A characterization of Clifford parallelism by automorphisms

Abstract

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'.MSC 2000: 51H10, 51A15, 51M30Consider R 4 as the quaternion skew field H. Then the orthogonal group SO(4, R) may be described as the product of two commuting copiesΛ,Φ of the unitary group U (2, C), consisting of the maps q → aq and q → qb, respectively, where a, b are quaternions of norm one and multiplication is quaternion multiplication. The intersection of the two factors is of order two, containing the map −id. Thus, passing to projective space, we get PSO(4, R) = Λ × Φ, a direct product of two copies of SO(3, R). The left and right Clifford parallelisms are defined as the equivalence relations on the line space of PG(3, R) formed by the orbits of Λ and Φ, respectively.The two Clifford parallelisms are equivalent under quaternion conjugation q →q; this is immediate from their definition in view of the fact that conjugation does not change the norm and is an anti-automorphism, i.e., that pq =qp. Note that both Λ and Φ are transitive on the point set of projective space. Since they centralize one another, each acts transitively on the parallelism defined by the other, and the group PSO(4, R) leaves both parallelisms invariant (we say that it consists of automorphisms of these parallelisms). For more information on Clifford parallels, see [1], [8], [4]. For generalizations to other dimensions, compare also [10].The notion of a topological parallelism on real projective 3-space PG(3, R) generalizes this example. A spread is a set C of lines such that every point is incident with exactly one of them, and a topological parallelism may be defined as a compact set Π of compact spreads such that every line belongs to exactly one of them; see, e.g., [5] for details. Many examples of different topological parallelisms have been constructed in a series of papers by Betten and Riesinger, see, e.g., [2] and references given therein.The group Σ = Aut Π of automorphisms of a topological parallelism is a closed subgroup of PGL(4, R). The identity component ∆ = Σ 1 is compact [7], and hence (conjugate to) a subgroup of PSO(4, R) ∼ = SO(3, R)×SO(3, R). Hence, in the case of the Clifford parallelism, ∆ is the 6-dimensional group PSO(4, R) that we used to define the parallelism.

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

“…Here we show that only Clifford parallelism admits the irreducible SO 3 R-action. This implies the result of [6] mentioned above, because every subgroup Φ ≤ PSO(4, R) with dim Φ ≥ 4 contains a subgroup SO 3 R with irreducible action. Non-Clifford examples of regular ordinary parallelisms with a two-dimensional (torus) group have been given in [3], compare also [5] for a fresh view of the construction.…”

“…Proof. 1) It is known that dim Aut Π ≥ 4 characterizes Clifford parallelism, see [6] and, for the orientable case, [8]. However, we do not need this fact.…”

“…In particular, ∆ is contained in the maximal compact PO(4, R) ≤ PGL(4, R). It is known that only (oriented or non-oriented) Clifford parallelism has dim ∆ ≥ 4, see [6], [8] (in fact, ∆ = PSO(4, R) is 6-dimensional in this case).…”

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

“…The automorphism groups of an ordinary parallelism and of its associated oriented parallelism are the same, so by [7] we get part (a) of the next theorem; compare also [13], Corollary 1.2. The proof of part (b) is again virtually the same as in the oriented case, see [12].…”

“…Thus, they are among the most homogeneous non-classical topological parallelisms. Indeed, according to [12] the 'classical' Clifford parallelism is the only one with a group of dimension ≥ 4, and the only other possible 3-dimensional group is SO 3 R with its action induced by the irreducible Spin 3 action on R 4 . This follows from the fact that the automorphism group of a topological parallelism is compact [9], [13].…”