Clifford Parallelism Research Articles

Characterising Clifford parallelisms among Clifford-like parallelisms

We recall the notions of Clifford and Clifford-like parallelisms in a 3-dimensional projective double space. In a previous paper the authors proved that the linear part of the full automorphism group of a Clifford parallelism is the same for all Clifford-like parallelisms which can be associated to it. In this paper, instead, we study the action of such group on parallel classes thus achieving our main results on characterisation of the Clifford parallelisms among Clifford-like ones.

Rotational spreads and rotational parallelisms and oriented parallelisms of $${\mathrm{PG}(3,{\mathbb {R}})}$$ PG ( 3 , R )

We introduce topological parallelisms of oriented lines (briefly called oriented parallelisms). Every topological parallelism (of lines) on PG(3,R) gives rise to a parallelism of oriented lines, but we show that even the most homogeneous parallelisms of oriented lines other than the Clifford parallelism do not necessarily arise in this way. In fact we determine all parallelisms of both types that admit a reducible SO(3)-action. Only the Clifford parallelism admits a larger group Surprisingly, it turns out that there are far more oriented parallelisms of this kind than ordinary parallelisms. In particular, we show that the SO(3)-invariant ordinary parallelisms constructed by Betten and Riesinger are the only ones compatible with this group action. We also study the rotational Betten spreads used in this construction and their automorphisms. The automorphism group of the (oriented) parallelisms is always SO(3), no matter how large the automorphism group of the non-regular spread is. There are far more isomorphism types of parallelisms than types of spreads.

A characterization of Clifford parallelism by automorphisms

Betten and Riesinger have shown that Clifford parallelism on real projective space is the only topological parallelism that is left invariant by a group of dimension at least 5. We improve the bound to 4. Examples of different parallelisms admitting a group of dimension 3 are known, so 3 is the "critical dimension".

Clifford-like parallelisms

Given two parallelisms of a projective space we describe a construction, called blending, that yields a (possibly new) parallelism of this space. For a projective double space ({mathbb P},{mathrel {parallel _{ell }}},{mathrel {parallel _{r}}}) over a quaternion skew field we characterise the “Clifford-like” parallelisms, i.e. the blends of the Clifford parallelisms mathrel {parallel _{ell }} and mathrel {parallel _{r}}, in a geometric and an algebraic way. Finally, we establish necessary and sufficient conditions for the existence of Clifford-like parallelisms that are not Clifford.

Parallelisms of $$\mathrm{PG}(3,\mathbb R)$$ PG ( 3 , R ) admitting a 3-dimensional group

Betten and Riesinger constructed Parallelisms of $\mathop{\rm PG}(3,\mathbb R)$ with automorphism group $\mathop{\rm SO}(3,\mathbb R)$ by applying the reducible $\mathop{\rm SO}(3,\mathbb R)$-action to a rotational Betten spread. This was generalized by the present author so as to include oriented parallelisms (i.e., parallelisms of oriented lines). In this way, a much larger class of examples was produced. Here we show that, apart from Clifford parallelism, these are the only topological parallelisms admitting an automorphism group of dimension 3 or larger. In particular, we show that a topological parallelism admitting the irreducible action of $\mathop{\rm SO}(3,\mathbb R)$ is Clifford.

Clifford parallelisms defined by octonions

We define (left and right) Clifford parallelisms on a seven-dimensional projective space algebraically, using an octonion division algebra. Thus, we generalize the two well-known Clifford parallelisms on a three-dimensional projective space, obtained from a quaternion division algebra. We determine (for both the octonion and quaternion case) the automorphism groups of these parallelisms. A geometric description of the parallel classes is given with the help of a hyperbolic quadric in a Baer superspace, obtained from the split octonion algebra over a quadratic extension of the ground field, again generalizing results that are known for the quaternion case. In contrast to the quaternion case, the orbits of the two Clifford parallelisms under the group of direct similitudes of the norm form of the algebra are non-trivial in the octonion case. The two spaces of parallelisms can be seen as the point sets of two point-line geometries, both isomorphic to the seven-dimensional projective space. Together with the original space, we thus have three versions of this projective space. We introduce a triality between them which is closely related to the triality of the polar space of split type $$\mathrm {D}_4$$ .

Pencilled regular parallelisms

Over any field \({\mathbb{K}}\), there is a bijection between regular spreads of the projective space \({{\rm PG}(3,\mathbb{K})}\) and 0-secant lines of the Klein quadric in \({{\rm PG}(5,\mathbb{K})}\). Under this bijection, regular parallelisms of \({{\rm PG}(3,\mathbb{K})}\) correspond to hyperflock determining line sets (hfd line sets) with respect to the Klein quadric. An hfd line set is defined to be pencilled if it is composed of pencils of lines. We present a construction of pencilled hfd line sets, which is then shown to determine all such sets. Based on these results, we describe the corresponding regular parallelisms. These are also termed as being pencilled. Any Clifford parallelism is regular and pencilled. From this, we derive necessary and sufficient algebraic conditions for the existence of pencilled hfd line sets.

Clifford parallelisms and external planes to the Klein quadric

For any three-dimensional projective space ${\mathbb P}(V)$, where $V$ is a vector space over a field $F$ of arbitrary characteristic, we establish a one-one correspondence between the Clifford parallelisms of ${\mathbb P}(V)$ and those planes of ${\mathbb P}(V\wedge V)$ that are external to the Klein quadric representing the lines of ${\mathbb P}(V)$. We also give two characterisations of a Clifford parallelism of ${\mathbb P}(V)$, both of which avoid the ambient space of the Klein quadric.

A note on Clifford parallelisms in characteristic two

It is well known that a purely inseparable field extension L/F with some extra property and degree [L : F] = 4 determines a Clifford parallelism on the set of lines of the three-dimensional projective space over F. By extending the ground field of this space from F to L, we establish the following geometric description of such a parallelism in terms of a distinguished ‘absolute pencil of lines’ of the extended space: Two lines are Clifford parallel if, and only if, there exists a line of the absolute pencil that meets both of them. Mathematics Subject Classification (2010): 51A15, 51A40, 51J10, 51J15.

Collineation groups of topological parallelisms

Abstract Let P be a topological parallelism of the real projective 3-space PG3ℝ. We investigate the group AutP of all collineations of PG3ℝ leaving P invariant. We prove that AutP is a Lie group whose dimension is at most 6. The main result of the article says: if the dimension equals 6, then P is a Clifford parallelism. Furthermore, we show that there are no topological parallelisms in PG3ℝ with a 5-dimensional group.

Clifford parallelism: old and new definitions, and their use

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 $${{\rm PG}(3,\mathbb{R})}$$ whose (complex) focal lines (=directrices) form a regulus 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 $${{\rm PG}(3,\mathbb{R})}$$ onto those lines of $${{\rm PG}(5,\mathbb{R})}$$ having no common point with the Klein quadric; a regular parallelism P of $${{\rm PG}(3,\mathbb{R})}$$ is Clifford, if the spreads of P are mapped by γ onto a plane of lines (D2 = planarity definition). We prove the equivalence of (D1) and (D2). Associated with γ is a simple dimension concept for regular parallelisms which allows us to say instead of (D2): the 2-dimensional regular parallelisms of $${{\rm PG}(3,\mathbb{R})}$$ are Clifford (D3 = dimensionality definition). Submission of (D2) to λ−1 yields a complexification free definition of a Clifford parallelism which uses only elements of $${{\rm PG}(3,\mathbb{R})}$$ : 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) simultaneously 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 submitted to the group of all rotations about o; we prove, that a Clifford parallelism is generated. Example 2. We determine the group $${Aut_e({\bf P}_{\bf C})}$$ of all automorphic collineations and dualities of the Clifford parallelism P C and show $${Aut_e({\bf P}_{\bf C})\hspace{1.5mm} \cong ({\rm SO}_3\mathbb{R} \times {\rm SO}_3\mathbb{R})\rtimes \mathbb{Z}_2}$$ .

Generalized Clifford parallelisms

We define generalized Clifford parallelisms in PG ( 3 , F ) with the help of a quaternion skew field H over a field F of arbitrary characteristic. Moreover we give a geometric description of such parallelisms involving hyperbolic quadrics in projective spaces over suitable quadratic extensions of F .

Hyperflock determining line sets and totally regular parallelisms of PG $${(3,\,{\mathbb{R}})}$$

By a totally regular parallelism of the real projective 3-space \({\Pi_3:={{\rm PG}}(3, \mathbb {R})}\) we mean a family T of regular spreads such that each line of Π3 is contained in exactly one spread of T. For the investigation of totally regular parallelisms the authors mainly employ Klein’s correspondence λ of line geometry and the polarity π5 associated with the Klein quadric H5 (for details see Chaps. 1 and 3). The λ-image of a totally regular parallelism T is a hyperflock of H5, i.e., a family H of elliptic subquadrics of H5 such that each point of H5 is on exactly one subquadric of H. Moreover, \({\{\pi_5({{\rm span}} \,\lambda(\mathcal {X}))\vert\mathcal {X}\in\bf{T}\}=:\mathcal {H}_{\bf{T}}}\) is a hyperflock determining line set, i.e., a set \({\mathcal {Z}}\) of 0-secants of H5 such that each tangential hyperplane of H5 contains exactly one line of \({\mathcal {Z}}\) . We say that \({{{\rm dim}}({{\rm span}}\,\mathcal {H}_{\bf{T}})=:d_{\bf{T}}}\) is the dimension of T and that T is a dT- parallelism. Clifford parallelisms and 2-parallelisms coincide. The examples of non-Clifford parallelisms exhibited in Betten and Riesinger [Result Math 47:226–241, 2004; Adv Geom 8:11–32, 2008; J Geom (to appear)] are totally regular and of dimension 3. If \({\mathcal{G}}\) is a hyperflock determining line set, then \({\{\lambda^{-1}\,{\rm (}\pi_5(X){\,\cap H_5)\,|\, X\in\mathcal{G}\}}}\) is a totally regular parallelism. In the present paper the authors construct examples of topological (see Definition 1.1) 4- and 5-parallelisms via hyperflock determining line sets.

Topological parallelisms of the real projective 3-space

A parallelism of a projective 3-space Π is a family P of spreads such that each line of Π is contained in exactly one spread of P. A parallelism is said to be totally regular, if all its members are regular spreads. By a generalized line star with respect to an elliptic quadric Q of a classical projective 3-space we understand a set \(\cal A\) of 2-secants of Q such that each non-interior point of Q is incident with exactly one line of \(\cal A\). From each generalized line star we can construct a totally regular parallelism which we do in essential by the Thas-Walker construction. A parallelisms of the real projective 3-space PG(3, ℝ) is called topological, if the operation of drawing a line parallel to a given line through a given point is continuous. Clifford parallelisms are topological. Using generalized line stars we exhibit examples of non-Clifford topological parallelisms and of non-topological parallelisms.

J. A. Tyrrell and J. G. Semple, Generalized Clifford Parallelism. (Cambridge Tracts in Mathematics and Mathematical Physics, 61). VII + 140 S. Cambridge 1971. University Press. Preis geb. £3.60

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und MechanikVolume 52, Issue 9 p. 557-557 Buchbesprechung J. A. Tyrrell and J. G. Semple, Generalized Clifford Parallelism. (Cambridge Tracts in Mathematics and Mathematical Physics, 61). VII + 140 S. Cambridge 1971. University Press. Preis geb. £3.60 H. Keller, H. Keller Halle/SaaleSearch for more papers by this author H. Keller, H. Keller Halle/SaaleSearch for more papers by this author First published: 1972 https://doi.org/10.1002/zamm.19720520919AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume52, Issue91972Pages 557-557 RelatedInformation

On Riemannian manifolds of four dimensions

Introduction. It is well known that in three-dimensional elliptic or spherical geometry the so-called Clifford's parallelism or parataxy has many interesting properties. A group-theoretic reason for the most important of these properties is the fact that the universal covering group of the proper orthogonal group in four variables is the direct product of the universal covering groups of two proper orthogonal groups in three variables. This last-mentioned property has no analogue for orthogonal groups in n ( > 4 ) variables. On the other hand, a knowledge of three-dimensional elliptic or spherical geometry is useful for the study of orientable Riemannian manifolds of four dimensions, because their tangent spaces possess a geometry of this kind. I t is the purpose of this note to give a study of a compact orientable Riemannian manifold of four dimensions at each point of which is attached a three-dimensional spherical space. This necessitates a more careful study of spherical geometry than hitherto given in the literature, except, so far as the writer is aware, in a paper by E. Study [2] . 2 0ur main result consists of two formulas, which express two topological invariants of a compact orientable differentiable manifold of four dimensions as integrals over the manifold of differential invariants constructed from a Riemannian metric previously given on the manifold. These two topological invariants have a linear combination which is the Euler-Poincare characteristic.