Regular parallelisms on PG(3,R) admitting a 2-torus action
aa r X i v : . [ m a t h . G T ] J a n Regular parallelisms on PG(3 , R ) admitting a 2-torusaction Rainer L¨owen and G¨unter F. Steinke
Abstract
A regular parallelism of real projective 3-space PG(3 , R ) is an equivalence rela-tion on the line space such that every class is equivalent to the set of 1-dimensionalcomplex subspaces of C = R . We shall assume that the set of classes is compact,and characterize those regular parallelisms that admit an action of a 2-dimensionaltorus group. We prove that there is a one-dimensional subtorus fixing every par-allel class. From this property alone we deduce that the parallelism is a 2- or 3-dimensional regular parallelism in the sense of Betten and Riesinger [6]. If a 2-torusacts, then the parallelism can be described using a so-called generalized line star or gl star which admits a 1-torus action. We also study examples of such parallelismsby constructing gl stars. In particular, we prove a claim which was presented in [6]with an incorrect proof. The present article continues a series of papers by the firstauthor on parallelisms with large groups.MSC 2010: 51H10, 51A15, 51M30 A spread in real projective 3-space PG(3 , R ) is a set of mutually disjoint lines coveringthe point set. A parallelism on PG(3 , R ) is a set Π of mutually disjoint spreads coveringthe set of lines. A parallelism is usually considered as an equivalence relation on theline set. We shall always assume that spreads and parallelisms are topological in thesense explained in Section 2. That property is equivalent to compactness in some naturaltopology. The classical Clifford parallelism consists of the orbits of one of the two factorsof SO(4 , R ) = SO(3 , R ) · SO(3 , R ). The pioneering work of Betten and Riesinger (e.g., [1])has produced numerous ways of constructing nonclassical examples.The automorphism group Aut Π ≤ PGL(4 , R ) of a topological parallelism is a compactLie group [7], [13] of dimension at most 6. There are plenty examples with a smallgroup (of dimension ≤ >
3, then the parallelism is Clifford, see [14],and its group is 6-dimensional. Parallelisms with a 3-dimensional group are completelyknown [15], [16]. Passing from dimension 3 to 2, matters suddenly become extremely L¨owen and Steinkedifficult. So it seems reasonable at first to restrict attention to a particularly nice type ofparallelisms, the regular ones, where every parallel class is a regular spread, i.e. isomorphicto the set of complex one-dimensional subspaces of C . This has the advantage thatthese parallelisms can be handeled conveniently using the Klein quadric K in PG(5 , R ),where regular spreads appear as quadrics obtained by intersecting K with 3-dimensionalsubspaces of PG(5 , R ). Moreover, the group of automorphisms and dualities of PG(3 , R ),which contains the group of the parallelism, corresponds to the group PO(6 , K . Equivalently, one can specify the polar lines of those 3-spaces withrespect to the polarity π which defines the Klein quadric. These lines are 0-secants of K , i.e., disjoint from K . A set H of 0-secants of K defines a parallelism in this way ifand only if it is a so called hfd line set, i.e., if every tangent hyperplane π ( p ), p ∈ K ,contains exactly one line in H , see [4] or [12]. The parallelism is topological if and only ifthe set H is compact. Betten and Riesinger introduced the term dimension of Π to meanthe dimension of the subspace of PG(5 , R ) generated by the hfd line set H correspondingto Π, dim Π = dim span H . Examples exist for parallelisms of dimensions 2, 3, 4 and 5, see [3], [4]. Dimension 2charcterizes Clifford parallelism; in this case, H consists of all lines of the plane generatedby H . Our main result is THEOREM 1.1
A regular topological parallelism with a 2-dimensional automorphismgroup is at most 3-dimensional.The only 2-dimensional compact connected Lie group is the 2-torus. Examples of regularparallelisms with 2-torus action are known only in very special cases. We shall givesimplified proofs of the existing results in Sections 5 and 6, and we shall prove stronggeneral existence results in Section 7.In the 3-dimensional case, the hfd line set H defining a regular parallelism can be replacedby a simpler object, namely, by a compact so-called generalized line star or gl star S . Thegl star is obtained by applying to the set H of lines the polarity π induced by π on the3-space P generated by H , S = π π (Π) . This is due to Betten and Riesinger [2]; see [12] for a simpler approach and proof. Thedefining property of a gl star, which is equivalent to the fact that the gl star correspondsto a 3-dimensional parallelism, is this: S is a set of lines in a 3-dimensional subspace P of PG(5 , R ), and Q = P ∩ K is an elliptic quadric such that every line in S meets Q intwo points, and every point of P not in the interior of Q is incident with precisely oneline from S . Here, a point is called exterior if it is incident with an exterior line, thatis, a 0-secant of Q . An interior point is a point p such that every line containing p is a2-secant (meeting Q in two points). The remaining points are precisely those of Q .egular parallelisms admitting a 2-torus 3We shall show (Theorem 5.1) that for compact S the defining property of gl-stars can besimplified: it suffices that every point on Q is on some line of S and two lines from S never meet in a non-interior point with respect to Q .A compact gl star can be described by a fixed point free involutory homeomorphism σ of Q ; the star consists of the lines q ∨ σ ( q ) for all q ∈ Q , and conversely, σ sends a point q ∈ Q to the second point of intersection of Q with the line L ∈ S containing q . The glstar corresponding to Clifford parallelism consists of all lines passing to some fixed point p in the interior of Q , it is an ordinary line star. The corresponding involution is theantipodal map with respect to p . We shall freely use the continuity properties of the topological projective space PG(3 , R ),see, e.g., [11].We recall the definition and the properties of the Klein correspondence in the case of realprojective 3-space PG(3 , R ), following Pickert [19]; compare also [10], Section 2.1. Bydefinition, PG(3 , R ) is the lattice of subspaces of R . In the 6-dimensional vector spaceof non-degenerate alternating forms on R , we can represent a line L of PG(3 , R ) by theone-dimensional subspace of forms containing L in their radical. This maps the line set L of PG(3 , R ) bijectively onto a quadric K in PG(5 , R ), called the Klein quadric, definedby a nondegenerate quadratic form g of index 3. The space R contains two families ofmaximal isotropic subspaces (of projective dimension 2), representing the points and thehyperplanes of PG(3 , R ), respectively. Incidence between lines and points or hyperplanesin PG(3 , R ) corresponds to inclusion in the Klein model of PG(3 , R ). Automorphisms of PG(3 , R ) correspond to orthogonal maps with respect to g . For us, itsuffices to know that the connected component PSL( R ,
4) of Aut PG(3 , R ) corresponds tothe projective orthogonal group PSO(6 ,
3) with respect to g . The automorphism groupAut Π of a regular parallelism appears in the Klein model as a subgroup of this group.According to [7], [13], this group is compact and hence is contained in the maximalcompact subgroup O(3 , R ) · O(3 , R ) (product with amalgamated central involutions). Thefactors of this product are induced by the orthogonal groups on the factors of R = R × R ,where the quadratic form defining the Klein quadric is given by g ( u, v ) = k u k − k v k ,with k u k denoting the Euclidean norm. A 2-torus in Aut Π is therefore a product T · T ,where T i ∼ = SO(2 , R ) consists of the rotations of the i th factor R about some fixed axis.Clearly, the g -orthogonal group Aut Π leaves the 3-space P = span H and the gl star S associated with Π invariant.By definition, a spread of PG(3 , R ) is regular if together with any triple of distinct linesit contains the entire regulus determined by this triple. As pointed out earlier, amongspreads in PG(3 , R ) this property charcterizes the complex spread, consisting of the one-dimensional complex subspaces with respect to some complex structure on R , see [10],4.15. A spread of PG(3 , R ) is regular if and only if its image under the Klein corre-spondence is an elliptic subquadric of K spanning a 3-space, see, e.g., [6], Proposition L¨owen and Steinke13. Topological spreads, topological regular parallelisms, topological gl stars and topologicalhfd line sets are discussed thoroughly in [12], Section 3; here we give a brief summary.In the case of a parallelism, being topological means that the unique line parallel (i.e.,equivalent) to a given line L and passing through a given point p depends continuouslyon the pair ( p, L ). For a regular parallelism, this is equivalent to compactness of the setof 3-spaces in PG(5 , R ) defining the regular spreads belonging to the parallelism. Thetopology to be considered on this set is the topology of the Grassmann manifolds of all3-spaces in PG(5 , R ). For the other objects enumerated above, the definition of a topo-logical object is similar; e.g., for a spread it is required that the line containing a givenpoint depends continuously on that point. Again, compactness in the topology inducedby the relevant Grassmann manifold is equivalent to this property.About the possibilities for automophism groups of regular parallelisms, the followingresults are known. We begin with large groups. THEOREM 2.1
A regular parallelism with an automorphism group of dimension atleast 3 is Clifford.Proof.
This is a corollary of the first author’s results on general parallelisms with a 3-dimensional group [16], Corollary 3.2. For 3-dimensional regular parallelisms, a simpleproof is available, compare [6], Theorem 33. At the time, the compactness of automophismgroups was not known, therefore we update the argument here: A compact subgroup ofPO(6 ,
3) of dimension at least 3 contains a copy of SO(3 , R ), which acts on the quadric Q ∼ = S associated with the gl star S in the ordinary way. The rotations fixing a point q ∈ Q fix the unique line L = q ∨ σ ( q ) ∈ S that contains q , hence L coincides with therotation axis and passes through the universal fixed point of the rotation group. Thus wehave an ordinary gl star and a Clifford parallelism.On the other hand, a 3-dimensional regular parallelism always has an automorphism groupof positive dimension: THEOREM 2.2 [6] Every at most 3-dimensional regular parallelism admits an actionof a 1-torus T that fixes every parallel class. Actually, the same proof yields an action of the orthogonal group O(2 , R ) with the sameproperty. Proof.
We may assume that the given regular parallelism Π is not Clifford. Hence P =span H is a 3-space, and on the underlying vector space U = R the form g induces aform of signature (3 , g -orthogonal complement C of U in R is of signature(0 , g -orthogonal effective torus action that induces SO(2 , R ) on C and istrivial on U . Therefore it fixes all elements of S and, hence, all parallel classes.egular parallelisms admitting a 2-torus 5In Section 3, we shall prove the converse of this theorem, and this will be the key to theproof of our main result 1.1.Betten and Riesinger have given numerous examples of non-Clifford regular parallelismsand their automorphism groups. In [3] they construct 3-dimensional regular parallelismsadmitting a 2-torus. These examples are of a very special kind called axial. A muchsimpler treatment of these examples and their properties will be given in Section 5. In[6], Theorem 42, they give a construction which they claim yields more general exampleswith 2-torus action. Their proof is invalid, but we shall prove an improved version oftheir claim (Theorem 7.1). In [5], Betten and Riesinger present regular parallelisms ofdimension 3 and 4 whose full automorphism group is 1-dimensional, as well as regular par-allelisms of dimension 4 and 5 whose group is 0-dimensional (hence finite). In [5] they askwhether groups of dimension 2 can act on regular parallelisms of dimension 4 or 5. Ourmain result Theorem 1.1 answers this question in the negative. They also ask the following Question:
Can a 1-torus act on some 5-dimensional regular parallelism?This question remains open.
We need the following well-known fact.
PROPOSITION 3.1
A 2-torus T cannot act effectively on a connected surface otherthan a torus.Proof. Let S be the surface and assume first that T has a 2-dimensional orbit B . Then B is homeomorphic to a torus and is open and closed in the connected surface S , hence S = B is a torus. If all orbits of T are of dimension at most 1, we can argue in twoindependent ways. First, all stabilizers are of dimension at least one, and there are onlycountably many such subgroups in the torus. By Baire’s theorem, one of them has afixed point set with nonempty interior. By Newman’s theorem [18], [8], this subgroupacts trivially.The other argument uses the results of Mostert [17] on compact group actions with orbitsof codimension one. The orbits with minimal stabilizers (called principal orbits) cover adense open set, and all stabilizers of points on principal orbits are conjugate, hence infact equal. This subgroup then acts trivially on S .Note that a 2-dimensional factor group of a 2-torus is again a 2-torus, hence the kernelof ineffectivity has positive dimension if a 2-torus acts on a connected surface which isnot a torus. A parallelism Π is homeomorphic to a star of lines in PG(3 , R ) via themap that sends a line to its unique parallel passing through some fixed point. Thus Π ishomeomorphic to the real projective plane and certainly not to a torus. It follows that L¨owen and Steinkea 2-torus cannot act effectively on Π, and our Theorem 1.1 becomes a corollary to theconverse of Theorem 2.2, which we prove next. THEOREM 3.2
Let Π be a topological regular parallelism. Then Aut Π contains a 1-torus T that fixes all parallel classes if and only if Π is at most 3-dimensional.Proof. From the description of tori given in Section 2 we see that the action of T leavesa decomposition R = R × R invariant. On both factors, T induces (possibly trivial)rotations about some axis. So we may regroup factors and obtain the action of T on R = C = C × C × C consisting of the maps τ t : ( u, v, w ) → ( u, e kt v, e lt w ) , where t ∈ [0 , π ] and k, l are relatively prime integers (possibly zero). We claim that0 ∈ { k, l } . We assume that this is not the case and aim for a contradiction.The fact that T fixes all parallel classes means that T acts trivially on the hfd line set H associated to Π. In other words, H is contained in the set F of all fixed lines of T inPG(5 , R ). In C , these lines appear as T -invariant 2-dimensional real vector subspaces.So we wish to determine all such subspaces L .If the action of T on L is trivial, then L is the first factor of C . So assume that this actionis nontrivial. Then the T -equivariant projection of L to the first factor C is not bijective,because the actions are different. On the other hand, the kernel of this projection is T -invariant, so the projection must be zero, and L is contained in the product C × C ofthe second and third factor. These factors are invariant. If there is another invariant 2-dimensional subspace L , then by combining the projections of L onto the factors of C × C we obtain an equivariant bijection between these factors, which means that k = ± l . Since T is effective on R , we have { k, l } ⊆ { , − } . Applying complex conjugation to thefactors as necessary, we may obtain k = l = 1. Then T acts on the complex vector space C via multiplication by complex scalars, and F consists of the first factor of C plus theelements of the complex (regular) spread of C . Now this spread is homeomorphic to the2-sphere (the Riemann number sphere). On the other hand, H is homeomorphic to Π,i.e., to the real projective plane. Hence the inclusion H ⊆ F is impossible.The only remaining possibility is that k = 0 and l = ± F consists of the third factor togetherwith all 2-dimensional real subspaces of C × C ×
0. Since H is connected, the third factordoes not belong to H , and span H is contained in C × C × If Π is a non-Clifford regular parallelism admitting an action of a 2-torus T , then we knowfrom Proposition 3.1 that some 1-dimensional subtorus acts trivially on Π, and likewise onegular parallelisms admitting a 2-torus 7the associated gl star S and on the hfd line set H . From the proof of Theorem 3.2 we inferthat it even acts trivially on the 4-dimensional vector space U corresponding to span H ,on which g has signature (3 , g -orthogonal maps, all such subspaces of R areequivalent by a theorem of Witt, see, e.g., [9], 6.1. The entire group T cannot act triviallyon the space U , because it cannot act effectively on its 2-dimensional g -orthogonal spacein R , which is in fact a vector space complement.Thus there is a 1-torus Φ contained in T which acts nontrivially on the 3-space span H and commutes with the polarity π induced on it by π . Moreover, Φ leaves the quadric Q and the gl star S invariant. Following Betten and Riesinger [6], in this situation wecall S a rotational gl star . Up to conjugacy, there is only one 1-torus on span H whichleaves π invariant, so we can assume the following Standard position for rotational gl stars: Q is the unit sphere S in the affine part R ofPG(3 , R ). The group Φ is the group of rotations of R about the z-axis Z , and leaves thegl star S invariant.We summarize: PROPOSITION 4.1 [6] Let Π be a non-Clifford regular parallelism. Then dim Aut Π =2 if and only if Π is defined by a rotational gl star, which we shall always assume instandard position. Of course, Clifford parallelism corresponds to a rotational gl star, as well. Note also thatΦ acts on S if and only if Φ commutes with the involution σ : S → S defining S . Thiswill be used in the next proof. PROPOSITION 4.2 [6] Let S be a rotational gl star in standard position.a) The rotation axis Z belongs to S .b) There is a point p ∈ Z in the interior of S such that all lines perpendicular to Z andpassing through p belong to S .c) Up to isomorphism, we may assume that the point p is the origin.Proof. a) The defining involution σ commutes with Φ, hence it interchanges the two fixedpoints n = (0 , ,
1) and s = − n of Φ. Thus Z = n ∨ s ∈ S .b) For the same reason, σ acts on the space S / Φ ≈ [0 ,
1] of orbits, interchanging the endpoints. Therefore, σ sends some Φ-orbit to itself. This implies (b).c) Recall the polarity π defining the quadric Q = S . The orthogonal group Ψ =PSO(3 ,
1) with respect to π is triply transitive on S . The stabilizer Ψ n,s is transitive onthe interior open segment on Z defined by these points. Moreover, the stabilizer is abelian(isomorphic to C × ) and contains Φ. Hence we can use it to move the orbit constructedin (b) into the plane { ( x, y, | x, y ∈ R } without disturbing the action of Φ.In [2], a gl star S is called axial if there is a line A , called axis, which meets all lines in S . We observe: L¨owen and Steinke PROPOSITION 4.3
If a non-ordinary (topological) rotational gl star is axial, then theaxis as defined above coincides with the rotation axis Z .Proof. If A / ∈ S , then the map h : S → A sending L ∈ S to L ∧ A is continuous, andits image contains the open subset B ⊆ A of all exterior points on A . On B , there is acontinuous inverse map j : B → h − ( B ), because the gl star S is topological. Then thesurface S contains an open subset homeomorphic to the real line, a contradiction. Thus A belongs to S and meets the rotation axis Z ∈ S . Every line ϕ ( A ), ϕ ∈ Φ, has the sameproperties. If A = Z , then these lines cover either a cone or a plane. Now let L ∈ S notbe contained in a plane containing Φ A . Then L meets all lines ϕ ( A ), and hence mustpass through their only common point A ∧ Z , which lies in the interior of S . By density,the remaining lines of S also pass through this point, and S is an ordinary line star.Rotational gl stars such that every line meets the rotation axis have been called latitudinal [3]. Justified by the above proposition, we shall often use the term axial instead. THEOREM 4.4
A rotational gl star S is axial if and only if the reflection ζ E about anyplane E containing the axis Z acts on S .Proof. The reflection ζ E fixes all lines contained in E . For ϕ ∈ Φ, the line ϕ ( L ) containedin ϕ ( E ) is mapped to ζ E ϕ ( L ) = ζ E ϕζ E ( L ) = ϕ − ( L ) ∈ S . Conversely, assume that S = ζ E ( S ) is not axial. Then there is a line L ∈ S not meeting Z . Choose ϕ ∈ Φ suchthat ϕ ( L ) is parallel to E in the Euclidean sense. Then ϕ ( L ) and ζ E ϕ ( L ) meet at infinity,a contradiction since both lines belong to S . First we simplify the task of recognizing a gl star.
PROPOSITION 5.1
Let σ : S → S be a continuous fixed point free involution. Theset S = { p ∨ σ ( p ) | p ∈ S } of lines in PG(3 , R ) is a topological gl star if two distinctlines of S never meet in a point that is exterior with respect to S . In fact, it suffices toassume that every line in S has a neighborhood in which this condition holds.Proof. Clearly, S is compact. The quotient map S → S /σ is a two sheeted coveringmap. Hence the quotient space is a compact connected surface (in fact, a real projectiveplane). First suppose that the condition on intersections holds globally. Then the onlyproperty that remains to be verified is that every exterior point p lies on some line of S .Let E be a plane containing p , disjoint from S . The map i E : S → E : p → ( p ∨ σ ( p )) ∧ E egular parallelisms admitting a 2-torus 9induces a continuous injection S /σ → E of compact connected surfaces. Such a mapis surjective by domain invariance. If the condition on intersections holds only locally,then the map i E is locally injective, hence a covering map. Every covering map betweentwo copies of the real projective plane is bijective, hence the global intersection propertyfollows.Now we turn to rotational gl stars S that are axial. Let E be a plane containing therotation axis Z . Then S induces in E a gl pencil P , the 2-dimensional analog of a gl star.That is, P covers the plane E , and two lines of P never meet except in the interior of thecircle S ∩ E . Moreover, P is symmetric, i.e., invariant under orthogonal reflection ρ inthe line Z .As for rotational gl stars, we shall mostly assume a standard position for symmetric glpencils: the affine part of E is R = { ( x, z ) | x, z ∈ R } , the circle is the unit circle, and(as in Proposition 4.2) the coordinate axes X and Z belong to the pencil.In order to give a complete description of all symmetric gl pencils, we introduce thefollowing notation. The intersection of the unit circle with the first quadrant, i.e., theshort segment from (1 ,
0) to (0 , A . The opposite segment − A joinsthe points ( − ,
0) and (0 , − THEOREM 5.2 a) Let A ⊆ S be the segment defined above, and let µ : A → − A bea continuous bijection which sends (1 , to ( − , . Then µ uniquely extends to a fixedpoint free continuous involution σ : S → S that commutes with ρ , the reflection in thethe z -axis Z . The set P = { p ∨ σ ( p ) | p ∈ S } is a topological symmetric gl pencil.b) Every symmetric gl pencil in standard position arises in this way. In particular, everysymmetric gl pencil is topological. Remark:
A slightly adapted version of this result holds for non-symmetric gl pencils,with virtually the same proof. We have chosen this version because symmetric gl pencilsarise from gl stars that are axial and rotational.
Proof. a) If p, q are two points on the circle, then the pair ( p, σ ( p )) separates the pair( q, σ ( q )). Indeed, otherwise some segment B of S bounded by p and σ ( p ) contains q and σ ( q ), hence B is mapped to itself and contains a fixed point of σ , a contradiction.Therefore, any two lines of P meet in the interior of the circle and not in the exterior.The proof that every exterior point is covered by P is virtullay the same as in Proposition5.1.b) Conversely, if P is a symmetric gl pencil (not assumed to be topological) then theseparation property encountered in (a) holds for P , because otherwise some pair of P -lines would meet in the exterior of S . This means that the bijection µ : A → − A sendinga point p to the other intersection point of S with the P -line containing p is strictly0 L¨owen and Steinkemonotone with respect to a natural ordering on S \ { √ (1 , − } ≈ R . Hence µ is ahomeomorphism and extends uniquely to a fixed point free continuous involution σ on S that commutes with ρ . Then P is the gl pencil defined by σ , and P is compact, hencetopological.Instead of using the involution σ , the authors of [3] describe the lines of a gl pencil bytheir intersections with the segment A and with the z -axis. The pencils are thus givenby a function f which expresses the z -coordinate of the latter point in terms of the z -coordinate of the former. This results in rather tricky conditions that f has to satisfyand makes it hard to construct examples. In our description, no such conditions appear.Now we can describe all latitudinal (i.e., axial and rotational) gl stars in standard position. THEOREM 5.3 [3] a) Let S be a latitudinal gl star in standard position and let E be aplane containing the rotation axis Z . In E then S induces a symmetric gl pencil P , fromwhich S can be recovered as S = Φ( P ).b) In particular, every latitudinal gl star is topological.c) Conversely, every symmetric gl pencil P in E yields a latitudinal gl star S = Φ( P ).Proof. Assertions a) and c) are obvious. Note that the reflection of E about Z is inducedby the element of order 2 in Φ. Part b) follows from compactness of P (Theorem 5.2 (b))and of Φ, together with continuity of the surjective map Φ × P → S : ( ϕ, L ) → ϕ ( L ).The proof that latitudinal gl stars are topological given in [3] uses four pages in print.This is because it consists of an explicit description of the map sending a point-line pair( p, L ) to the line parallel to L and containing p . Verification of the continuity of severalimplicit maps is required in the course of this proof. In order to specify a rotational gl star S in standard position one has to enumerate theΦ-orbits of lines in S . The orbit consisting of Z alone and the orbit consisting of thelines perpendicular to Z and passing through the origin are always present. There maybe other lines L ∈ S meeting Z . In this case, the orbit of L consists of all lines containedin some Φ-invariant cone. If L ∈ S misses Z , then its orbit is one of the two regulicontained in some hyperboloid invariant under Φ. In order to distinguish between thetwo reguli, it is convenient to adopt this convention: We call Φ( L ) a right regulus if L may be parametrized by a function v : R → L with v ( t ) = ( x ( t ) , y ( t ) , z ( t )) such thatthe lines R ( x ( t ) , y ( t )) in the ( x, y )-plane rotate counterclockwise and the function z ( t ) isincreasing. If z is decreasing, then we have a left regulus. Now S may be described by exhibiting the points σ ( p t ) or the lines L t = p t ∨ σ ( p t ) for p t = ( √ − t , , t ) , t ∈ [0 , . egular parallelisms admitting a 2-torus 11A necessary condition is that L = Z and that L passes through the origin and contains p ; hence, L is the x -axis X . Instead of exhibiting these lines, we may specify the conesand hyperboloids carrying their orbits, for example by writing down the hyperbolas orline pairs obtained by intersection with the plane y = 0. In this case, it is necessary todecide for each hyperpoloid whether the right or left regulus should be taken. In order toget a continuous involution σ and a topological gl star, the choice of left or right regulushas to obey a rule: If no lines L t meeting Z occur for t ≤ t ≤ t , then the choice of leftor right must not be changed within the interval [ t , t ]. In other words, a change fromleft to right or vice versa is only possible at parameters t corresponding to a cone.A special case is that of symmetric rotational gl stars. By symmetric, we mean symmetryof each of the cones and hyperboloids with respect to the ( x, y )-plane. This means thatthe third coordinate of σ ( p t ) is always equal to − t , or that the line pairs or hyperbolasseen in the plane y = 0 all have symmetry axes Z and X . Note, however, that the gl staritself is usually not symmetric with respect to the ( x, y )-plane, because this symmetrymap exchanges the right and left reguli in each hyperboloid. Instead, we have symmetryof the gl star about the y -axis (or in fact about any line passing through 0 and orthogonalto Z ).We remark here that the full automorphism group of the parallelism from a symmetricand rotational gl star is a degree four extension of T and is isomorphic to the correspond-ing group in the case of a latitudinal gl star. However, the actions are different. In thesymmetric case, we have the reflection about the y -axis, while in the latitudinal case thereis the reflection about any plane containing Z , see 4.4. More details on full group actionsare given in [6]. Note also that only Clifford paralllelism is both latitudinal and symmetric.In the symmetric case, it is possible to describe all examples in an easy manner.Let a : [0 , → [0 , ∞ [ be an increasing bijection satisfying the conditions t ≤ a ( t ) a ( t ) for all t ∈ [0 , , and (1)lim 0. For t > 0, let H t be the hyperbola or line pairdefined by the equation a ( t ) x − z = c ( t ) . This yields a family of Φ-invariant cones and hyperboloids. THEOREM 6.1 [6] For any admissible choice of reguli, the cones and hyperboloids de-fined above yield a topological, symmetric and rotational gl star, and all such gl stars instandard position are obtained in this way. Proof. The hyperbola or line pair H t passes through the points p ± t = ( √ − t , ± t ) onthe right half of the circle S and intersects the positive x -axis at v t = ( c ( t ) a ( t ) , ± a ( t ), and by condition (2), the points v t converge to the origin for t → 0. This shows that the lines L t contained in Φ( H t ) and passing through p + t convergeto X or Z for t → t → 1, respectively. If any two lines in the resulting set S meetoutside the unit sphere, then there is a pair of hyperbolas H t , H t with t < t whoseupper right branches meet outside the circle. Now the asymptotes of these branches haveslopes a ( t ) < a ( t ), and the upper right intersection point of H t with the circle is belowthe corresponding point of the other hyperbola. This shows that the right upper branchesof the two hyperbolas meet twice outside the circle, where we count a point of touchingas two points of intersection. Now the hyperbolas are symmetric about both X and Z ,hence they have 8 points in common and coincide, a contradiction. Now Theorem 5.1implies that we have obtained a symmetric rotational gl star.The necessity of the conditions imposed on the function a are easily checked, and thisproves the last statement of the theorem. The special gl stars of the previous two sections correspond to some set of functions be-tween real intervals. For rotational gl stars in general, one expects that pairs of suchfunctions are needed, because the points σ ( p t ) from the introduction to Section 6 have tobe specified. In fact, Betten and Riesinger claim a result concerning a family of rotationalgl-stars corresponding to pairs of such functions ([6], Theorem 42), inclucing examplesthat are neither symmetric nor axial. However, their proof is invalid because it uses theargument about intersecting hyperbolas from the proof of Theorem 6.1 in a situationwhere the two hyperbolas are not known to share two axes of symmetry. Here we give aproof which has virtually no overlap with their attempt. Also we have corrected a ratherobvious omission (surjectivity of the function f ), and we admit both left and right reguliby introducing a sign function ε .Let f : [0 , → [0 , 1] be an increasing (continuous) bijection and let g : [0 , → [ − , 0] bea continuous non-decreasing function satisfying g (0) = − g (1) = 0 and the inequalities − p − f ( t ) ≤ g ( t ) ≤ t . Moreover let ε : [0 , → { , − } be any function that is constant on every interval I ⊆ [0 , 1] such that − p − f ( t ) < g ( t ) holds on I . For the points p t = ( √ − t , , t ), t ∈ [0 , σ f,g ( p t ) = ( g ( t ) , ε ( t ) p − f ( t ) − g ( t ) , − f ( t )) , and complete σ to a Φ-invariant involution of S in the unique way. Let S f,g be theresulting set of lines, S f,g = { u ∨ σ f,g ( u ) | u ∈ S } . egular parallelisms admitting a 2-torus 13 THEOREM 7.1 The set S f,g defined above is a topological, rotational gl star.Proof. 1. Note that σ is surjective and in fact an involution of S because we have assumedthat f is surjective. This condition is missing in [6]. Note also that S = S f,g contains thelines L t = p t ∨ σ ( p t ) and, in particular, contains the z -axis Z = L , the x -axis L , andthe horizontal lines ϕ ( L ), ϕ ∈ Φ. In particular, on the equator S ∩ Z ⊥ the involution σ induces the antipodal map p → − p .2. We want to apply Proposition 5.1. First we need to check that σ is continuous. Thefunction ε is constant except at zeros of p − f ( z ) − g ( t ) , hence σ is continuous on thehalf meridian M = { p t | t ∈ [0 , } and on σ ( M ), and hence everywhere.3. By definition, no line in S meets either the Z-axis or one of the horizontal lines ϕ ( L )outside the sphere. So we only have to deal with exterior points of intersection betweentwo of the remaining lines. By rotation invariance, we may always assume that one of thetwo lines is among the lines L t .4. The sign factor ε serves to choose between left and right reguli. Since swapping left andright reguli does not affect the properties of a gl star, we may assume that ε is constant,say ε = − f = id , then S is a symmetric rotational gl star by Theorem 6.1, and if g ( t ) = − p − f ( t ) holds for all t , then S is an axial gl star by Theorem 5.3. Now supposethat the lines L t and ϕ ( L s ) intersect outside S . Then we can choose a number r > max { s, t } and change the functions f and g on the interval [ r, 1] such that the conditionfor symmetric or axial gl stars is satisfied on some subinterval [ u, 1] with u > r . Thenew line set S ′ still contains the same pair of intersecting lines. In order to prove thatsuch a pair cannot occur, we may therefore assume that the local intersection conditionof Theorem 5.1 is satisfied near the z -axis Z .6. We proceed to show that the local intersection condition is satisfied near every line L t , t > 0. By rotational symmetry, it suffices to show that L t does not meet any line ϕL s outside the sphere, where ϕ ∈ Φ rotates through an angle less than π/ t < s . Suppose thatthese lines meet in a point v outside the sphere. We shall discuss the case that v lies inthe upper half space defined by z ≥ 0. The case z ≤ v as projection centerand a suitable image plane E , see Figure 1. For a point p , the projection is given by p ∗ = ( p ∨ v ) ∧ E . The fact that v ∈ L t means that p ∗ t = q ∗ t . Similarly, v ∈ ϕL s means that( ϕp s ) ∗ = ( ϕq s ) ∗ . (3)Now by the conditions on f and g , we know that q s lies in a region R on the sphere boundedabove by the orbit circle Q t = Φ q t , bounded backwards by the circle B : x = g ( t ) and infront by the circle F : x = 0, and bounded below by the circle D : y = 0. Thus q ∗ s liesin the region R ∗ bounded by the conics Q ∗ t , B ∗ , F ∗ and D ∗ . In particular, q s lies to theleft of B ∗ . From equation (3) we see that the orbits P s = Φ p s and Q s have intersectingimages P ∗ s and Q ∗ s . Then the orbits P t and Q t , which lie between the former two orbits,4 L¨owen and Steinkehave images intersecting in two points. These are p ∗ t and one further point, situated tothe right of p ∗ t on the lower arc of P ∗ t . Since ϕ rotates through an angle less than π/ ϕ is a clockwise rotation, so that q ∗ s is moved to the right. Then p ∗ s is movedto the left, and cannot hit one of the intersection points of P ∗ s and Q ∗ s , which lie betweenthose of P ∗ t and Q ∗ t . This shows that an intersection of L t and ϕL s in the upper half spaceis not possible. yx zp ∗ t q ∗ s p ∗ s B ∗ Q ∗ t F ∗ D ∗ R ∗ Figure 1: Central projection from v onto the plane v ⊥ 8. Neihgborhoods of L are special. Such a neighborhood contains lines ϕL s with s ≥ ϕ either rotates through an angle near 0 or through an angle near π . Lines of thefirst kind obviously do not meet lines of the second kind outside the sphere. Lines of thesame kind do not meet outside the sphere by what we have shown in the preceding steps.This completes the proof. Remark. The class of gl stars obtained in Theorem 7.1 is substantial but does notcomprise all rotational gl stars. The restrictions arise from the monotonicity conditionimposed on g , which is needed to make this proof work. Together with the necessarycondition g (1) = 0, it enforces that g is always negative and thus excludes many regulifrom being part of the gl stars constructed here. However, we shall show later that everyregulus consisting of 2-secants of the sphere can be part of a rotational gl star, see Theo-rem 7.4 below.We return to the method of describing a topological rotational gl star presented at thebeginning of Section 6. One starts from a set H of cones and hyperboloids, each of theminvariant under the action of the rotation group Φ, and the gl star S then consists ofegular parallelisms admitting a 2-torus 15the axis Z , all horizontal lines containing the origin, and all lines contained in one of thecones, plus one of the two reguli contained in every one of the hyperboloids. THEOREM 7.2 In the ( x, z ) -plane R × × R , consider a family of subsets H a definedby equations a x − ( z − b ) = c , (4) for all a > , where b = b ( a ) and ≤ c = c ( a ) are continuous functions of a . Assumethat(1) b ( a ) + c ( a ) < a for all a .(2) lim a → b ( a ) = 0 = lim a → c ( a ) a .(3) Every point p t = ( √ − t , , t ) , = t ∈ ] − , , belongs to exactly one set H a .(4) Every point ( x, , z ) with x > , z = 0 and x + z ≥ belongs to at most one set H a .Then we obtain a topological, rotational gl star S , consisting of the z -axis Z , the orbit Φ X of the x -axis X , and all lines contained in cones Φ H a , c ( a ) = 0 , plus the right regulicontained in the hyperboloids Φ H a for c ( a ) = 0 . On intervals bounded by consecutive zerosof the function c , right reguli may be replaced by left reguli.Conversely, every topological, rotational gl star in standard position is obtained in thisway. Remark. The axial and symmetric cases are given by c ≡ b ≡ 0, respectively.It is not hard to recover the earlier Theorems 5.3 and 6.1 from the present result in thosespecial cases. Proof. 1. Condition (1) says that every set H a intersects X inside the unit circle. Con-sequently, H a contains a point p t ( a ) with t ( a ) > 0. The function t : ]0 , ∞ [ → ]0 , 1[ iscontinuous and, by (3), bijective. Condition (4) implies that t is not decreasing, so itis an increasing function of a . Every point p t , t ∈ ]0 , t . There is a unique line L t ∈ S whichsatisfies the same equation and belongs to the right regulus of Φ H a if c = 0. Therefore,the map L : t → L t is continuous. By (1), L t meets the interior of the sphere S , hence L t intersects the sphere in a second point q t =: σ ( p t ).In step 2, we shall show that the map L : ]0 , → S is continuously extended to [0 , 1] bysetting L = X and L = Z . Then it will follow that σ extends uniquely to a Φ-invariantmap σ : S +2 → S − from the (closed) upper hemisphere to the lower hemisphere. By (4)and by Φ-invariance, σ is injective. On the equator S +2 ∩ S − , σ induces the antipodal map − id . It follows that σ is surjective, because the antipodal map is not null homotopic inany proper subset of the lower hemisphere, or alternatively, because the connected imageof σ is Φ-invariant and contains both the equator and σ ( p ) = − p . Thus we obtain6 L¨owen and Steinkea fixed point free involution σ : S → S extending σ , such that S consists of the lines p ∨ σ ( p ), p ∈ S .In order to complete the proof by applying Proposition 5.1, we need to know that twolines in S never meet outside the sphere. This follows from condition (4) if the lines arecontained in two distinct sets Φ H a . It follows from (1) if one of the lines belongs to Φ X .Lines of the same regulus are always disjoint. The last remaining case arises for cones H a . Two lines of the cone intersect in the vertex, which moreover belongs to Z ∈ S . Sowe have to show that all vertices of cones are inside the sphere. This follows from (4)together with our claim that L t → Z if t → t → p t → p ∈ Z . The line L t contains p t . Since a ( t ) → ∞ , the pointat infinity of L t belongs to a Φ-orbit that converges to the point at infinity of Z . Thisproves that L t → Z . For t → 0, we have that p t → p ∈ X . Moreover, if c = 0, then L t contains the vertex of the cone containing p t , and the vertex converges to the orgin since b → 0. For those t with c = 0, the line L t contains a point from the Φ-orbit of the vertex v = ( b, ca ) of the hyperbola H a . Hence, the distance of L t from the origin is at most k v k ,and L t → X follows from (2).3. Conversely, suppose that we ave a topological rotational gl star S . Then we obtaina system of hyperboloids and cones and a family of equations (4). The necessity ofconditions (3) and (4) is obvious, as they are part of the definition of a gl star. Condition(1) expresses the fact that H a intersects X inside the unit circle, which follows from X ∈ S . Condition (2) is obtained by reversing the argument in step 2. This proves thelast assertion.We wish to reshape the preceding result, describing the sets H a in terms of their inter-section points with the unit circle rather than by the coefficients of their equation. Thismight help to guess examples.Consider the situation of Theorem 7.2 and retain the notation used above. The set H a intersects the unit circle in 4 points. Those with a positive x -coordinate are p t ( a ) and p − s ( a ) , where t ( a ) = b + p b + ( a + 1)( a − b − c ) a + 1 , (5) − s ( a ) = b − p b + ( a + 1)( a − b − c ) a + 1 . (6)As before, the functions t and s are homeomorphisms ]0 , ∞ [ → ]0 , b and c are expressed in terms of t and s by the equations b ( a ) = ( a + 1) (cid:18) t − s (cid:19) , (7) c ( a ) = a − ( a + 1) (cid:18) t + s (cid:19) + a (cid:18) t − s (cid:19) ! . (8)egular parallelisms admitting a 2-torus 17This makes sense if the expression for c is nonnegative. Thus, the functions t and s arerestricted by the condition a − ( a + 1) (cid:18) t + s (cid:19) + a (cid:18) t − s (cid:19) ! = − (cid:0) ( t − s ) a − − t − s ) a + ( t + s ) (cid:1) ≥ a > 0. This inequality can be rewritten as a a + 1 − ts ≥ ( a + 1) (cid:18) t − s (cid:19) . (9)With the formulas (7) and (8) for b and c the equation for H a becomes h x,z ( a ) := ( a + 1)( x − t − s ) z + ts ) + 1 − x − z = 0 . (10)Using these results, we may rewrite Theorem 7.2 as follows. COROLLARY 7.3 Let t, s : [0 , ∞ [ → [0 , be homeomorphisms such that(1) lim = a → t ( a )+ s ( a )2 a = 1 .(2) a a +1 − t ( a ) s ( a ) ≥ ( a + 1) (cid:16) t ( a ) − s ( a )2 (cid:17) for all a > .(3) The map h x,z given by a ( a + 1)( x − t ( a ) − s ( a )) z + t ( a ) s ( a )) + 1 − x − z = a ( x + z − 1) + ( a + 1)( t ( a ) − z )( s ( a ) + z ) has at most one positive root for each ( x, z ) such that x > , z = 0 and x + z ≥ .Then we obtain a topological, rotational gl star by applying Theorem 7.2 to the values b ( a ) and c ( a ) defined by equations (7) and (8) above.Conversely, every topological, rotational gl-star in standard position can be obtained inthis way for suitable functions t and s as above. Remark. Here, the symmetric case is given by s = t and the axial case by c ≡ Proof. Note first that we have required continuity of t ( a ) and s ( a ) at a = 0, whereascontinuity of b ( a ) and c ( a ) in Theorem 7.2 is required only for a > 0. Therefore, lim a → b =0 holds. Moreover, lim = a → ca = 0 follows easily from the present condition (1). Fromequations (7) and (8) we get 0 ≤ b + c = a − ( a +1) ts ≤ a , which gives condition (1) ofTheorem 7.2. Conditions (4) of the theorem is condition (3) of the corollary, and condition8 L¨owen and Steinke(3) of the theorem is guaranteed because the surjective functions t and s describe, by theirdefinition, the z -coordinates of the intersections of H a with the unit circle. Example. Checking the properties of the functions t and s can be very hard. Wesucceded in one concrete example. For each r > ϕ r : [0 , ∞ [ → [0 , 1[ be the mapdefined by ϕ r ( a ) = a ( a + r ) a + ra + r = 1 − ra + ra + r . If we set t = ϕ and s = ϕ , then after some lengthy computation we obtain h x,z ( a ) = l ( a )(2 a + 3 a + 3)( a + 2 a + 2) , where l ( a ) = 2 x a + 7 x a + (13 x − z + z − a + (12 x − z − a + (6 x − z + z ) a − az − z , Using Descartes’ rule of signs, it can be shown that this function has exactly one positiveroot for each admissible pair ( x, z ). The other conditions of the Corollary can be checkedwithout difficulty and one gets a rotational gl star.By continuity, the choice t = ϕ r and s = ϕ r ′ , where ( r, r ′ ) is sufficiently close to ( , Remark. Here is a different perspective on condition (3) of the corollary. We focusattention on a horizontal line z = z = 0. If we let a z = ∞ , if | z | ≥ ,t − ( z ) , if 0 < z < ,s − ( − z ) , if − < z < , and p z ( a ) = a + 1 a ( z − t ( a ))( z + s ( a ))where z = 0, then condition (3) from Corollary 7.3 is equivalent to p z being injective (infact, strictly decreasing) on the interval ]0 , a z [ for all z = 0.Indeed, h x,z ( a ) = 0 yields x + z − a +1 a ( z − t ( a ))( z + s ( a )), and every ( x, z ) with x > x > − z is covered by the sets H a at most once. Only those a for which( z − t ( a ))( z + s ( a )) > a z .We conclude with a fairly versatile construction based on Theorem 7.2. We shall obtaina large set of examples which shows, in particular, that the family of examples providedby Theorem 7.1 is far from exhausting all possibilities.egular parallelisms admitting a 2-torus 19We shall use a technique of linear interpolation which is awkward to handle for hyperbolas.Therefore we use a transformation which translates families of hyperbolas into familiesof parabolas. As always, we allow degenerate hyperbolas (i.e., line pairs). So let ω : R × × R → R be given by ( u, v ) = ω ( x, , z ) = ( z, x ) . This maps the hyperbola (or line pair) H with equation H : a x − ( z − b ) = c (where 0 ≤ c and 0 < a ) to the parabola P = ωH with equation P : v = α ( u − β ) + γ, where α = 1 a β = b, γ = c a . Moreover, ω sends the vertices ( b, ± ca ) of H to the vertex ( β, γ ) of ωH . The z -axis Z is mapped to the u -axis U , and the right half ( x ≥ 0) of the unit circle C goes to theparabolic arc D = { ( u, − u ) | − ≤ u ≤ } . Let R be the bounded region bounded by U ∪ D .We choose a two-way infinite sequence of parabolas P i indexed by integers i , defined byequations v = p i ( u ) = α i ( u − β i ) + γ i with α i > γ i ≥ 0, such that the followingconditions hold, compare Figure 2:(1) The sequence { a i } i is strictly increasing with lim i →−∞ a i = 0 and lim i →∞ a i = ∞ .(2) lim i →−∞ γ i = 0 . (3) P i ∩ D consists of two points separated by the v -axis V , and each point of P i +1 ∩ D separates the points in P i ∩ D on the arc D .(4) P i and P i +1 do not intersect outside R .(5) The vertices ( β i , γ i ) of P i converge to the origin as i → ∞ .Now we interpolate i → p i in order to obtain a one-parameter family P r , r ∈ R , ofparabolas. That is, for t ∈ [0 , 1] we define P i + t by p i + t := (1 − t ) p i + tp i +1 . These parabolas simply cover the two regions of R \ R bounded by P i and P i +1 , and thelimit conditions (1) and (3) now hold for r → ±∞ . This is obvious for (1), and for (3)we use the following observation. For u < min { β i , β i +1 } , the functions p i and p i +1 areboth decreasing, and the same holds for p i + t , t ∈ [0 , u t , v t ) of p i + t satisfies u t ≥ min { β i , β i +1 } , and likewise, u t ≤ max { β i , β i +1 } . Moreover, the function p i + t lies between the functions p i and p i +1 , hence its minimum value is bounded above by thelarger one of their minimum values. This proves our claim, and we have the following.0 L¨owen and Steinke uvP i P i +1 ( β i , γ i )( β i +1 , γ i +1 ) RD uvP i +1 P i ( β i +1 , γ i +1 )( β i , γ i ) R D Figure 2: The parabolas P i and P i +1 THEOREM 7.4 Given a sequence P i of parabolas satisfying conditions (1), (2) and (3)as above, taking ω -inverse images of the interpolated family P r , r ∈ R , we obtain a familyof hyperbolas or line pairs as in Theorem 7.2, and thus we obtain a topological, rotationalgl star.Proof. Remember that α = a is a strictly increasing function of r ∈ R , hence the familyof hyperbolas may be parametrized by a > 0. Only condition (1) of 7.2 then needs anexplanation: this condition expresses that the hyperbolas intersect the axis X inside theunit circle. Under ω , this corresponds to the property that the parabolas intersect the v -axis V inside the region R bounded by the parabolic curve D and the axis U . This isguaranteed by our condition (2) above. Remarks. 1. As announced earlier, this result shows that Theorem 7.1 is far from givinga complete view of all rotational gl stars in standard position. We see here that everyhyperboloid meeting the ( x, y )-plane inside the unit sphere carries two reguli that maybe embedded in rotational gl stars. By contrast, in Theorem 7.1 the hyperboloids arerestricted by the incisive condition g ( t ) ≤ a . They may even tendto ∞ as a → ∞ .2. It is easy to modify the construction so that one may start with a finite sequence ora one-way infinite sequence of parabolas. One just completes the family by adding tp i , t ∈ [0 , i , and by adding tp j , t ≥ 1, if the sequenceends with index j and the vertex of P j is the origin. References [1] D. Betten and R. 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