2/6-10

- 6(2)) + 1 = 2.

F r o m this and the definition o f F it follows 1/2 = F-1(2) = 2 -1/r ; hence r = 1. In particular F = G.

66 4

G.E Steinke Three Desarguesian Points of which Precisely Two Are (-)-Parallel

Suppose that ~ contains three desarguesian points pi (i = 0, 1, 2) such that pl and P2 are (-)-parallel and p0 is not parallel to Pl (and P2). We may assume that in the desarguesian coordinates of the derived affine plane d o at Po the points pl,p2 have the coordinates (0,0) and (a,0), a ~ 0, 0o, respectively. Again, there are parameters r, s > 0 such that circles passing through the point pt = (0, 0) have the form {(x,y) E P l y = f - l ( m F ( x ) + t ) }

(m, t E N . , m--riO)

where F denotes the extension of fr to ~ U {co}, and circles passing through the point P2 = (a, 0) have the form {(x,y) e P l y = G - l ( m G (

x-a)+t)}

(m, t e lR, m r

where G denotes the extension of fs to I1 u {co}. Without loss of generality we may assume that r < s (otherwise we interchange the role of PI and p2). Contrary to the case in the previous chapter there is no circle through Pl and p2 because these two points are parallel; therefore we cannot compare different descriptions of the same circle in order to prove F = G. Instead, we consider the circles through the points ( - a , co), (co, 1), (0, 0) and ( - a , o0), (0o, 1), (a, 0) respectively. From the condition that these two circles have only the points ( - a , co) and (co, 1) in common we shall infer the desired result F=G. The circles K~ passing through ( - a , co), (co, l) and Pl = (0,0) and Kz passing through ( - a , co), (co, 1) and p2 = (a,0) are described by K1

=

{(x,y) e P [Y = F - l ( F ( x / a ) + t)}

K2

=

{(x,y) e P i Y = G - I ( G ( ( x / a ) - 1)/G(2) + 1)}.

Replacing x / a by u we then have the condition that the equation (5)

F-I(F(u) + 1) = G-I(G((u - 1)/2) + 1)

has the only solutions u = 0o and u = - 1 . Let H = F o G-l . Then H(u) = u. [u[t-1 for u 6 R , and H ( ~ ) = oo ; here t = r/s, thus 0 < t < 1. With this notation and the coordinate transformation u ~ G(u) we obtain from (5) that the equation

H - I ( H ( z ) + 1) = 1 + G((G-I(z) - 1)/2) has the only solutions z = - 1 and z = 0. We abbreviate the left side by L(z) and the right side by R(z) and consider the restrictions of these two mappings on the interval I = [0, 1). They are continuous in 1 with image c R and they are continuously differentiabte in I' = (0, 1). An easy conputation shows L(O) -- R(O) = 1

2-dimensional Minkowski Planes L(z) = (z t + l) l/t,

L(1) = H -l (2) E IR,

R(z) = l + 2S(z - 1 ~ s - l ) -s

67 (zEI')

lim R(z) = +oo z---~ 1

L'(z) = (z t

+ 1)-l+l/tzt-l,

R'(z) = 2s(1 -- z-l/s) -s-1

(z E I')

R'(O) = 2 ~ > 1

lira U ( z ) = lim z t-1 = { 1 z-~0 z--,0 +oo

for

t= 1 0< t< 1

If 0 < t < 1, then L(0) = R(0) and L'(0) > R'(0) ; hence L(z) > R(z) for all z > 0 sufficiently close to 0. But L(z) < R(z) for z < 1 sufficiently close to 1. By continuity of L and R there is a point z0 E I t such that L(zo) = R(zo) contrary to the condition that the equation L(z) = R(z) has only the solutions z=-I andz=0. Therefore t = 1, thus r = s and F = G. 5

Proof of the Theorem

Suppose that ~ contains a parallel class H of desarguesian points and a further desarguesian point P0 r 17. Without loss of generality we may assume that 11 is the (-)-parallel class of the point p~ I1+ P0 (otherwise we interchange the role of II+ und 1[_). We choose one point pl ~ l-I \ {p~} and coordinatize the derived affine plane at p0 as in 2 such that P0 becomes the point (~, ~ ) and Pl becomes the point (0,0). Then the parallel class 11 consists of all points (a,0), a E R U {~}. Let F be the extension of fr to R U {~} which corresponds to the coordinate transformation of the derived affine planes at p0 and pl (compare 2). From 4 we know that up to translation the same F corresponds to the coordinate transformation of the derived affine planes at P0 and p E I I \ {p~}. Therefore circles not passing through the point poo have the form {(x,y) E P [ y = F - l ( m F ( x - a ) + t ) }

(rn, t E R ,

[email protected]).

It remains to determine the form of the circles through p~ = (~,0) or equivalently the lines of the derived affine plane at p~. After the coordinate transformation (x, y), , (x, F(y)) the point p~ becomes (o% oo) and circles not passing through (0% oo) are of the form K(a,t,m)

=

{(x,y) E ( ~ U { o o } ) • (a,t, m E R ,

mr

We fix m, t E ~-., m 5~ 0, and choose z ~ R \ {0, - 1 , + 1}. Then we consider the two points q0 = (0, t) and qz = ( z , - ( m / r ) F ( z - 1) + t - re~r) on the circle K = K ( 1 , t - - ( m / r ) , - - m / r ) and the third point PI/~ = ( 1 / z , ~ ) not on K. Since

68

G.F. Steinke

z 4: 0, ~ , __+1 these three points are pairwise nonparallel, hence there exists a unique circle Lz through these points; moreover, because of z :~ 0, the circle passes not through ( ~ , ~ ) and therefore Lz is of the form K(a',t',m') for suitable a', t', m' E R , m' ~ O. An easy computation shows that a'

=

1/z

t'

=

t - - ( m / r ) . ( F ( z - - 1) + 1)/(F(z 2 - 1) + 1)

m'

=

--(m/r) . r ( z ) . ( V ( z - - 1) + 1)/(F(z 2 - 1) + 1).

By definition Lz intersects K in the two points q0 and qz where qz converges to qo for z --* 0; equally Pl/z converges to poo for z ~ 0. Since the point space P is compact and connected, the Minkowski plane J / is K 1-coherent (see [7] 3.16) which means in our situation that the circles Lz converge to the unique circle L through poo = lim pl/z which is tangential to K at the z---*0

l

point q0 = lira qz (i.e. L and K have only the point qo in common); for a z---*0

precise definition of Kl-coherence see [7] 2.1. We calculate the limit circle L pointwise; it is the graph {(x, h(x)) I x 6 R U {oo}} of a suitable mapping h : I1 u {oo} --. R u {oo}. Since L passes through p~ = (oo,oo) and q0 = (0, t), we have h(oo) = oo and h(0) = t. Let Hz(x) = m'F(x - a') + t', i.e. Lz = {(x, Hz(x)) I I1 U {oe}}. Using the formulae for a', t', m' we rewrite Hz (x) for x 7~ 0, oo as follows H~ (x) =

m

F(z-1)+l

r

z2

F(xz-1)+l

F(z 2 -- 1) + 1

z

xz

x + t.

Since F is continuously differentiable in - 1 , de l'Hospital's rule may be applied to obtain ~i~(F(u - 1) + 1)/u = u--,01imF'(u -- 1) = F'(--1) = --r.

Therefore h(x) = lim Hz(x) = m . x + t ; z-"*O

hence in the given coordinate system lines of the derived affine plane at poo are just the usual euclidean lines. So we have the following description of the circles of J / : they are of the form {(x,y) E P l y = m x + t }

(m, tEP,~, m4=O)

or of the form {(x,y) E P l y = m F ( x - a ) + b }

(a,b, m E R ,

m~O)

where F(x)

=

x" ]xl -r-1

F(0)

=

oo

F(oo)

=

0.

for

x E I 1 \ {0},

2-dimensional Minkowski Planes

69

This shows that J / / i s isomorphic to the Minkowski plane ~/(r) of HARTMANN described in the introduction. If ~ contains a further desarguesian point p E I I not (+)-paraUel to p0, then ~ contains three pairwise nonparallel points (viz. p, P0 and a point p' E I-I). As shown in 3 then r = 1, and M/ is isomorphic to J//(1) which is the classical real Minkowski plane. References

[1] T. BUCHANAN,Ovale und Kegelschnitte in der komplexen projektive Ebene. Math. Phys. Sem. Ber. 26 (1979), 244-260. [2] Y. CHEN, G. KAERLEIN,Eine Bemerkung fiber endliche Laguerre- und MinkowskiEbenen, Geom. Dedicata 2 (1973), 193-194. [3] E. HARTMANN,Beispiele nicht einbettbarer reeller Minkowskiebenen, Geom. Dedicata 10 (1973), 155-159. [4] E. HARTMANN,Minkowski-Ebenen mit Transitivitiitseigenschaften, Result. Math. 5 (1982), 136-148. [5] W. HEISE,Minkowski-Ebenen gerader Ordnung, J. Geom. 5 (1974), 83. [6] R. L6WEN, Projectivities and the Geometric Structure of Topological Planes, In: Geometry - - von Staudt's Point of View (ed. by P. Plaumann and K. Strambach), Bad Windsheim 1981, pp. 339-372. [7] A. SCHENKEL,Topologische Minkowski-Ebenen. Dissertation, Erlangen-Niirnberg 1980. [8] G.F. STEINKE,Topological Properties of Locally Compact Connected Minkowski Planes and their Derived Affine Planes, Geom. Dedicata 32 (1989), 341-351.

Eingegangen am 05.05.1989 in revidierter Fassung am 12.06.1989

Author's address: Giinter E Steinke, Mathematisches Seminar der Universitiit Kiel, Ludewig-Meyn-Str. 4, D-2300 Kiel 1, Federal Republic of Germany

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