J. Geom. 103 (2012), 31–73 c 2012 Springer Basel AG 0047-2468/12/010031-43 published online June 27, 2012 DOI 10.1007/s00022-012-0118-2 Journal of Geometry Clifford parallelism: old and new definitions, and their use Dieter Betten and Rolf Riesinger Abstract. Parallelity in the real elliptic 3-space was defined by W. K. Clifford in 1873 and by F. Klein in 1890; we compare the two concepts. A Clifford parallelism consists of all regular spreads of the real projective 3-space PG(3, R) whose (complex) focal lines (=directrices) form a reg- ulus contained in an imaginary quadric (D1 = Klein’s definition). Our new access to the topic ‘Clifford parallelism’ is free of complexification and involves Klein’s correspondence λ of line geometry together with a bijective map γ from all regular spreads of PG(3, R) onto those lines of PG(5, R) having no common point with the Klein quadric; a regular par- allelism P of PG(3, R) is Clifford, if the spreads of P are mapped by γ onto a plane of lines (D2 = planarity definition). We prove the equiv- alence of (D1) and (D2). Associated with γ is a simple dimension con- cept for regular parallelisms which allows us to say instead of (D2): the 2-dimensional regular parallelisms of PG(3, R) are Clifford (D3 = dimen- sionality definition). Submission of (D2) to λ -1 yields a complexifica- tion free definition of a Clifford parallelism which uses only elements of PG(3, R): A regular parallelism P is Clifford, if the union of any two distinct spreads of P is contained in a general linear complex of lines (D4 = line geometric definition). In order to see (D1) and (D2) simul- taneously at work we discuss the following two examples using, at the one hand, complexification and (D1) and, at the other hand, (D2) under avoidance of complexification. Example 1. In the projectively extended real Euclidean 3-space a rotational regular spread with center o is sub- mitted to the group of all rotations about o; we prove, that a Clifford parallelism is generated. Example 2. We determine the group Aute(P C ) of all automorphic collineations and dualities of the Clifford parallelism P C and show Aute(P C ) ∼ = (SO3R × SO3R) Z2. Mathematics Subject Classification (2010). 51A15, 51M30, 51H10. Keywords. Topological parallelism, complexification, focal line of a regular spread, imaginary quadric, Klein’s correspondence of line geometry, elliptic displacement.
Clifford parallelism: old and new definitions,and their use
Dieter Betten and Rolf Riesinger
Abstract. Parallelity in the real elliptic 3-space was defined by W. K.Clifford in 1873 and by F. Klein in 1890; we compare the two concepts.A Clifford parallelism consists of all regular spreads of the real projective3-space PG(3, R) whose (complex) focal lines (=directrices) form a reg-ulus contained in an imaginary quadric (D1 = Klein’s definition). Ournew access to the topic ‘Clifford parallelism’ is free of complexificationand involves Klein’s correspondence λ of line geometry together with abijective map γ from all regular spreads of PG(3, R) onto those lines ofPG(5, R) having no common point with the Klein quadric; a regular par-allelism P of PG(3, R) is Clifford, if the spreads of P are mapped by γonto a plane of lines (D2 = planarity definition). We prove the equiv-alence of (D1) and (D2). Associated with γ is a simple dimension con-cept for regular parallelisms which allows us to say instead of (D2): the2-dimensional regular parallelisms of PG(3, R) are Clifford (D3 = dimen-sionality definition). Submission of (D2) to λ−1 yields a complexifica-tion free definition of a Clifford parallelism which uses only elements ofPG(3, R): A regular parallelism P is Clifford, if the union of any twodistinct spreads of P is contained in a general linear complex of lines(D4 = line geometric definition). In order to see (D1) and (D2) simul-taneously at work we discuss the following two examples using, at theone hand, complexification and (D1) and, at the other hand, (D2) underavoidance of complexification. Example 1. In the projectively extendedreal Euclidean 3-space a rotational regular spread with center o is sub-mitted to the group of all rotations about o; we prove, that a Cliffordparallelism is generated. Example 2. We determine the group Aute(PC)of all automorphic collineations and dualities of the Clifford parallelismPC and show Aute(PC) ∼= (SO3R × SO3R) � Z2.
Keywords. Topological parallelism, complexification, focal line of aregular spread, imaginary quadric, Klein’s correspondence of linegeometry, elliptic displacement.
32 D. Betten and R. Riesinger J. Geom.
1. Introductory survey
1.1. Explaining the intention
In a series of articles [2–6] we constructed mainly topological regular parallel-isms of the real projective 3-space PG(3, R) and used Klein’s correspondence λof line geometry and the polarity π5 of the Klein hyperquadric as helpful tools.At the bottom of all these regular parallelisms stands the classical, that is theClifford parallelism. In order to classify the topological parallelisms accordingto the dimension of their automorphism group we want to give as preparationan introductory summary of the topic “Clifford parallelism”. For the determi-nation of the 6-dimensional automorphism group of a Clifford parallelism weoffer two ways.
1. The historic way begins with W. K. Clifford and F. Klein in the nineteenthcentury and extensively applies the complexification of PG(3, R).
2. The (new) planarity way Among the regular parallelisms the Clifford paral-lelism is the simplest, it is in a certain sense linear as the planarity definition(see Definition 1.10) shows. In Sect. 9.2 we determine the automorphism groupof the Clifford parallelism via the new planarity definition and on this way weavoid complexification completely. The hasty reader, who is interested only inthe new way and who is familiar with Klein’s correspondence of line geome-try, can start with the planarity definition (together with the paragraph rightbefore the Definition) and jump immediately to the Sects. 9.2 and 9.3.
Of course, the equivalence of the Klein definition (see Definition 1.8) and theplanarity definition is shown (see Sect. 3). The exhibited two ways to determinethe automorphism group of a Clifford parallelism are logically independent, butclearly both ways yield the same result.
To deepen our imaginative faculties we consider star mappings induced bythe Clifford parallelism in Sect. 7 and give a very illustrative generation of aClifford parallelism in Sect. 8.
We consider PG(n, R) = (Pn,Ln) and its complexification PG(n, R ⊂ C) =(˜Pn, ˜Ln) as point-line geometries and make a clear distinction between ele-ments and maps of PG(n, R) and their complex extensions, n ∈ {3, 5}. Thisimplies a more elaborate, but worthwhile notation. The same is true when wewrite EL and EP for the set of lines resp. points in a plane E of PG(5, R).
1.2. Two parallelities in the elliptic 3-space
In the meeting of the British Association of 1873 W. K. Clifford referred in anunpublished lecture on the real elliptic 3-space and a concept of geometric par-allelity (cf. Zentralblatt MATH JFM 27.0370.02); Clifford gave the following
Definition 1.1. (Clifford’s definition of parallelity) In the real projective 3-spacePG(3, R) an absolute elliptic polarity α is given. Two distinct lines G and Hof PG(3, R) are said to be Clifford parallel, if the four lines G, H, α(G), and
Vol. 103 (2012) Clifford parallelism 33
α(H) are elements of the same regulus (cf. [11, S.257, Bemerkung 4], [30, p.5] or [10]).
In the audience of the lecture was among others F. Klein who revived Clif-ford’s ideas in 1890 (cf. Zentralblatt MATH JFM 35.0668.03) using the com-plexification PG(3, R ⊂ C) = (˜P3,˜L3) of PG(3, R) = (P3,L3); for a detailedintroduction of the complexification see [9, Chpt.7.7]. Recall: Let x ∈ ˜P3 andlet x be the conjugate complex point, then the mapping ˜P3 → ˜P3 with x �→ xis a non-projective collineation, occasionally called anticollineation. F. Kleindefined
Definition 1.2. (Klein’s definition of parallelity) (For details compare Sect. 3.1)Assume that A ⊂ ˜P3 is the absolute quadric belonging to the given absoluteelliptic polarity α of PG(3, R); note A = A, but A ∩ P3 = ∅. The quadric A
carries two reguli, say E (⊂ ˜L3) and F (⊂ ˜L3), with E = E and F = F . Areal line G is said to be Klein E-parallel to the real line H, if their complexextensions ˜G and ˜H meet the same conjugate complex lines of the regulus E .Klein F-parallel is defined analogously (Cf. [32, §142, p. 373–375]1).
Remark 1.3. It is not determinable which of the two reguli is denoted by Eand which by F since there exists a collineation κ of PG(3, R) whose complexextension κ interchanges E and F ; compare Corollary 5.1.
In Sect. 6 we prove the subsequent two Lemmas which show how Clifford’sand Klein’s definition of parallelity are connected.
Lemma 1.4. If two disjoint lines G and H of PG(3, R) are Clifford parallel,then they are either Klein E-parallel or Klein F-parallel.
Lemma 1.5. If two different lines G and H of PG(3, R) are Klein E-parallel,then they are Clifford parallel. The same holds for Klein F-parallel lines.
Clifford’s and Klein’s parallelity are subsets of L3 × L3, e.g. relations on L3.
Definition 1.6. An equivalence relation on L3 satisfying the Euclidean parallelpostulate is called a parallelism of PG(3, R). (Cf. [16, p. xvi, Def. 1]).
In our investigations we use the following equivalent
Definition 1.7. A family of spreads of PG(3, R) which simply covers L3 isnamed a parallelism of PG(3, R). (Cf. [16, p. xvi, Def. 2 and Remark 1]).
If p ∈ P3 and L ∈ L3, then there exists exactly one line LE which is E-parallelto L and runs through p, and there exists exactly one line LF which is F-par-allel to L and runs through p. In general holds: LE �= LF (cf. [32, p. 374] or[22, p. 273]), hence Clifford’s parallelity is no parallelism of PG(3, R).
Klein’s parallelity is a parallelism and the equivalence classes (=parallel clas-ses) are those sets of real lines whose complex extensions meet a given line
1 The concept that two lines of an elliptic 3-space are parallel is discussed by many authorsas the numerous bibliographic items in [11, p. 257] confirm.
34 D. Betten and R. Riesinger J. Geom.
E ∈ E (⊂ ˜L3); these line sets are regular spread which we denote by NE ,in symbols: NE = {X ∈ L3 | ˜X ∩ E �= ∅}. All complex extensions of the(real) lines of NE meet also E, i.e. NE = NE . The imaginary lines E and Eare named focal lines (or directrices, in German: “Brennlinien”) of the regu-lar spread NE = NE . (E ∩ E = ∅). To the Clifford parallelity according toDefinition 1.2 there correspond two parallelisms, namely
Pα, E := {NE | E ∈ E} and Pα, F := {NF | F ∈ F}. (1.1)
By a Clifford parallelism in the narrow sense we mean either Pα, E or Pα, Ffrom (1.1) where α is a fixedly given elliptic polarity; in other words: a Clif-ford parallelism in the narrow sense consists of all regular spreads of the realprojective 3-space PG(3, R) whose (complex) focal lines (=directrices) form aregulus contained in the absolute imaginary quadric of a fixedly given ellipticpolarity.
1.3. Four equivalent definitions of a Clifford parallelism
In the following Definition we relinquish the fixation of the given elliptic struc-ture.
Definition 1.8. (Klein’s definition of a Clifford parallelism) A Clifford parallel-ism consists of all regular spreads of the real projective 3-space PG(3, R) whose(complex) focal lines (=directrices) form a regulus contained in an imaginaryquadric.
The subsequent three definitions of a Clifford parallelism have two advanta-ges: (1) they avoid complexification and (2) they work without a given ellipticstructure.
In [2, Lemma 14] we gave the following characterization of the (according toKlein defined) Clifford parallelisms:
A regular, pairwise cosymplectic parallelism is a Clifford parallelism (in thesense of Klein).
We use this to define
Definition 1.9. (Line geometric definition) A parallelism P of PG(3, R) is Clif-ford, iff the following two properties hold:
(i) (“Regular”:) Each spread C ∈ P is regular.(ii) (“Pairwise cosymplectic”:) If C1, C2 ∈ P are different, then C1 ∪ C2 is
contained in a linear complex of lines.
For the derivation of the above characterization of Clifford parallelisms (in thesense of Klein) we used the composition γ of Klein’s correspondence λ of linegeometry and of the polarity π5 of the Klein quadric H5 (cf. Sect. 2.2) appliedto the set C of all regular spreads (cf. [2, (4)]):
γ : C→ Z; C �→ π5
(
span λ(C))
=: γ(C) (1.2)
Vol. 103 (2012) Clifford parallelism 35
where Z is the set of all 0-secants2 of H5 (cf. for instance [31, chpt. 111]).Equivalent to Definition 1.9 are the subsequent two Definitions (compare [2,Lemma 11] and [5, Lemma 2.7]):
Definition 1.10. (Planarity definition) A regular parallelism P of PG(3, R) isClifford, if {γ(X ) | X ∈ P} is a plane EL of lines.
Remark 1.11. As any spread X ∈ P is regular, so each line γ(X ) is a 0-secantof H5 and therefore the plane EP := span(EL) of points has no common pointwith the Klein quadric H5, in symbols: EP ∩ H5 = ∅.Definition 1.12. (Dimensionality definition) In PG(3, R) a regular parallelismsis Clifford, if it is 2-dimensional; in [5, Def. 2.4] the dimension of a regular par-allelisms T of PG(3, R) is defined as the dimension of
span({γ(X ) | X ∈ T}) (⊂ P5).
For our investigations the planarity definition plays a central role. Though theequivalence of Definition 1.8 and Definition 1.10 is already known from [2], itis proved in the Sects. 3.1 and 3.2 again, but in greater detail. Furthermore werecall from [2, p. 231]:
Any two Clifford parallelisms (in the sense of Klein) of PG(3, R) are projec-tively equivalent.
Each Clifford parallelism (in the sense of Klein) of PG(3, R) is topological (cf.Definition 2.1). In [2–6] we exhibit many examples of topological non-Cliffordparallelisms.
If a Clifford parallelism is given according to Definition 1.10, then the ellipticstructure is implicitly determined because of
Lemma 1.13. Let P be a Clifford parallelism of PG(3, R) according Defini-tion 1.10, then there exists a unique elliptic polarity α of PG(3, R) such thatP coincides either with Pα, E or Pα, F from (1.1).
For the Proof see Sect. 6.
Definition 1.14. We say that the elliptic polarity α from Lemma 1.13 is asso-ciated with the Clifford parallelism P.
Lemma 1.13 shows that also Clifford parallelisms in planarity sense appear inpairs. As two elliptic polarities of PG(3, R) are projectively equivalent, so twoClifford parallelisms according to Definition 1.10 are projectively equivalentby Lemma 1.13. Thus we are allowed to speak of the Clifford parallelism ofPG(3, R) denoted in the following by PC.
2 By an n-secant, n ∈ {0, 1, 2}, of a quadric Q we mean a line of span Q which has exactlyn common points with Q.
36 D. Betten and R. Riesinger J. Geom.
1.4. Surveying the Chapters 7, 8, 9, and 10
In Sect. 7 we deal with star mappings induced by the Clifford parallelism.
In Sect. 8 a very illustrative generation of a Clifford parallelism in a projec-tively extended Euclidean 3-space E
3 is taken from [6]: If a rotational regularspread S of E
3 with center o is submitted to the group of all rotations about o,then a Clifford parallelism V is generated, in other words, V is the orbit of Sunder the group SO3R. For this assertion we give two proofs thus contraposing,at the one hand (Sect. 8.1), the argumentation via complexification to verifythe Klein definition (cf. Definition 1.8) and, at the other hand (Sect. 8.2), acomplexification free approach to show the validity of the planarity definition(cf. Definition 1.10).
By AutPC we denote the group of all the automorphic collineations of theClifford parallelism PC of PG(3, R), that is κ ∈ AutPC ⇔ κ(PC) = PC;in Sect. 9 we determine AutPC. In Sect. 9.1 we start with the Klein defi-nition of PC and show that AutPC coincides with the group of all ellipticdisplacements of PG(3, R) where the elliptic structure is given by the ellipticpolarity associated with PC. Hence we can take over the results from [32, p.335, §126]; we remark that these results are computed also under applicationof complexification. In Sect. 9.2 we use the new planarity definition of PC anddetermine AutPC without applying complexification. In Sect. 9.3 we showthat AutPC is isomorphic to
(
Spin3R/Z2
) × (Spin3R/Z2
)
; furthermore, weconsider the group Aute(PC) of all automorphic collineations and dualitiesof the Clifford parallelism PC and show that Aute(PC) is isomorphic to thesemidirect product of (SO3R× SO3R) and Z2.
In Sect. 10 we exhibit some references concerning the generalization of theconcept ‘Clifford parallelism’.
2. Preliminaries
2.1. Regulus, spread, parallelism
The set of all lines of the projective 3-space PG(3, K), K is a field, which meetthree pairwise disjoint lines is called regulus. A regulus R is uniquely deter-mined by three different lines A,B,C ∈ R, we say R is generated by A,B,Cand write R(A,B,C). A line set S of PG(3, K) is called spread, if each point ofPG(3, K) is incident with exactly one line of S. A spread S is called regular, iffor any three different lines A,B,C ∈ S follows: R(A,B,C) ⊂ S. The regularspreads coincide with the elliptic linear congruences of lines.3 A parallelism isa family P of spreads such that each line of PG(3, K) is contained in exactlyone spread of P. A parallelism of PG(3, K) all whose members are regularspreads is called (totally) regular. Two lines L1, L2 of PG(3, K) are P-paral-lel, in symbols L1 ‖P L2, iff L1 and L2 are members of the same spread of P.
3 In German also called “elliptisches Netz”.
Vol. 103 (2012) Clifford parallelism 37
Clearly, the parallel axiom holds: For each line L and each point a of PG(3, K)there exists a unique line, say L‖, which passes through a and is P-parallel toL. Thus we have the mapping
pP
: L × P → L; (L, a) �→ L‖. (2.1)
Throughout the rest of this paper, we assume that the coordinate field K
is either R, the field of reals, or C, the field of complex numbers, and thatR ⊂ C. As in Chapter 1 we write P3 for the point set and L3 for the line set ofPG(3, R), and ˜P3 denotes the point set and ˜L3 the line set of the complexifi-cation PG(3, R ⊂ C) of PG(3, R). Considered in the complexification a regularspread S consists of all real lines whose complex extensions meet a pair, say B,B, of skew complex conjugate lines named the focal lines of S (cf. [11, p. 231]);according to [32, p. 281–283] a line G ∈ ˜L3 is said to be imaginary of secondkind (German: “hochimaginar”, cf. [11, p. 231]), if its conjugate complex lineG is skew to G, i.e. G ∩G = ∅.The field R it is endowed with the natural topology defined by the norm‖r‖ =
√r2, r ∈ R. In [27, Chpt. 64.3] (see also [12, Chpt. 23, p. 1260]) from
the natural topology of R a topological structure of P3 and one of L3 is derivedsuch that P3 and L3 are the Grassmann manifolds G4,1 and G4,2, respectively.Hence it makes sense to define
Definition 2.1. A parallelism P of PG(3, R) is called topological, if the mappingpP
from (2.1) is continuous.
2.2. Plucker coordinates and Klein’s correspondence of line geometry
(Cf. [31, Chap. XI, §109], [13, p. 28–32], [23, p. 363–374], or [25]) In the fol-lowing sections we always assume that PG(3, R) = (P3,L3) and PG(5, R) =(P5,L5) are the projective spaces on the right vector spaces R
4 and R6, respec-
tively, and that the complexifications PG(3, R ⊂ C) = (˜P3,˜L3) and PG(5, R ⊂C) = (˜P5,˜L5) are the projective spaces on the right vector spaces C
4 and C6,
respectively; the complexification is done by the mappings with
and Ω describes the polarity π5 of H5 which is an antiautomorphism of thelattice of subspaces of PG(5, R). The λ-images of two intersecting lines areπ5-conjugate. If we interpret a hyperplane of PG(5, R) with equation x0a0 +· · · + x5a5 = 0 as point R(a0, . . . , a5) of the dual space to PG(5, R), then wehave the following simple formula:
note that the dual space to PG(5, R) is the projective space over the left vectorspaces R
6.
If we describe a line L ∈ L3 as intersection of two different planes, say C andD, where C and D have the equations c0x0 + c1x1 + c2x2 + c3x3 = 0 andd0x0 +d1x1 +d2x2 +d3x3 = 0, respectively, then we compute the dual Pluckercoordinates (p∗
01, p∗02, p
∗03, p
∗23, p
∗31, p
∗12) according to: p∗
jk = cjdk − ckdj . ThePlucker coordinates (p01, p02, p03, p23, p31, p12) of a line L and the dual Pluc-ker coordinates (p∗
01, p∗02, p
∗03, p
∗23, p
∗31, p
∗12) of the same line L are connected as
follows:
(p01, p02, p03, p23, p31, p12)R = (p∗23, p
∗31, p
∗12, p
∗01, p
∗02, p
∗03)R; (2.8)
compare with [31, Theorem 28] and take the different indexing in [23, p. 366](=(2.4)) and [31, p. 327] into account:
(j, k) in [23, p. 366] ←→ (j + 1, k + 1) in [31, p. 327]. (2.9)
The Klein quadric H5 contains two disjoint families of planes: The λ-imageof a plane of lines is a Greek plane contained in H5, the λ-image of a star oflines is a Latin plane contained in H5. The Klein image of a regulus is an (irre-ducible) conic, the Klein images of a pair of opposite reguli are conics whosecarrier planes are π5-conjugate. The Klein image λ(C) of a regular spread Cof PG(3, R) is an elliptic subquadric of H5 which is contained in the 3-spacespanλ(C) and the line π5
(
spanλ(C))
1.2= γ(C) is a 0-secant of H5. Let C be theset of all regular spreads of PG(3, R) and let Z be the set of all 0-secants ofH5, then the mapping γ from (1.2) is a bijective map.
Vol. 103 (2012) Clifford parallelism 39
If H is a hyperbolic linear congruence of lines4 of PG(3, K) (K is a commu-tative field, especially in the present paper K ∈ {R, C}) with the focal linesB1 and B2, then λ(H) is a hyperbolic subquadric of H5 which spans a 3-spaceand π5
(
spanλ(H))
is a 2-secant of H5 joining the points λ(B1) ∈ H5 andλ(B2) ∈ H5.
In the same way the complex extensions ˜λ, ˜H5, and π5 of λ, H5, and π5
are defined; the authors refrain from writing down further details. Obviouslyholds: If L ∈ ˜L3 and L ∈ ˜L3 are conjugate complex lines, then ˜λ(L) ∈ ˜P5 and˜λ(L) ∈ ˜P5 are conjugate complex points.
2.3. Collineations and dualities under the Klein mapping λ
Each collineation κ of PG(3, R) induces a collineation κλ of PG(5, R) whichleaves H5 and the family of Greek planes invariant (and consequently thefamily of Latin planes invariant), and it holds:
(a)κ|L = λ−1 ◦ κλ|H5 ◦ λ (b) κλ(H5) = H5
(c) κλ ◦ π5 = π5 ◦ κλ; (2.10)
(note that we apply mappings from right to left). Any duality δ of PG(3, R)induces a collineation δλ of PG(5, R) which leaves H5 invariant and inter-changes the family of Greek planes and the family of Latin planes and wehave:
Each automorphic collineation of H5 is induced either by a collineation or aduality of H5. The group AutH5 of all automorphic collineations of the Kleinquadric H5 is isomorphic to the extended projective group PGL e(4, R) whichconsists of all collineations and all dualities of PG(3, R); cf. [23, p. 373].
Any collineation of PG(3, R) with (x0, x1, x2, x3)R �→ (y0, y1, y2, y3)R isdescribed by
yjμ =3∑
k=0
ajkxk with ajk, μ ∈ R (j = 0, 1, 2, 3) and det(ajk) �= 0;
(2.12)
the collineation determines the corresponding 4× 4-matrix⎛
⎜
⎜
⎜
⎜
⎝
a00 a01 a02 a03
a10 a11 a12 a13
a20 a21 a22 a23
a30 a31 a32 a33
⎞
⎟
⎟
⎟
⎟
⎠
=: K4 (2.13)
4 In German also called “hyperbolisches Netz”.
40 D. Betten and R. Riesinger J. Geom.
uniquely, up to proportionality. If the collineation κ is described by (2.12) or(2.13), then the induced collineation κλ with (p0, . . . , p5)R �→ (q0, . . . , q5)R ischaracterized by
qj kμ =3∑
l,m=0
aj lakmp lm (j, k = 0, 1, 2, 3) (2.14)
(cf. [23, p. 368]); hence we get for the corresponding (6× 6)-matrix5
K6 :=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
a00a11 − a01a10 a00a12 − a02a10 a00a13 − a03a10
a00a21 − a01a20 a00a22 − a02a20 a00a23 − a03a20
a00a31 − a01a30 a00a32 − a02a30 a00a33 − a03a30
a20a31 − a21a30 a20a32 − a22a30 a20a33 − a23a30
a30a11 − a31a10 a30a12 − a32a10 a30a13 − a33a10
a10a21 − a11a20 a10a22 − a12a20 a10a23 − a13a20
a02a13 − a03a12 −a01a13 + a03a11 a01a12 − a02a11
a02a23 − a03a22 −a01a23 + a03a21 a01a22 − a02a21
a02a33 − a03a32 −a01a33 + a03a31 a01a32 − a02a31
a22a33 − a23a32 −a21a33 + a23a31 a21a32 − a22a31
a32a13 − a33a12 −a31a13 + a33a11 a31a12 − a32a11
a12a23 − a13a22 −a11a23 + a13a21 a11a22 − a12a21
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
(2.15)
If a duality δ of PG(3, R) with (x0, x1, x2, x3)R �→ R(y0, y1, y2, y3) is describedby (2.12), then δλ is also characterized by (2.14) but on the left side of (2.14)we now have instead of qj k the dual Plucker coordinates q∗
j k.
Immediately we see: the anticollineation − : ˜P3 → ˜P3 induces the anticollin-eation of ˜P5.
In the Sects. 4 and 5 we provide examples of special collineations and dualitieswhich we will need later.
3. Klein and planarity definition are equivalent
3.1. Properties of an imaginary quadric
Any line of L3 is called real. A line L ∈ ˜L3 is the complex extension of a realline if, and only if, L = L.
As any two elliptic polarities of PG(3, R) are equivalent w.r.t. PGL(4, R) (cf.[11, p. 62]), so in the following we may assume without loss of generality
5 For space reasons we split the matrix along the mid-line running from top to bottom.
Vol. 103 (2012) Clifford parallelism 41
that the imaginary quadric is the absolute quadric of the elliptic structure inPG(3, R) defined by the elliptic polarity α with:
for all (x0, x1, x2, x3)R, (y0, y1, y2, y3)R ∈ P3. The absolute quadric A of α isdescribed by
x20 + x2
1 + x22 + x2
3 = 0; (3.2)
hence one speaks also of the “absolute sphere” A. We give some simple prop-erties of A:
(A1) From (3.2) follows: A contains no real point, i.e. A∩P3 = ∅. (This factis meant when one speaks in German of the “nullteilige Quadrik” A.)
(A2) From (3.2) we deduce: If any point x ∈ ˜P3 satisfies x ∈ A, then also theconjugate complex point x belongs to A. Consequently holds: A = A.
(A3) Let G be an arbitrary line with G ⊂ A, then follows G ⊂ A(A2)= A and
G∩G = ∅ because of (A1), i.e. (G,G) is a pair of conjugate imaginarylines of second kind.
Proposition 3.1. All lines contained in A form a pair, say (E ,F), of comple-mentary reguli.
Proof. The projective classification of quadrics in finite dimensional complexprojective spaces shows that (3.2) is the only normal form for (regular) quad-rics in PG(3, C); cf. [28, p. 233, Satz 24.4 and the table on p. 234]. In otherwords, for any (regular) quadric Q of PG(3, C) there exists a collineation κ ofPGL(4, C) with κ(Q) = A. The line
〈(1, i, 0, 0), (0, 0, 1, i)〉C = {((1, i, 0, 0)μ + (0, 0, 1, i))
C | μ ∈ C} ∪ {(0, 0, 1, i)C}with i =
√−1 is part of the quadric A, hence A is of index 1 and all lines ofA form a pair of complementary reguli; cf. [8, p. 208–210]. �Proposition 3.2. (a) The anticollineation permutates the lines of the regulus
E on A, in symbols : E = E. For the complementary regulus F on A holdsalso: F = F .
(b) If the complex extension ˜G of a real line G ∈ L3 meets the line E ∈ E,then ˜G meets also E ∈ E.
Proof.
(a) Let X is any line of E , then follows by (A3): X ∈ E . Consequently,E = E . In the same way we deduce: F = F .
(b) For the real line holds ˜G = ˜G. Assume that p ∈ ˜P3 is the common pointof ˜G and E, i.e. p ∈ ˜G and p ∈ E. Application of the anticollineationyields: p ∈ ˜G = ˜G and p ∈ E, i.e. p is a common point of ˜G and E. By(a) E belongs to E . �
42 D. Betten and R. Riesinger J. Geom.
Before we describe the regulus E on the quadric A analytically, we recall fromthe theory of projective Pappian 3-spaces:
A pencil of planes or an axial pencil is the set of all planes incident with thesame line which is called the axis of the pencil; cf. [31, p. 55].
The set of all lines of intersection of pairs of homologous planes of two projec-tively linked pencils of planes with skew axes is a regulus; cf. [31, p. 299] or[9, p. 181, Satz 10.2].
Planes of PG(3, C) can be seen as hyperplanes of PG(3, C), hence they aredescribable by (non-trivial) linear homogenous equations with coefficientsin C.
In the following first step the regulus E on A is generated by a projectivitybetween two pencils of planes having skew axes. In the second second step aparameter representation of E in Plucker coordinates is computed.
Step 1. We proceed as in [29, p. 289] where the real case is presented. Wedecompose x2
0 + x21 and x2
2 + x23 from (3.2) into linear factors, this is possible
All planes through the lines V1, 0 ∩ V0,1 and W1, 0 ∩ W0,1 form the pencils{
Vρ, σ | (ρ, σ) ∈ C2\{(0, 0)}
}
=: EV resp.{
Wρ, σ | (ρ, σ) ∈ C2\{(0, 0)}
}
=: EW .
The map with Vρ, σ �→ Wρ, σ is a projectivity from EV onto EW and theintersection lines Vρ, σ ∩ Wρ, σ of homologous planes form the regulus
{
Vρ, σ ∩ Wρ, σ | (ρ, σ) ∈ C2\{(0, 0)}} =: RV W .
With (3.3) we immediately check that the line Vρ, σ ∩ Wρ, σ is contained inthe imaginary quadric A for all (ρ, σ) ∈ C
2\{(0, 0)}. Hence the regulus RV W
is carried by A. We can choose the notation such that RV W = E .Step 2. To determine the dual Plucker coordinates of the line Vρ, σ ∩ Wρ, σ weinterpret Vρ, σ as point C(ρ, iρ, σ, iσ) and Wρ, σ as point C(−σ, iσ, ρ,−iρ) ofthe dual projective 3-space and compute the determinants of certain (2× 2)-submatrices of the following (2× 4)-matrix:
(
ρ iρ σ iσ
−σ iσ ρ −iρ
)
.
Vol. 103 (2012) Clifford parallelism 43
Thus we get the subsequent parameter representations of the regulus E in dualPlucker coordinates:{
wherefrom the Plucker coordinates of F can be computed as in Step 2.
3.2. A detailed Proof of a characterization of a Cliffordparallelism in the sense of Klein
Proposition 3.4. A regular parallelism T of PG(3, R) is a Clifford parallelismin the sense of Klein, iff {γ(S) | S ∈ T} is a plane TL of lines in PG(5, R)with TP ∩ H5 = ∅ where TP denotes the plane of points belonging to TL, insymbols: TP = span(TL).
Proof.
(a) The Klein image ˜λ(E) of E is described by (3.6) being the parameterdescription of a conic CE with carrier plane
For the carrier plane of the conic ˜λ(F) see Remark 3.6.
(b) Let P be the Clifford parallelism in the sense of Klein which is deter-mined by the elliptic polarity α and the regulus E . We choose an arbi-trary spread X of P. Then all lines X ∈ X have complex extensions ˜Xwhich meet a pair B, B of conjugate complex lines with B,B ∈ E , i.e.{ ˜X | X ∈ X} ⊂ ˜L3 is subset of a hyperbolic congruence, say H, of lineswith the focal lines B, B. Hence the 3-space span ˜λ(H) is mapped byπ5 to the line ˜λ(B) ∨ ˜λ(B) with ˜λ(B), ˜λ(B) ∈ CE ⊂ ˜EP , in symbolsγ(H) = (π5 ◦ ˜λ)(H) = ˜λ(B) ∨ ˜λ(B) ⊂ ˜EP . Hence
(
˜λ(B) ∨ ˜λ(B)) ∩ P5 =
γ(X ) is a line in the plane EP .
44 D. Betten and R. Riesinger J. Geom.
(c) We have to show that each line Y of EP is the γ-image of a spread ofP. Put ˜Y for the complex extension of Y . Then ˜Y ∩ CE consists of apair, say BY , BY of conjugate complex points. The regular spread Ywith the conjugate complex focal lines ˜λ−1(BY ) ∈ E and ˜λ−1(BY ) ∈ Eis a member of P and has the wished effect.
(d) Conversely, let T be a regular parallelism of PG(3, R) such that γ(T) ={γ(X ) | X ∈ T} is a plane TL of lines in PG(5, R) with TP ∩ H5 = ∅.Then ˜TP ∩ ˜H5 is an (irreducible) conic, say CT , with CT = CT and˜λ−1(CT ) =: R is a regulus with R = R which defines a Clifford paral-lelism in the sense of Klein; compare Definition 1.8. �
Corollary 3.5. Definitions 1.8 and 1.10 are equivalent.
Remark 3.6. The Klein image ˜λ(F) of the complementary regulus F of E is aconic CF ⊂ ˜P5 in the plane π5(˜EP) =: ˜FP and with (2.7) we compute˜FP = {(p0, . . . , p5)C ∈ ˜P5 | p0 − p3 = p1 − p4 = p2 − p5 = 0}; (3.11)
4. The absolute polarity α and the collineation αλ
By α we mean the elliptic polarity from 3.1 (3.1) which maps the point(y0, y1, y2, y3)R onto the dual point(=plane) R(y0, y1, y2, y3), hence α isdescribed by the (4× 4)-matrix diag(1, 1, 1, 1) to which corresponds accordingto (2.15) the (6×6)-matrix diag(1, 1, 1, 1, 1, 1) and a change from dual Pluckercoordinates back to ordinary Plucker coordinates yields the matrix
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=: A6.
We recall EP and FP from (3.9) resp. (3.12), and using the matrix A6 wecompute immediately:
(P1) Exactly the points of EP ∪ FP are fixed under the involutory collin-eation αλ : P5 → P5.
Vol. 103 (2012) Clifford parallelism 45
(P2) If x ∈ P5\(EP∪ FP), then αλ(x) is incident with the unique line Mx
through x meeting EP and FP ; put Mx∩ EP =: xE and Mx∩ FP =: xF ,then (x, xE , αλ(x), xF ) is a harmonic quadruple 6 of points.
For the following properties (P3), (P4), and (P5) we have to give a proofs.
(P3) If X ⊂ P5 is a plane fixed by αλ and different from EP and FP ,then X intersects either EP or FP along a line.
Proof. To a point x ∈ X with x �∈ EP ∪ FP we construct αλ(x) accordingto (P2). From αλ(x) ∈ X follows Mx ⊂ X, hence xE ∈ X and xF ∈ X.Analogously we deduce for y ∈ X\(EP ∪ FP) with y �∈ Mx: My ⊂ X withMx �= My and yE ∈ X, yF ∈ X. Because of Mx �= My the case with xE = yE
and xF = yF is impossible. Therefore we consider the following cases:
Case: xE �= yE. Then the line xE ∨ yE is the intersection of EP and X. NowxE �= yE implies xF = yF ; the assumption xF �= yF yields that the two linesxE ∨ yE and xF ∨ yF of the plane X have a common point lying in EP ∩ FP ,a contradiction to EP ∩ FP = ∅.Case: xF �= yF . As above follows: xE = yE and xF ∨ yF = FP ∩ X. �For the set of all lines of PG(5, R) meeting EP and FP we put:
{X ∈ L5 | #(X ∩ EP) = #(X ∩ FP) = 1} =:MEF .
(P4) The collineation αλ : P5 → P5 fixes exactly the lines in the planesEP and FP and all lines of MEF ; in symbols
(a) From (P1) follows that the set on the right side of (4.1) is contained inthe set on the left side of (4.1).
(b) Let Y ∈ L5 with αλ(Y ) = Y . Assume Y �⊂ EP ∪ FP . Then thereexists a point y ∈ Y with y �∈ EP ∪ FP . By (P2), αλ(y) is on the lineMy ∈ MEF carrying y. From αλ(Y ) = Y we deduce αλ(y) ∈ Y , and,consequently, Y = y ∨ αλ(y) = My ∈MEF . �
(P5) Each line of MEF is a 2-secant of the Klein quadric H5.
Proof. We span a line S of MEF by the points (r, s, t,−r,−s,−t)R ∈EP and a := (u, v, w, u, v, w)R ∈ FP �⊂ H5 with (r, s, t), (u, v, w) ∈R
3\{(0, 0, 0}. Any point of S\{a} is described by sR with
s := (r + μu, s + μ v, t + μ w,−r + μ u,−s + μ v,−t + μ w)and μ ∈ R.
6 In other words: the cross ratio of (x, xE , αλ(x), xF ) is −1.
46 D. Betten and R. Riesinger J. Geom.
The condition sR ∈ H5 is equivalent with
μ2(u2 + v2 + w2)− (r2 + s2 + t2) = 0.
The quadratic equation above in the unknown μ has two different realsolutions because of (u2 + v2 + w2) > 0 and (r2 + s2 + t2) > 0. �Immediately from (3.9), (3.12), (P4) and (P5) we deduce:
(P6) Apart from the lines in the planes EP and FP , there exist no 0-secants of the Klein quadric H5 which are fixed by αλ.
5. Elliptic orthogonal reflections σx and (σx)λ
Again we may assume that the elliptic structure is given by the elliptic polarityα from 3.1 (3.1) and that E and F are the reguli on the imaginary absolutequadric A from (3.2).
Let σx : P3 → P3 be an involutory homology with center x ∈ P3 and axisα(x); we call σx an elliptic orthogonal reflection.
It holds σx ◦ α = α ◦ σx, hence σx(E ∪ F) = E ∪ F . In order to get a moredetailed statement we consider σo with o = (1, 0, 0, 0)R:
σo
(
(x0, x1, x2, x3)R)
= (x0,−x1,−x2,−x3)R
for all (x0, x1, x2, x3) ∈ R4\{(0, 0, 0, 0)}. Application of (2.15) yields the (6×6)-
matrix diag(−1,−1,−1, 1, 1, 1) =: So which describes (σo)λ and also (σo)λ.With the help of So and (3.9), and (3.12) we compute: (σo)λ(EP) = FP (⇒(˜σo)λ(˜EP) = ˜FP) wherefrom we deduce:
σo interchanges the reguli E and F .
There exists an elliptic orthogonal reflection σy with σy(o) = x and togetherwith the above follows
Corollary 5.1. The complexification σx of any elliptic orthogonal reflection σx
interchanges the reguli E and F .
Corollary 5.2. If σx and σy are elliptic orthogonal reflections, then
σy ◦ σx(E) = E and σy ◦ σx(F) = F .
Corollary 5.3. If σx and σy are elliptic orthogonal reflections and if the linesG,H ∈ L3 are Klein E-parallel, then the lines (σy ◦ σx)(G) and (σy ◦ σx)(H)are Klein E-parallel, too.
We recall some concepts. A collineation of a quadric (also imaginary quadric)which transforms each regulus on the quadric into itself is said to be direct,otherwise non-direct; cf. [32, p. 260, §102]. A direct (real) collineation of theabsolute sphere of an elliptic 3-space is called an elliptic displacement; anelliptic symmetry is defined as a non-direct (real) collineation of the absolutesphere; cf. [32, p. 373, §142]. An orthogonal elliptic reflection is an elliptic
Vol. 103 (2012) Clifford parallelism 47
symmetry, the product of two orthogonal elliptic reflections is an elliptic dis-placement.
6. Proofs of the Lemmas 1.4, 1.5, and 1.13
Again we may assume that the elliptic structure is given by the elliptic polarityα from 3.1 (3.1) and that E and F are the reguli on the imaginary absolutequadric A from (3.2).
Proof. (of Lemma 1.4) Assume that the lines G and H of PG(3, R) are Cliffordparallel (see Definition 1.1). Then G, H, α(G), and α(H) are elements of aregulus, say R. Hence the four points λ(G) =: g, λ(H) =: h, αλ(G) =: g′, andαλ(H) =: h′ belong to a plane R ⊂ P5, namely the carrier plane of the conicλ(R). From R = g ∨ h ∨ g′ ∨ h′ we deduce
αλ(R) = g′ ∨ h′ ∨ α2λ(g) ∨ α2
λ(h) = g′ ∨ h′ ∨ g ∨ h = R
because of α2λ = idP5 . Hence we may apply (P3) from Sect. 4 to the plane R
and get that R intersects either EP or FP along a line.
Case 1. R∩ EP is a line. Let Rc be the complementary regulus of R, then theplanes span λ(Rc) =: S ⊂ P5 and the plane R are π5-conjugate, i.e. S = π5(R).Hence we have:
in words: the plane S intersects the plane FP along a line, say L. We considerthis in the complexification PG(5, R ⊂ C) of PG(5, R): the line ˜L intersectsthe conic ˜λ(F) in a pair of complex-conjugate points, say fS and fS . Hencethe lines ˜λ−1(fS) ∈ F and ˜λ−1(fS) ∈ F are met by ˜G and ˜H, with the wordsof Definition 1.2: G and H are Klein F-parallel.
Case 2. R ∩ F is a line. As above we deduce that G and H are KleinE-parallel. �Proof. (of Lemma 1.5.)
(a) Put
G0 := (1, 0, 0, 0)R ∨ (0, 1, 0, 0)R = λ−1(
(1, 0, 0, 0, 0, 0)R)
.
If G �= G0, then there exists a pair (σx, σy) of elliptic reflections with(σy ◦σx)(G) = G0 as the reader checks easily. According to Corollary 5.3it suffices to prove: If G0 and H are Klein E-parallel, then G0, α(G0),H, and α(H) are lines of a regulus.
(b) The line ˜G0 meets the absolute quadric A in the point (1, i, 0, 0)C =: g0.In the formulas (3.4) and (3.5), which indirectly describe the regulusE , we choose (ρ, σ) = (1, 0) and get a line, say E0 ∈ E , which is theintersection of the two planes with x0 + ix1 = 0 and x2 − ix3 = 0, i.e.
E0 = {(m, im, 1, i)C ∈ ˜P3 | m ∈ C} ∪ {1, i, 0, 0)C}.
48 D. Betten and R. Riesinger J. Geom.
Clearly, g0 ∈ E0. The line H with H = H meets E0 in the general point(m, im, 1, i)C, hence
H = (m, im, 1, i)C ∨ (m,−im, 1,−i)C = ˜λ−1(
(m2, 0,m, 1, 0,−m)C)
.
Using the (6× 6)-matrix A6 from Sect. 4 we compute:
In order to determine the situation of G0, α(G0), H, and α(H) we discuss thesubsequent (4× 6)-matrix:
⎛
⎜
⎜
⎜
⎜
⎝
1 0 0 0 0 0
0 0 0 1 0 0
m2 0 m 1 0 −m
1 0 −m m2 0 m
⎞
⎟
⎟
⎟
⎟
⎠
=: M4,6.
The matrix M4,6 contains two columns consisting only of zeroes, we deletethese two columns and get the following (4× 4)-matrix:
⎛
⎜
⎜
⎜
⎜
⎝
1 0 0 0
0 0 1 0
m2 m 1 −m
1 −m m2 m
⎞
⎟
⎟
⎟
⎟
⎠
=: M4,4.
Because of det(M4,4) = 0 we have rank(M4,6) ≤ 3, and the statement isproved. (If m = 0, then H = α(G) (⇒ α(H) = G) and the regulus is notdetermined uniquely). �Proof. (of Lemma 1.13.) Because of the planarity Definition from Sect. 1, theset {γ(X ) | X ∈ P} is a plane KL of lines with KP ∩ H5 = ∅. Considered inthe complex extension PG(5, R ⊂ C) of PG(5, R), we have that ˜KP ∩ ˜H5 isan (irreducible) conic CK which contains no real point, hence ˜λ−1(CK) is aregulus, say K ⊂ ˜L3 which contains no real line. The carrier quadric AK of Kcontains no real point and is therefore associated with an elliptic polarity ofPG(3, R) with the asserted effect. �
7. The Clifford parallelism and certain star mappings
We may assume that for the Clifford parallelism holds: P = Pα, E where αand E are from Sect. 3.1, (3.1) and (3.6), respectively. We recall that the planeγ(P) = EL of lines is described by EP from (3.9).
In the following three Proposition we start with the most general assertionand go step by step to simpler assertions; in the subsequent Proofs we takethe reverse way.
Vol. 103 (2012) Clifford parallelism 49
Proposition 7.1. Let L3[u] and L3[v] be the stars of lines of PG(3, R) withvertices u ∈ P3 and v ∈ P3, respectively. If πuv : L3[u] → L3[v] maps eachline of L3[u] onto the P-parallel line (compare Sect. 2.1) of L3[v], then πuv
is a (projective) collineation.
Proposition 7.2. Let EL be the plane of lines carried by the plane EP from(3.9). Each line Y of the star L3[y], y ∈ P3, belongs to exactly one regularspread CY of the Clifford parallelism P. The mapping
εy : L3[y]→ EL ⊂ L5 with εy(Y ) = γ(CY ) (7.1)
is a (projective) collineation.
Proposition 7.3. Let o = (1, 0, 0, 0)R be the origin. Then εo is a (projective)collineation.
Proof. (of Proposition 7.3.) We describe the line X ∈ L3[o] as join of o anda point (0, r1, r2, r3)R of the plane Ω with x0 = 0: X = (1, 0, 0, 0)R ∨(0, r1, r2, r3)R = λ−1
(
(r1, r2, r3, 0, 0, 0)R)
. According to [2, p. 228, Lemma 5]the regular spread CX contains X, iff the tangent hyperplane π5(λ(X)) of H5
In doubly primed coordinates the plane EP is described by p′′0= p′′
1= p′′2= 0 and
the hyperplane π5
(
λ(X))
by r1(p′′0 − p′′
3 ) + r2(p′′1 − p′′
4 ) + r3(p′′2 − p′′
5 ) = 0 andX∗ can be seen as point R(r1, r2, r3) of the dual plane E∗ of EP (also here E∗
is seen as projective space over the left vector space R3). Thus we have the
collineation (note that in the following formula the first point is in unprimedand the second point in doubly primed coordinates):
ω : Ω→ E∗; (0, r1, r2, r3)R �→ R(r1, r2, r3).
Using the perspectivity πo Ω : L3[o] → Ω; X �→ X ∩ Ω we get εo = ω ◦ πo Ω
(to be read from right to left) which shows the validity of the assertion. �Proof. (of Proposition 7.2.) If o �= y, then there exists a pair (σ1, σ2) of orthog-onal elliptic reflections with (σ2◦σ1)(o) = y where σ2◦σ1 =: ϕ is a collineation.From Corollary 5.2 follows ϕλ(EP) = EP . We submit
εo(X) = EP ∩ π5
(
λ(X))
for all X ∈ L3[o]
to the collineation ϕλ satisfying ϕλ(H5) = H5 (⇔ ϕλ ◦π5 = π5 ◦ϕλ) and have
(ϕλ ◦ εo)(X) = EP ∩ (ϕλ ◦ π5)(
λ(X))
= EP ∩ π5
(
ϕλ
(
λ(X))
)
= EP ∩ π5
(
λ(
ϕ(X))
)
= εy(Y )
for all Y := ϕ(X). Hence εy = ϕλ ◦ εo. �
50 D. Betten and R. Riesinger J. Geom.
Proof. (of Proposition 7.1.) The lines U ∈ L3[u] and V ∈ L3[v] are P-parallel,iff they belong to the same regular spread CUV ∈ P, hence εu(U) = εv(V ) isthe same line γ(CUV ) ⊂ EP . This implies εu ◦ ε−1
v = πuv. �
8. Generating a Clifford parallelism in the projectivelyextended Euclidean 3-space
This section can be seen at the one hand as an application of Klein’s andthe planarity definition of a Clifford parallelism and at the other hand as anaddition to [6, Sect. 1–3].
Using a continuous strictly monotonic function F : R≥0 → R with F (0) =
0 and F (t) → ∞ for t → ∞ Betten constructs in [1] a partition BF ofthe real 4-dimensional vector space R
4. To this partition there corresponds aspread S(BF ) of the real projective 3-space PG(3, R). Riesinger [26, Section3.2] endows PG(3, R) with a Euclidean metric and demonstrates how to com-pose S(BF ) of reguli contained in coaxial one-sheeted rotational hyperboloidswith common center m, hence the spreads S(BF ) are called rotational Bettenspreads.
Especially if F is linear, that is F (t) = kt with k ∈ R>0, then S(BF ) is a
regular spread which we denote by S(Bk).
Theorem 8.1. (a) Let S(BF ) be a rotational Betten spread with center m andlet SO3R be the group of all rotations about m, then the family PF :={g(S(BF )) | g ∈ SO3R} of spreads is a topological parallelism.
(b) If F is linear, then PF is a Clifford parallelism,(c) otherwise PF is irregular.
Part (a) and (c) of Theorem 8.1 are proved in [6], a proof of part (b) is miss-ing. As we now have a clear idea of the concept ‘Clifford parallelism’, so wecan give belated proofs of part (b) of Theorem 8.1. We exhibit a proof usingKlein’s definition and another one based on the planarity definition.
From [6, (5)] we take the following description of S(Bk) in R3:
(u, v, s)=(
x
C3,
y
C3,−kt(sin ϕ)x + kt(cos ϕ)y
C3
)
, C3 := t(cos ϕ)x+t(sin ϕ)y.
(8.1)
We read off from the formula above:
Lemma 8.2. The similarity Σ with u �→ u, v �→ v, s �→ ks maps S(B1) ontoS(Bk).
Clearly, Σ commutes with any rotation g about (0, 0, 0), in symbols:
Σ ◦ g = g ◦ Σ for all g ∈ SO3R. (8.2)
In each of the following two proofs we first show the assertion for S(B1) andapply in a final step the similarity Σ.
Vol. 103 (2012) Clifford parallelism 51
8.1. Proof of Theorem 8.1 part (b) with Klein’s definition
We take the description (4) from [6].
The partition B1 of R4 = {(x, y, w, z) | x, y, w, z ∈ R} is composed of the
following 2-dimensional subspaces:
(1) the vertical plane S = {(0, 0, w, z) | w, z ∈ R}(2) the 2-dimensional subspaces described by the equations
t(cos ϕ)x + t(sin ϕ)y − w = 0−t(sin ϕ)x + t(cos ϕ)y − z = 0,
t ≥ 0, 0 ≤ ϕ < 2π. (8.3)
We embed R4 into C
4 and consider the 2-dimensional subspace, say UE0 ,described by the subsequent two equations:
x + iy = 0w + iz = 0. (8.4)
For the determinant of the homogeneous system of linear equations formed by(8.3) and (8.4) we compute:
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
t cos (ϕ) t sin (ϕ) −1 0
−t sin (ϕ) t cos (ϕ) 0 −1
1 i 0 0
0 0 1 i
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
= 0 (8.5)
for all t ∈ R≥ 0 and for all ϕ ∈ (0, 2π). This means that any subspace of B1
has a 1-dimensional subspace with UE0 in common, or interpreted projectivelyin PG(3, R ⊂ C): the line described by UE0 is a directrix (focal line) of theregular spread S(B1).
To the 2-dimensional subspace UE0 of C4 there corresponds the line, say E0,
of C3 ⊂ PG(3, R ⊂ C) with:
(u, v, s) = (0, 0, i) + (1, i, 0)c with c ∈ C. (8.6)
comparison with (3.2) shows that Aa is the affine part of the absolute sphereA when x0 = 0 describes the plane at infinity. Hence we can use the propertiesof A from Sect. 3.1 also for Aa. The imaginary quadric Aa carries a pair ofopposite reguli, say E and F . Immediately we check E0 ⊂ Aa, and we maychoose the notation such that E0 ∈ E . A central role within this proof plays
Proposition 8.3. Let g be any rotation about the point m = (0, 0, 0), that isg ∈ SO3R, then the complex extension g of g leaves invariant Aa and trans-forms each regulus on Aa into itself, in symbols: g(Aa) = Aa, g(E) = E,g(F) = F . In other words: g is a direct transformation of Aa.
Proof. (of Proposition 8.3.)
52 D. Betten and R. Riesinger J. Geom.
(a) According to Euler’s rotation theorem any rotation g about the origin(0, 0, 0) can be given as a composition of rotations about the three coor-dinate axes; cf. [33]. We describe the rotations about the u-, v-, ands-axis by the (3× 3)-matrices
Rρ :=
⎛
⎜
⎝
1 0 0
0 cos (ρ) sin (ρ)
0 − sin (ρ) cos (ρ)
⎞
⎟
⎠, Sσ :=
⎛
⎜
⎝
cos (σ) 0 sin (σ)
0 1 0
− sin (σ) 0 cos (σ)
⎞
⎟
⎠,
and Tτ :=
⎛
⎜
⎝
cos (τ) sin (τ) 0
− sin (τ) cos (τ) 0
0 0 1
⎞
⎟
⎠, respectively. (8.8)
(b) Let p = (up, vp, sp) ∈ C3 be an arbitrary point and r a rotation of R
3
about the u-axis through an angle of measure ρ ∈ (0, 2π). With the helpof the matrix Rρ we compute
r(p) =(
up, cos (ρ) vp + sin (ρ) sp,− sin (ρ) vp + cos (ρ) sp
)
and get
r(p) ∈ Aa(8.7)⇐⇒u2
p + v2p + s2
p + 1 = 0(8.7)⇐⇒ p ∈ Aa.
Hence r(Ap) = Ap. In the same way we prove that the complex exten-sions of rotations about the v-axis and of rotations about the s-axis leavethe imaginary quadric Aa invariant. Consequently, for each g ∈ SO3R
holds g(Aa) = Aa.
(c) Clearly, g({E ,F}) = {E ,F}. To determine the type of an affinity a ofR
3 with a(Aa) = Aa it suffices to consider a line E ∈ E which is notfixed under a. If a is direct, then E and a(E) are contained in the sameregulus E on Aa, and therefore E and a(E) are skew. If a is non-direct,then E and a(E) are elements of different reguli on Aa, and therefore Eand a(E) are intersecting.
(d) We determine the type of a rotation r of R3 about the u-axis through
an angle of measure ρ ∈ (0, 2π). As test line we use the line E0 ∈ Efrom (8.6) which is spanned by the points b1 := (0, 0, i) (c = 0) andb2 := (1, i, i) (c = 1) and via the matrix Rρ we compute
r(b1) =(
0, i sin (ρ) , i cos (ρ))
=: h1,
r(b2) =(
1, i cos (ρ) + i sin (ρ) ,−i sin (ρ) + i cos (ρ))
=: h2.
The mutual situation of b1, b2, h1, h2 we can read off from
det(b2 − b1, h1 − b1, h2 − b1) = 2(
1− cos(ρ))
.
As(
1− cos(ρ))
does not vanish in the open interval (0, 2π), so the linesE0 = b1∨ b2 and r(E0) = h1∨ h2 are skew, and r is proved to be direct.
Vol. 103 (2012) Clifford parallelism 53
(e) Also for a rotation s of R3 about the v-axis through an angle of measure
σ ∈ (0, 2π) we can use the line E0 ∈ E from (8.6) as test line. Via thematrix Sσ we compute
s(b1) =(
i sin (σ) , 0, i cos (σ))
=: k1,
s(b2) =(
cos (σ) + i sin (σ) , i,− sin (σ) + i cos (σ))
=: k2.
Because of det(b2− b1, k1− b1, k2− b1) = 2(
cos(σ)− 1)
the rotation s isdirect.
(f) For a rotation t of R3 about the s-axis through an angle of measure
τ ∈ (0, 2π) we have to employ another test line, namely
E1 := �1 ∨ �2 with �1 := (1, i,−i), �2 := (−1, i, i).
We check E1 ⊂ Aa. Since det(b2 − b1, �1 − b1, �2 − b1) = −4 the linesE0 ∈ E and E1 are skew and thus E1 ∈ E . Via the matrix Tτ we compute
s(�1) =(
cos (τ) + i sin (τ) ,− sin (τ) + i cos (τ) ,−i)
=: m1,
s(�2) =(− cos (τ) + i sin (τ) , sin (τ) + i cos (τ) , i
)
=: m2.
Now det(�2−�1,m1−�1,m2−�1) = 8(
cos(τ)−1)
shows that the rotationt is direct.
Combination of (a), (d), (e), and (f) yields that any rotation from SO3R isdirect. Thus the proof of Proposition 8.3 is complete. �We continue the proof of Theorem 8.1 part (b). By Proposition 3.2 holds E = Eimplying E0 ∈ E . If g ∈ SO3R, then g({E0, E0}) ∈ E because g is direct byProposition 8.3. Consequently,
{
g({E0, E0}) | g ∈ SO3R} ⊆ E . (8.9)
Next we prove the reverse inclusion of the formula above. Let E be any lineof E , then E ∈ E , too. The conjugate complex lines E and E are focal lines ofa unique regular spread NE . By the way, S(B1) = NE0 .
Proposition 8.4. The regular spread NE is a member of the family
P1 :={
g(S(B1)
) | g ∈ SO3R}
.
Proof. (of Proposition 8.4.) In the spread NE there exists exactly one (real)line, say H ′, through (0, 0, 0) and ˜H ′ meets E and E; ˜H ′ ∩ E =: h′, hence˜H ′ ∩ E = h′ and h′, h′ ∈ Aa. We denote the carrier line of the s-axis by H
and put ˜H ∩ E0 =: h′, hence ˜H ∩ E0 = h and h, h ∈ Aa. There exists a rota-tion χ ∈ SO3R with χ(H) = H ′. By Proposition 8.3 holds χ(Aa) = Aa andχ(E) = E and therefore χ({h, h}) = {h′, h′}. From the fact that E0, E0 and E,E are the only elements of the regulus E through h, h and h′, h′, respectively,we deduce: χ({E0, E0}) = {E,E}, that is χ(NE0) = NE implying NE ∈ P1.Now the proof of Proposition 8.4 is complete. �From the proof above we read off:
E ⊆ {g({E0, E0}) | g ∈ SO3R}
. (8.10)
54 D. Betten and R. Riesinger J. Geom.
We combine (8.9) and (8.10) and get:{
g({E0, E0}) | g ∈ SO3R}
= E . (8.11)
We express the formula above in words: The focal lines of the regular spreadsof the family P1 form the regulus E contained in the imaginary quadric Aa,this means that P1 is a Clifford parallelism according to Klein’s definition.
By Lemma 8.2 there exists a similarity Σ of R3 with Σ
(S(B1))
= S(Bk) and˜Σ maps the focal lines of S(B1) onto the focal lines of S(Bk). We take (8.2)into account, and submit the result above to ˜Σ and get that the focal lines ofthe regular spreads of the family Pk :=
{
g(S(Bk)
) | g ∈ SO3R}
form theregulus ˜Σ(E) on the imaginary quadric ˜Σ(Aa), hence Pk is a Clifford parallel-ism for each k ∈ R
>0. This completes the proof of Theorem 8.1 part (b) withKlein’s definition.
Remark 8.5. For any rotation t of R3 about the s-axis through an angle of
measure τ ∈ (0, 2π) holds: t(S(B1)
)
= S(B1), hence these rotations do notcontribute to the generation of P1. Nevertheless we can not omit part (f) inthe proof of Proposition 8.3.
8.2. Complexification free Proof of Theorem 8.1 part (b) with the planaritydefinition
To determine the γ-image of S(B1) we need a description of S(B1) in Pluckerand before in projective coordinates. For a rotation χ about the origin we haveto determine γ
(
χ(S(B1)
)
)
, therefore we begin with the simple
Lemma 8.6. For any given regular spread C of PG(3, R) and any collineationκ ∈ PGL(4, R) holds:
γ(
κ(C)) = κλ
(
γ(C)). (8.12)
Proof. (of Lemma 8.6.) We consider the following chain of equalities:
γ(
κ(C)) (1.2)= π5
(
span(λ ◦ κ)(C)) = π5
(
span(κλ ◦ λ)(C)) =
π5
(
κλ
(
spanλ(C)))
(2.10c)= κλ
(
π5
(
spanλ(C)))
(1.2)= κλ
(
γ(C).
The second sign of equality of the chain above is valid because (2.10a) implies:λ ◦ κ = κλ ◦ λ. The third sign of equality of the chain above holds becausethe span construction is a projective concept. This completes the proof of theLemma above. �Via −w =: x0, x =: x1, y =: x1, z =: x3 we change from (8.3) to the projectivecoordinates (x0, x1, x2, x3); we put V := {((x0, x1, x2, x3)R | x1 = x2 = 0}and get the following description of S(B1)\{V }:
By Euler’s rotation theorem any rotation g about the origin (1, 0, 0, 0)R canbe given as a composition of rotations about the three coordinate axes; cf.[33]. We describe the rotations of PG(3, R) about the x1-, x2-, and x3-axisaccording to (2.12) and (2.13) by the (4× 4)-matrices
Rρ, 4 :=
⎛
⎜
⎜
⎜
⎜
⎝
1 0 0 0
0 1 0 0
0 0 cos ρ sin ρ
0 0 − sin ρ cos ρ
⎞
⎟
⎟
⎟
⎟
⎠
resp. Sσ, 4 :=
⎛
⎜
⎜
⎜
⎜
⎝
1 0 0 0
0 cos σ 0 sinσ
0 0 1 0
0 − sin σ 0 cos σ
⎞
⎟
⎟
⎟
⎟
⎠
(8.15)
resp. Tτ, 4 :=
⎛
⎜
⎜
⎜
⎜
⎝
1 0 0 0
0 cos τ sin τ 0
0 − sin τ cos τ 0
0 0 0 1
⎞
⎟
⎟
⎟
⎟
⎠
with 0 < ρ, σ, τ < 2π.
(8.16)
Assume that r is a rotation of PG(3, R) about the x1-axis described by thematrix Rρ, 4 from (8.15), then by (2.14) (or (2.15)) the induced collineation rλ
is characterized by the (6× 6)-matrix
Rρ, 6 :=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
1 0 0 0 0 0
0 cos ρ sin ρ 0 0 0
0 − sin ρ cos ρ 0 0 0
0 0 0 1 0 0
0 0 0 0 cos ρ sin ρ
0 0 0 0 − sin ρ cos ρ
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
. (8.17)
56 D. Betten and R. Riesinger J. Geom.
Using Rρ, 6 we compute:
rλ(e0) = e0 and rλ(e1) =(
0, cos (ρ) ,− sin (ρ) , 0,− cos (ρ) , sin (ρ))
R.
Immediately we see that for all ρ ∈ (0, 2π) the point rλ(e1) belongs to thefollowing plane:
(8.18)with e2 = (0, 0, 1, 0, 0,−1)R; EL := {L ∈ L5 | L ⊂ EP}.
A central role within this second proof plays
Proposition 8.7. If χ is any rotation of PG(3, R) about (1, 0, 0, 0)R, then theplane EP is invariant under χλ.
Proof. (of Proposition 8.7.) We decompose χ according to Euler’s theorem.
(a) For a rotation r of PG(3, R) described by Rρ, 4 we already checked:rλ(ek) ∈ EP for k ∈ {0, 1}. Via Rρ, 6 we get:
rλ(e2) =(
0, sin (ρ) , cos (ρ) , 0,− sin (ρ) ,− cos (ρ))
R ∈ EP .
(b) For a rotation s of PG(3, R) described by Sσ, 4 we compute sλ(ek) fork ∈ {0, 1, 2} via the (6× 6)-matrix
Sσ, 6 :=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
cos σ 0 sin σ 0 0 0
0 1 0 0 0 0
− sin σ 0 cos σ 0 0 0
0 0 0 cos σ 0 sin σ
0 0 0 0 1 0
0 0 0 − sin σ 0 cos σ
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(8.19)
and easily check sλ(ek) ∈ EP for k ∈ {0, 1, 2}.(c) For a rotation t of PG(3, R) described by Tτ, 4 we compute tλ(ek) for
k ∈ {0, 1, 2} via the (6× 6)-matrix
Tτ, 6 :=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
cos τ sin τ 0 0 0 0
− sin τ cos τ 0 0 0 0
0 0 1 0 0 0
0 0 0 cos τ sin τ 0
0 0 0 − sin τ cos τ 0
0 0 0 0 0 1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
, (8.20)
and easily check tλ(ek) ∈ EP for k ∈ {0, 1, 2}.Combination of (a), (b), and (c) yields the validity of Proposition 8.7. �
Vol. 103 (2012) Clifford parallelism 57
We continue the complexification free proof of Theorem 8.1 part (b). Alto-gether we have:
γ(
g(S(B1)
)
)
Lemma 8.12= gλ
(
γ(S(B1)
)
)
(8.14)= gλ(e0 ∨ e1) (8.21)
for all g ∈ PSO3R; because of e0 ∨ e1 ∈ EL and
gλ(EL) = EL(⇐ gλ(EP)
Prop. 8.7= EP
)
we deduce{
γ(
g(S(B1)
)
)
| g ∈ PSO3R
}
⊆ EL. (8.22)
Next we prove the reverse inclusion of the formula above. According to (8.21)it suffices to show
EL ⊆ {gλ(e0 ∨ e1) | g ∈ PSO3R}. (8.23)
Put ek := (δ0k, δ1k, δ2k,−δ0k,−δ1k,−δ2k)R for k ∈ {0, 1, 2}; with δjk we denotethe Kronecker symbol. In the plane EP we use the quadrangle
{e0 = e0R, e1 = e1R, e2 = e2R, (e0 + e1 + e2)R}as frame of reference for homogeneous coordinates (y0, y1, y2). Any line G ∈EL is described by an equation a0y0 + a1y1 + a2y2 = 0 with (a0, a1, a2) ∈R
3\{(0, 0, 0)}. Assume that g ∈ PSO3R is composed by two rotations, say rρ
and sσ, described by (8.15); now it makes sense to include the measures ρ andσ of the rotation angles in the notation. Then (sσ ◦ rρ)λ is described by the(6× 6)-matrix Sσ, 6.Rρ, 6 and we compute
(
sσ ◦ rρ)λ(e0) =(
cos (σ) , 0, sin (σ) ,− cos (σ) , 0,− sin (σ))
R,
that is the point(
cos (σ) , 0, sin (σ))
R in (y0, y1, y2)-coordinates. Further-more,
(
sσ ◦ rρ)λ(e1) is the point(
sin (σ) sin (ρ) , cos (ρ) ,− cos (σ) sin (ρ)) in(y0, y1, y2)-coordinates. Thus we have
(
sσ ◦ rρ)λ(e0) ∈ G ⇔ σ = − arctan(
a0
a2
)
.
Now σ is known, and with this σ we can determine ρ via(
sσ ◦ rρ)λ(e1) ∈ G ⇔ ρ = arctan(
a1
−a0 sin (σ) + a2 cos (σ)
)
.
Hence (8.23) is valid.
Combining (8.23), (8.22), and (8.21) we get finally:{
γ(
g(S(B1)
)
)
| g ∈ PSO3R
}
= EL. (8.24)
The formula above says that the γ-images of the regular spreads of the familyP1 :=
{
g(S(B1)
)| g ∈ PSO3R}
form a plane of lines, hence P1 is a Cliffordparallelism according to the planarity definition.
By Lemma 8.2 there exists a similarity Σ of R3 with Σ
(S(B1))
= S(Bk). Wesubmit both sides of (8.24) to Σλ, take (8.2) and Lemma 8.6 into account, andget:
{
γ(
g(S(Bk)
)
)
| g ∈ PSO3R
}
= Σλ(EL).
58 D. Betten and R. Riesinger J. Geom.
The formula above says that the γ-images of the regular spreads of the familyPk :=
{
g(S(Bk)
) | g ∈ PSO3R}
form the plane Σλ(EL) of lines, hence Pk isa Clifford parallelism for each k ∈ R
>0 according to the planarity definition.This completes the second proof of Theorem 8.1 part (b).
9. Automorphic collineations and dualities of the Cliffordparallelism
9.1. Procedure with Klein’s definition and complexification
Again we assume that the elliptic structure in PG(3, R) is given by the ellipticpolarity α from Sect. 3.1 (3.1); then A from (3.2) is the absolute quadric. Forthe Clifford parallelism P of PG(3, R) we choose: P = Pα, E where the regulusE is from (3.6). By F we denoted the complementary regulus to E . We recallfrom Sect. 3.2 that the conics ˜λ(E) and ˜λ(F) span the complex extensions ofthe planes EP resp. FP described by (3.9) resp. (3.12).
9.1.1. Preparatory considerations.
Definition 9.1. Let τ be a collineation or duality of PG(3, R), then τ is calledan automorphic transformation of the Clifford parallelism P or a P-transfor-mation, if τ(P) = P.
Clearly, τ(Pα, E) = Pα, E ⇔ τ(Pα, F ) = Pα, F .
Lemma 9.2. Let κ : P3 → P3 be a collineation. Then the following statementsare equivalent:
(a) κ(Pα, E) = Pα, E(b) For all regular spreads X ∈ Pα, E holds κ(X ) ∈ Pα, E(c) For all lines γ(X ) ∈ EL holds κλ(γ(X )) ∈ EL(d) κλ(EP) = EP
(e) κλ(˜EP ∩ ˜H5) = ˜EP ∩ ˜H5
(f) κλ
(
˜λ(E))
= ˜λ(E)(g) κ(E) = E(h) κ is a direct collineation of the absolute quadric A
(i) κ is an elliptic displacement.
Corollary 9.3. The group Aut(P) of all automorphic collineations of the Clif-ford parallelism P of PG(3, R) coincides with the group of all elliptic displace-ments of PG(3, R) where the elliptic structure is given by the elliptic polarityassociated with P.
Remark 9.4. Property (P1) of Sect. 4 implies: αλ(X) = X for all lines X ∈EL. Hence α fixes each regular spread γ−1(X) of the Clifford parallelism P.
Vol. 103 (2012) Clifford parallelism 59
The elliptic polarity α associated with the Clifford parallelism P is an auto-morphic duality of P. For the group Aute(P) of all automorphic collinea-tions and all automorphic dualities of the Clifford parallelism P of PG(3, R)holds:
Aute(P) = Aut(P)× {idP3 , α}where α is the elliptic polarity associated with P.
Definition 9.5. A regular spread S of PG(3, R) is called α-invariant or α-spread,if α(S) = S.
If A is a Clifford parallelism of PG(3, R), then {γ(X ) | X ∈ A} =: AL is con-tained in the plane AP and the plane π5(AP) =: BP with BP ∩ H5 = ∅ deter-mines the Clifford parallelism {γ−1(X) | X ∈ L5 ∧ X ⊂ BP} of PG(3, R)called the reflected parallelism of A.
Proposition 9.6. Assume that the elliptic polarity α is associated with the Clif-ford parallelism A of PG(3, R). Then the set of all α-spreads coincides withthe spreads of A and the spreads of the reflected parallelism of A.
Proof. Follows immediately from property (P6) of Sect. 4. �Corollary 9.7. Two disjoint lines G and H of PG(3, R) are Clifford parallelin the sense of Definition 1.1 if, and only if, they belong to the same α-spread.
9.1.2. Citing Veblen and Young II. In [32, p. 335, §126] all direct projectivecollineations of the imaginary quadric A are determined, in other words: allelliptic displacements are determined; any displacement is represented by aproduct of skew-symmetric (4× 4)-matrix D1(α0, α1, α2, α3) and a skew-sym-metric (4× 4)-matrix D2(β0, β1, β2, β3) with
D1(α0, α1, α2, α3) :=
⎛
⎜
⎜
⎜
⎜
⎝
α0 α1 α2 α3
−α1 α0 −α3 α2
−α2 α3 α0 −α1
−α3 −α2 α1 α0
⎞
⎟
⎟
⎟
⎟
⎠
(9.1)
and
D2(β0, β1, β2, β3) :=
⎛
⎜
⎜
⎜
⎜
⎝
β0 β1 β2 β3
−β1 β0 β3 −β2
−β2 −β3 β0 β1
−β3 β2 −β1 β0
⎞
⎟
⎟
⎟
⎟
⎠
(9.2)
where α0, . . . , β3 ∈ R.
Remark 9.8. In [32, p. 335] complexification is hidden in the last sentencewhere the authors write: “. . . any point satisfying (46). . . ”, but (46) describesthe imaginary sphere having no real points.Corollary 9.9. The group of all automorphic collineations of the Clifford par-allelism is 6-dimensional.
60 D. Betten and R. Riesinger J. Geom.
Proof. Because of
det(D1(α0, α1, α2, α3)) =(
α02 + α1
2 + α32 + α2
2)2
(9.3)
each matrix D1(α0, α1, α2, α3) with (α0, α1, α2, α3) �= (0, 0, 0, 0) describes,up to proportionality, a unique autocollineation (=displacement), say δ1, ofPG(3, R). Because of
det(D2(β0, β1, β2, β3)) =(
β02 + β1
2 + β32 + β2
2)2
(9.4)
each matrix D2(β0, β1, β2, β3) with (β0, β1, β2, β3) �= (0, 0, 0, 0) describes, upto proportionality, a unique displacement, say δ2. �
9.1.3. On the effect of δ1, δ2, (δ1)λ, and (δ2)λ. Via (2.15) we get from
D1(α0, α1, α2, α3) the following (6× 6)-matrix 7
D1,λ :=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
α02 + α1
2 −α0α3 + α2α1 α0α2 + α3α1
α0α3 + α2α1 α02 + α2
2 −α0α1 + α3α2
−α0α2 + α3α1 α0α1 + α3α2 α02 + α3
2
α22 + α3
2 −α2α1 + α0α3 −α3α1 − α0α2
−α2α1 − α0α3 α12 + α3
2 α0α1 − α3α2
α0α2 − α3α1 −α0α1 − α3α2 α12 + α2
2
α22 + α3
2 −α2α1 + α0α3 −α3α1 − α0α2
−α2α1 − α0α3 α12 + α3
2 α0α1 − α3α2
α0α2 − α3α1 −α0α1 − α3α2 α12 + α2
2
α02 + α1
2 −α0α3 + α2α1 α0α2 + α3α1
α0α3 + α2α1 α02 + α2
2 −α0α1 + α3α2
−α0α2 + α3α1 α0α1 + α3α2 α02 + α3
2
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(9.5)
which describes (δ1)λ. If (u, v, w) ∈ R3 \ {0, 0, 0}, then (u, v, w, u, v, w)R = uR
is a point of FP by (3.12). Its image under (δ1)λ is described by the transposedof the vector
7 For space reasons we split this and the subsequent (6 × 6)-matrices along the mid-linerunning from top to bottom.
Vol. 103 (2012) Clifford parallelism 61
D1,λ.uT
=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
(
α02 + α1
2)
u + (−α0α3 + α2α1) v + (α0α2 + α3α1) w+
(α0α3 + α2α1) u +(
α02 + α2
2)
v + (−α0α1 + α3α2) w+
(−α0α2 + α3α1) u + (α0α1 + α3α2) v +(
α02 + α3
2)
w+(
α02 + α1
2)
u + (−α0α3 + α2α1) v + (α0α2 + α3α1) w+
(α0α3 + α2α1) u +(
α02 + α2
2)
v + (−α0α1 + α3α2) w+
(−α0α2 + α3α1) u + (α0α1 + α3α2) v +(
α02 + α3
2)
w+(
α22 + α3
2)
u + (−α2α1 + α0α3) v + (−α3α1 − α0α2) w
(−α2α1 − α0α3) u +(
α12 + α3
2)
v + (α0α1 − α3α2) w
(α0α2 − α3α1) u + (−α0α1 − α3α2) v +(
α12 + α2
2)
w(
α22 + α3
2)
u + (−α2α1 + α0α3) v + (−α3α1 − α0α2) w
(−α2α1 − α0α3) u +(
α12 + α3
2)
v + (α0α1 − α3α2) w
(α0α2 − α3α1) u + (−α0α1 − α3α2) v +(
α12 + α2
2)
w
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
=(
(u, v, w, u, v, w)(α20 + α2
1 + α22 + α2
3))T
,
this means that each point of the plane FP is fixed by (δ1)λ. In the same way,we compute that the plane EP is invariant under (δ1)λ as a whole. Hence wehave: The displacements δ1 fix each regular spread of the parallelism Pα, Fwhile they permute the regular spreads of the parallelism Pα, E .
Next we study the effect of (δ2)λ. Via (2.15) we get from D2(β0, β1, β2, β3) thefollowing (6× 6)-matrix:
D2,λ :=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
β02 + β1
2 β0β3 + β2β1 −β0β2 + β3β1
−β0β3 + β2β1 β02 + β2
2 β0β1 + β3β2
β0β2 + β3β1 −β0β1 + β3β2 β02 + β3
2
−β22 − β3
2 β0β3 + β2β1 −β0β2 + β3β1
−β0β3 + β2β1 −β12 − β3
2 β0β1 + β3β2
β0β2 + β3β1 −β0β1 + β3β2 −β12 − β2
2
−β22 − β3
2 β0β3 + β2β1 −β0β2 + β3β1
−β0β3 + β2β1 −β12 − β3
2 β0β1 + β3β2
β0β2 + β3β1 −β0β1 + β3β2 −β12 − β2
2
β02 + β1
2 β0β3 + β2β1 −β0β2 + β3β1
−β0β3 + β2β1 β02 + β2
2 β0β1 + β3β2
β0β2 + β3β1 −β0β1 + β3β2 β02 + β3
2
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(9.6)
62 D. Betten and R. Riesinger J. Geom.
which describes (δ2)λ. If (u, v, w) ∈ R3\{(0, 0, 0)}, then the point vR =
(u, v, w,−u,−v,−w)R belongs to EP by (3.9). We compute
D2,λ.vT =(
(u, v, w,−u,−v,−w)(β20 + β2
1 + β22 + β2
3))T
,
this means that each point of the plane EP is fixed by (δ2)λ. In the sameway, we compute that the plane FP is invariant under (δ2)λ as a whole. Hencewe have: The displacements δ2 fix each regular spread of the parallelism Pα, Ewhile they permute the regular spreads of the parallelism Pα, F .
9.2. Complexification free procedure with the planarity definition
In the following we exhibit a new access to the determination of the automor-phic collineations of a Clifford parallelism; we use only the planarity definition(together with the paragraph right before the Definition 1.10) and those prop-erties of Klein’s correspondence λ of line geometry which concern only realelements.
Let M be a Clifford parallelism of PG(3, R), then ML := {γ(X | X ∈M} isa plane of lines with MP ∩ H5 = ∅ where MP := span ML. If κ ∈ PGL(4, R)is an automorphic collineation of M, that is κ(M) = M, then κ(X ) ∈M forall X ∈M, hence ML is invariant under the induced collineation κλ, in sym-bols: κλ(ML) = ML (⇒ κλ(MP) = MP). As the Klein quadric H5 is invariantunder κλ, so the polar plane of MP with respect to H5 is also invariant underκλ, in symbols: κλ(NP) = NP with NP := π5(MP). Note NP ∩ H5 = ∅ and,furthermore, that (MP , NP) is a pair of complementary planes of PG(5, R).Therefore we are interested in those collineations of PGL(6, R) which leave anordered pair of complementary planes of PG(5, R) invariant.
Lemma 9.10. Assume that M and N are complementary planes (of points) inPG(5, R).
(a) For each collineation χ ∈ PGL(6, R) which leaves the ordered pair(M,N) invariant holds χ = χM ◦χN = χN ◦χM where χM ∈ PGL(6, R)fixes M pointwise and χN ∈ PGL(6, R) fixes N pointwise.
(b) There exists exactly one involutory collineation ι(M,N) ∈ PGL(6, R)which fixes M ∪ N pointwise.
Proof.
(a) Put
ej := (δj0, δj1, δj2, δj3, δj4, δj5) for j = 0, . . . , 5 (9.7)
(δjk denotes the Kronecker symbol). We may assume M = e0R ∨ e1R ∨e2R and N = e3R ∨ e4R ∨ e5R, that is M and N are described bythe equations p3 = p4 = p5 = 0 and p0 = p1 = p2 = 0, respectively.Any collineation χ ∈ PGL(6, R) is represented by a (6× 6)-matrix B :=
Vol. 103 (2012) Clifford parallelism 63
(bjk), j, k = 0, . . . , 5. For the image of the point e0R under χ we compute
χ(e0R) =(
B.(e0)T)T
R = (b00, b10, b20, b30, b40, b50)R.
From χ(M) = M and e0R ∈ M follows χ(e0R) ∈ M which impliesb30 = b40 = b50 = 0. In the same way we get:
and compute B6 = F1.F2 = F2.F1 wherefrom the assertion (a) can bededuced easily.
(b) The image x′ of a point x �∈ M ∪ N under a collineation satisfying thedemanded properties must be incident with the unique line T through
64 D. Betten and R. Riesinger J. Geom.
x meeting M and N and x′ is necessarily that point on T which is har-monic to x with respect to the point pair (T ∩ M,T ∩ N), hence thereis at most one collineation of the desired kind. On the other hand, wecheck immediately that the collineation with(
Note that in the Lemma above the Klein quadric H5 is insignificant, whereasH5 is essential in the subsequent Lemmas.
Lemma 9.11. Let A1 and A2 be arbitrary planes in PG(5, R) with A�∩ H5 = ∅,� = 1, 2. Then there exists an automorphic collineation σ (∈ PGL(6, R)) of theKlein quadric H5 which maps A1 onto A2.
Proof. The Klein quadric H5 induces in A� an elliptic polarity α� , � = 1, 2. Asthe basic field is real, so the two elliptic polarities α1 and α2 are projectivelyequivalent (cf. [9, p. 52]), that is, there exist a collineation κ12 : A1 → A2 beingcompatible with α1 and α2. If we regard the metric vector space M = (R6,Ω)defined by the bilinear form Ω from (2.6), then to κ12 corresponds an isomor-phism of M from a proper subspace onto another subspace of M which bythe theorem of Witt (cf. [22, p. 166]) is extendable to an isomorphism of M.Hence there exists an autocollineation κ of PG(5, R) leaving H5 invariant andextending κ12. �As preparation of the following Lemma we consider the mutual situation ofthe Latin and the Greek planes contained in the Klein quadric H5.
Proposition 9.12. Two distinct planes in H5 of the same kind have exactlyone point in common. Two distinct planes in H5 of different kind have eitherexactly one line in common or have empty intersection.
Proof. Apply λ to the following facts about stars and planes of lines inPG(3, R): Two distinct planes and also two distinct stars of lines of have aline in common; a star of lines and a plane of lines have either a pencil of linesor no line in common. �Lemma 9.13. Let L be a plane (of points) in PG(5, R) with L ∩ H5 = ∅ andπ5(L) the conjugate plane to L with respect to the Klein quadric H5. Then theinvolutory collineation ι(L, π5(L)) from Lemma 9.10 leaves H5 invariant andinterchanges the set of Greek planes and the set of Latin planes in H5.
Proof. Via (7.2) we change to double primed coordinates. The Klein quadricH5 is described by x′′
02 + x′′
12 + x′′
22 − x′′
32 − x′′
42 − x′′
52 = 0. According to
Lemma 9.11 we may assume without loss of generality that L is the planedescribed by x′′
0= x′′1= x′′
2= 0. Then π5(L) has the equations x′′3= x′′
4= x′′5= 0
and for ι(L, π5(L)) holds
Vol. 103 (2012) Clifford parallelism 65
(x′′0 , x′′
1 , x′′2 , x′′
3 , x′′4 , x′′
5 )R �→ (x′′0 , x′′
1 , x′′2 ,−x′′
3 ,−x′′4 ,−x′′
5 )R
wherefrom one sees ι(
L, π5(L))
(H5) = H5 immediately. We split the descrip-tion of H5 and get
(x′′0 + x′′
3 )(x′′0 − x′′
3 ) + (x′′1 + x′′
4 )(x′′1 − x′′
4 ) + (x′′2 + x′′
5 )(x′′2 − x′′
5 ) = 0,
therefore the plane, say Φ, with (x′′0 + x′′
3 ) = (x′′1 + x′′
4 ) = (x′′2 + x′′
5 ) = 0 iscontained in H5. It is easy to compute that ι
(
L, π5(L))
(Φ) =: Ψ is describedby (x′′
0 − x′′3 ) = (x′′
1 − x′′4 ) = (x′′
2 − x′′5 ) = 0. Because of Φ ∩ Ψ = ∅ and
Proposition 9.12 the planes Φ and Ψ are of different kind. �Theorem 9.14. Any two Clifford parallelisms of PG(3, R) are projectivelyequivalent.
Proof. Let A and B be arbitrary Clifford parallelisms of PG(3, R). Then
{γ(X ) | X ∈ A} =: AL and {γ(X ) | X ∈ B} =: BL are planes of lines inPG(5, R). By Lemma 9.11 there exists an autocollineation κ5 of PG(5, R) withκ5(H5) = H5 which maps AL onto BL. For κ5 two alternatives are possible:
Alternative 1. κ5 permutes the set of Greek planes. By [23, p. 373] there existsan autocollineation κ of PG(3, R) with κλ = κ5. Consequently, κ(A) = B.
Alternative 2. κ5 interchanges the set of Greek planes and the set of Latinplanes. We compose κ5 with ι
(
BP , π5(BP))
, BP := span BL. By Lemma 9.13the collineation ι
(
BP , π5(BP)) ◦ κ5 =: χ5 permutes the set of Greek planes on
H5 and maps AL onto BL such that χ5 satisfies the conditions of Alternative 1.�
Assume that A is any Clifford parallelisms of PG(3, R). We intend to deter-mine the group
Aut A :=(
{ξ ∈ PGL(4, R) | ξ(A) = A } , ◦)
of all automorphic collineations of A. By Theorem 9.14, we may assume with-out loss of generality that the plane {γ(X ) | X ∈ A} =: EL of lines is con-tained in the plane EP with the equation p0 + p3 = p1 + p4 = p2 + p5 = 0(compare (3.9)). Put π5(EP) =: FP , then FP is described by p0−p3 = p1−p4 =p2 − p5 = 0 (compare (3.12)).
In the first part we derive necessary conditions for a collineation κ ∈ Aut A.The induced collineation κλ of PG(6, R) satisfies κλ(EP) = EP and becauseof κλ(H5) = H5 also κλ(FP) = FP . From Lemma 9.10 and [23, p. 373] followsthat κ ∈ Aut A must be the composition of two collineations ε, ϕ ∈ PGL(4, R)whose induced collineations ελ, ϕλ of PGL(6, R) fulfill
ελ(x) = x for all x ∈ EP and ϕλ(y) = y for all y ∈ FP . (9.9)
Hence we have to discuss two types collineations of PGL(4, R).
Type A. Necessary and sufficient conditions for collineations ε of PGL(4, R)with (9.9).
Then EP = n0 ∨ n1 ∨ n2. From ελ(n�) = n� , � = 0, 1, 2, we deduce that εleaves the line set
λ−1(
π5(n�) ∩ H5
)
=: G�
of PG(3, R) invariant; G� is a general linear complex of lines (German:“Gewinde”) and G� determines a unique null polarity, say ν�, of PG(3, R).From ε(G�) = G� follows that “ ε commutes with ν� ” which means expressedin strict symbols
ν� ◦ ε = ε∗−1 ◦ ν� and ν∗−1� ◦ ε∗−1 = ε ◦ ν∗−1
� for � = 0, 1, 2 (9.10)
where ε∗ and ν∗ denote the dual transformations of ε and ν, respectively.
Lemma 9.15. A collineation ε of PGL(4, R) with (9.9) commutes with each ofthe following three collineations: ν∗−1
1 ◦ ν0, ν∗−12 ◦ ν1, ν∗−1
0 ◦ ν2.
Proof. (ν∗−11 ◦ ν0) ◦ ε = ν∗−1
1 ◦ (ν0 ◦ ε)(9.10)= ν∗−1
1 ◦ (ε∗−1 ◦ ν0) = (ν∗−11 ◦ ε∗−1) ◦
ν0(9.10)= (ε◦ν∗−1
1 )◦ν0 = ε◦ (ν∗−11 ◦ν0). Cyclic permutation of the indices yields
the remaining Proof. �
Because of π5(n0)(2.7)= R(e0 − e3) the line set G0 is described in Plucker coor-
dinates by p0−p3 = 0. We take (2.5) and (2.9) into account, apply [31, p. 331,Cor. 2] and get:
ν0
(
(x0, x1, x2, x3)R)
= R(x1,−x0,−x3, x2). Analogously,ν1
(
(x0, x1, x2, x3)R)
= R(x2, x3,−x0,−x1), andν2
(
(x0, x1, x2, x3)R)
= R(x3,−x2, x1,−x0).
Hence we have:
(ν∗−11 ◦ ν0)
(
(x0, x1, x2, x3)R)
= (x3,−x2, x1,−x0)R
(ν∗−12 ◦ ν1)
(
(x0, x1, x2, x3)R)
= (x1,−x0,−x3, x2)R
(ν∗−10 ◦ ν2)
(
(x0, x1, x2, x3)R)
= (x2, x3,−x0,−x1)R,
that is the collineations ν∗−11 ◦ ν0, ν∗−1
2 ◦ ν1, and ν∗−10 ◦ ν2 are represented
by the matrices
K01 :=
⎛
⎜
⎜
⎜
⎜
⎝
0 0 0 1
0 0 −1 0
0 1 0 0
−1 0 0 0
⎞
⎟
⎟
⎟
⎟
⎠
, K12 :=
⎛
⎜
⎜
⎜
⎜
⎝
0 1 0 0
−1 0 0 0
0 0 0 −1
0 0 1 0
⎞
⎟
⎟
⎟
⎟
⎠
, and
K20 :=
⎛
⎜
⎜
⎜
⎜
⎝
0 0 1 0
0 0 0 1
−1 0 0 0
0 −1 0 0
⎞
⎟
⎟
⎟
⎟
⎠
,
respectively.
Vol. 103 (2012) Clifford parallelism 67
Remark 9.16. The duality ν2 ◦ ν∗−11 ◦ ν0 maps the point (x0, x1, x2, x3)R onto
the plane R(x0, x1, x2, x3) and therefore coincides with the absolute polarityα from (3.1). The same is true for ν0 ◦ ν∗−1
2 ◦ ν1 and ν1 ◦ ν∗−10 ◦ ν2.
Assume that ε of PGL(4, R) with (9.9) is represented by the (4 × 4)-MatrixA4 := (ajk), j, k = 0, 1, 2, 3. Then according to Lemma 9.15 the following threeconditions must hold:
K01.A4 = ρ01A4.K01 with ρ01 ∈ R\{0}, (9.11)K12.A4 = ρ12A4.K12 with ρ12 ∈ R\{0}, (9.12)K20.A4 = ρ20A4.K20 with ρ20 ∈ R\{0}. (9.13)
From the 0th rows of the two matrices in (9.12), (9.13), and (9.12) follows:
respectively. We substitute the intermediate result (9.14), (9.15), (9.16) inthe matrix A4 and get a new matrix, say A′
4, satisfying also (9.11),(9.12),and (9.13). Computing these matrix products we deduce from the two 3rdrows of (9.11): (a00, a01, a02, a03) = (ρ01)2(a00, a01, a02, a03) which togetherwith det A4 �= 0 ⇒ (a00, a01, a02, a03) �= (0, 0, 0, 0) and ρ01 ∈ R\{0} impliesρ01 = 1. Analogously we deduce ρ12 = 1 and ρ20 = 1. Using these last threeconditions in (9.14), (9.15), (9.16) we see that (a00, a01, a02, a03) can be cho-sen freely in R
4\{(0, 0, 0, 0)}. We put (a00, a01, a02, a03) =: (β0, β1, β2, β3) andcheck that the matrix A4 of ε is nessarily of the form (9.2).
The sufficiency of the condition above can be taken over from Sect. 9.1.3 wherewe used only real elements.
The set of collineations ε of PGL(4, R) with (9.9) coincides with the set of alldisplacements δ2 from Sect. 9.1.2.
Type B. Necessary and sufficient conditions for collineations ϕ of PGL(4, R)with (9.9).
After the extensive treatment of Type A we may confine ourselves to writingdown only the essential orientation points. We put
that is FP = n3∨ n4∨ n5. Let ν� be the unique null polarity of PG(3, R) deter-mined by λ−1
(
π5(n�) ∩ H5
)
=: G�, � = 3, 4, 5. A collineation ϕ of PGL(4, R)with (9.9) commutes with each of the following three collineations: ν∗−1
4 ◦ ν3,
68 D. Betten and R. Riesinger J. Geom.
ν∗−15 ◦ ν4, ν∗−1
3 ◦ ν5 which are represented by the matrices
K34 :=
⎛
⎜
⎜
⎜
⎜
⎝
0 0 0 −1
0 0 −1 0
0 1 0 0
1 0 0 0
⎞
⎟
⎟
⎟
⎟
⎠
, K45 :=
⎛
⎜
⎜
⎜
⎜
⎝
0 −1 0 0
1 0 0 0
0 0 0 −1
0 0 1 0
⎞
⎟
⎟
⎟
⎟
⎠
, and
K53 :=
⎛
⎜
⎜
⎜
⎜
⎝
0 0 −1 0
0 0 0 1
1 0 0 0
0 −1 0 0
⎞
⎟
⎟
⎟
⎟
⎠
,
respectively. Assume that ϕ with (9.9) is represented by the (4 × 4)-MatrixB4 := (bjk), j, k = 0, 1, 2, 3. Then the following three conditions must hold:
with σ34, σ45, σ53 ∈ R\{0}. The discussion of (9.17) yields finally:
The set of collineations ϕ of PGL(4, R) with (9.9) coincides with the set of alldisplacements δ1 from Sect. 9.1.2.
9.3. Connection with the group Spin3R
In [32, p. 337, §127] the authors show that(
{D1(α0, α1, α2, α3) | (α0, α1, α2, α3) ∈ R4}, ◦,+
)
is8 isomorphic to the skew-field (H, .,+) of the Hamiltonian quaternions; inthe corresponding proof the following four matrices are used:
1 := diag(1, 1, 1, 1), i :=
⎛
⎜
⎜
⎜
⎜
⎝
0 1 0 0
−1 0 0 0
0 0 0 −1
0 0 1 0
⎞
⎟
⎟
⎟
⎟
⎠
,
j :=
⎛
⎜
⎜
⎜
⎜
⎝
0 0 1 0
0 0 0 1
−1 0 0 0
0 −1 0 0
⎞
⎟
⎟
⎟
⎟
⎠
, and k :=
⎛
⎜
⎜
⎜
⎜
⎝
0 0 0 1
0 0 −1 0
0 1 0 0
−1 0 0 0
⎞
⎟
⎟
⎟
⎟
⎠
.
8 By ◦ we denote the multiplication of matrices.
Vol. 103 (2012) Clifford parallelism 69
For a displacement δ1 with matrix D1(α0, α1, α2, α3) holds (α0, α1, α2, α3) �=(0, 0, 0, 0) hence we can change to normed matrices
(
{
D1(α0, α1, α2, α3) |
(α0, α1, α2, α3) ∈ R4 and
√
α20 + α2
1 + α22 + α2
3 = 1}
, ◦)
and normed quaternions. Provided that D1(α0, α1, α2, α3) is normed, thenD1(−α0,−α1,−α2,−α3) is normed, too, and the two matrices describe thesame displacement, while the two corresponding quaternions are different. Thegroup (H1, .) of all quaternions with norm 1 is also called Spin(3), hence wehave:
The group of all displacements with matrix (9.1) is isomorphic to Spin3R/Z2.
Lemma 9.17.(
{D2(β0, β1, β2, β3) | (β0, β1, β2, β3) ∈ R4}, ◦,+
)
is also iso-morphic to (H, .,+).
Proof. Put
U := diag(1, 1, 1, 1), I :=
⎛
⎜
⎜
⎜
⎜
⎝
0 −1 0 0
1 0 0 0
0 0 0 −1
0 0 1 0
⎞
⎟
⎟
⎟
⎟
⎠
,
J :=
⎛
⎜
⎜
⎜
⎜
⎝
0 0 −1 0
0 0 0 1
1 0 0 0
0 −1 0 0
⎞
⎟
⎟
⎟
⎟
⎠
, and K :=
⎛
⎜
⎜
⎜
⎜
⎝
0 0 0 −1
0 0 −1 0
0 1 0 0
1 0 0 0
⎞
⎟
⎟
⎟
⎟
⎠
.
Then D2(β0, β1, β2, β3) = Uβ0 − Iβ1 − Jβ2 −Kβ3 and
I ◦ I = −U, I ◦ J = K, I ◦ K = −J,
J ◦ I = −K, J ◦ J = −U, J ◦ K = I,
K ◦ I = J, K ◦ J = −I, K ◦ K = −U.
This is the multiplication table of the quaternions. �As above we can change to normed matrices and normed quaternions and get:
The group of all displacements with matrix (9.2) is isomorphic to Spin3R/Z2.
We sum up
Theorem 9.18. For the group Aut(P) of all automorphic collineations of theClifford parallelism P of PG(3, R) holds:
Aut(P) ∼= (
Spin3R/Z2
)× (Spin3R/Z2
)
. (9.18)
Property (P1) of Sect. 4 implies: αλ(X) = X for all lines X ∈ EL. Hence αfixes each regular spread γ−1(X) of the Clifford parallelism P, e.g. the elliptic
70 D. Betten and R. Riesinger J. Geom.
polarity α is an automorphic duality of P. Together with Spin3R/Z2∼= SO3R
we get
Theorem 9.19. The group Aute(P) of all automorphic collineations and allautomorphic dualities of the Clifford parallelism P of PG(3, R) is isomorphicto the semidirect product of SO3R× SO3R and Z2, in symbols:
Aute(P) ∼= (SO3R× SO3R) � Z2. (9.19)
Remark 9.20. We compare the (4 × 4)-matrices from the Sects. 9.2 and 9.3and get:
i = K12, j = K20, k = K01, and I = K45, J = K53, K = K34.
Hence the matrices i, j, k and I, J,K applied in the algebraical proofs can beinterpreted geometrically via the Sect. 9.2.
10. A glimpse at generalized Clifford parallelisms
In literature we find at least three directions of generalization of the concept“Clifford parallelism in the real elliptic 3-space”.
The first one concerns the dimension. The investigation of the Clifford paralle-lity of two (q−1)-subspaces of a real elliptic space of odd dimension 2q−1 ≥ 3was initiated by the article [34] and continued in the book [30] whose approachwas taken over in [11, Kap.12].
The second kind of generalization concerns the coordinatizing field. In [7] theauthors define a generalized Clifford parallelisms in PG(3, F ) with the help ofa quaternion skew field H over any (commutative) field F of arbitrary char-acteristic. We mention that the consideration of finite coordinatizing fieldsyields no Clifford parallelism in the sense of the planarity definition becauseeach polarity of a finite projective plane of order n has at least n + 1 absolutepoints (cf. e.g. [14, p. 241, Theorem 12.5]); from this theorem follows that afinite projective 3-space Π3 admits no elliptic polarity, hence in Π3 there existsno Clifford parallelism in the sense of the Klein definition.
The third generalizations belong to the theory of incidence groups. H. Karzeland his research group dealt with the structures called “projective doublespace”, “prism space”, and “kinematic space”, which we explain below guidedby the clear and elegant survey [15, p. 112–115].
A projective double space is a projective space endowed with two parallelisms,‖�, ‖r, such that the condition of closing parallelograms holds, that is for anythree non-collinear points a, b, c the left-parallel line to a ∨ c through b andthe right-parallel line to a∨ b through c have non-empty intersection. Under aleft prism space we understand a projective double space (P,L, ‖� , ‖r) whichsatisfies the following “left prism axiom”: Let Lk ∈ L, k = 1, 2, 3, be dis-tinct and mutually right parallel and let ak, bk ∈ Lk be distinct points with
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a1 ∨ a2 ‖� b1 ∨ b2 and a2 ∨ a3 ‖� b2 ∨ b3, then 9 a3 ∨ a1 ‖� b3 ∨ b1. We getthe “right prism axiom” by interchanging “left” and “right” in the left prismaxiom. A projective double space which fulfills the left and the right prismaxiom is called prism space. The Klein E-parallelity and the Klein F-paralle-lity from Def. 1.2 make the real elliptic 3-space to a projective double space(cf. [18, p. 75–78], [19]) which is also a prism space (cf. [19,20]) as well as akinematic space which is defined as follows.
Let (G, ·) be a group and (G,L) be a projective space, then (G,L, ·) is calleda kinematic space provided that (i) the associated left and right translationsa� : x �→ a · x and ar : x �→ x · a are collineations of (G,L) for each a ∈ G and(ii) the lines in L which are incident with the identity element 1 are subgroupsof (G, ·). The kinematic space (G,L, ·) becomes a projective double space, ifleft and right parallelity of two lines L,M ∈ L of (G,L, ·) are defined by
L ‖� M :⇔ L = g · M for some g ∈ G
L ‖r M :⇔ L = M · g for some g ∈ G
(“kinematic definition of left and right Clifford parallelisms”). The results ofchapter 9 concerning the automorphism group of a Clifford parallelism suggestto embed the real numbers R into the Hamiltonian quaternions H, which forma 4-dimensional vector space over R. We consider the corresponding projec-tive space PG(3, R). We put H
∗ := (H\{0}, ·) and R∗ := (R\{0}, ·) for the
multiplicative groups of H and R, respectively, then H∗/R
∗ is the point set ofPG(3, R) and at the same time a group; by [19] the generated structure is akinematic space. This procedure done for R ⊂ H can be generalized to divisionK-algebras (A, K) (K is a field) satisfying r2 ∈ K+K r for all r ∈ A (kinematicdivision algebra). To such an algebra corresponds a projective kinematic spacecalled kinematic derivation. By [19], (1) kinematic spaces and prism spacescoincide and (2) every prism space with ‖� �=‖r is a kinematic derivation of a(general) quaternion skew field. By [20], (1) any two different left parallel linesof a projective double space span a 3-dimensional projective double space, (2)any 3-dimensional projective double space is a prism space, and (3) there existno finite projective double spaces. By [17], every projective double space withdistinct left and right parallelisms has dimension 3. By [21], every projectivedouble space is a prism space.
In [24] the author proposes a definition of regular parallelisms in a linear spacenot necessarily embedded into a projective space.
