Spreads admitting net generating regulizations

Riesinger, Rolf

doi:10.1007/bf00147806

Cited by 8 publications

(5 citation statements)

References 19 publications

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“…The following two theorems are due to Heimbeck in the Laguerre case; see [9]. Riesinger proves the general case with di¤erent techniques and terminology; see [18]. Proof.…”

Section: Flocks Of Miquelian Circle Planes and Translation Planesmentioning

confidence: 98%

Flocks of topological circle planes

Rosehr¹

2005

Advg

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We prove that every flock of a finite-dimensional locally compact connected circle plane is homeomorphic to R or S 1 and that every flock of a real Miquelian circle plane defines a compact 4-dimensional translation plane. Furthermore we investigate (topological) properties of regulizations. These properties are used to relate the automorphism group of a flock to the automorphism group of the corresponding translation plane.

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“…The following two theorems are due to Heimbeck in the Laguerre case; see [9]. Riesinger proves the general case with di¤erent techniques and terminology; see [18]. Proof.…”

Section: Flocks Of Miquelian Circle Planes and Translation Planesmentioning

confidence: 98%

Flocks of topological circle planes

Rosehr¹

2005

Advg

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“…We solve (20) along u 2 and substitute the solution in (22) and get for the unknown s-coordinate of the common points of h(ξ) and h (ξ) the following quadratic equation:…”

Section: Existence Of Rotational Spreads With a System Of Coaxial Acementioning

confidence: 99%

“…(See [20,Section 3].) In the present paper we need the concept 'flock' only in the elliptic case and only for real basic field.…”

Section: The Thas-walker Constructionmentioning

confidence: 99%

Parallelisms of $${{\rm PG}(3, {\mathbb R})}$$ composed of non-regular spreads

Betten

Riesinger²

2011

Aequat. Math.

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Any continuous strictly monotonic function F : R ≥0 → R with F (0) = 0 and F (t) → ∞ for t → ∞ gives rise to a topological rotational spread of PG (3, R); this spread is non-regular, if F is not linear. The action of the group SO 3 (R) on this spread yields a topological parallelism of PG (3, R). The article also contains a short investigation on rotational spreads. Moreover, we construct a parallelism P 72 of PG (3, R) which is composed of piecewise regular spreads each consisting of two segments which are tacked together along a common regulus. Using Klein's correspondence of line geometry and the Thas-Walker construction we represent every parallel class of P 72 via two parallel half-lines being noninterior to a given sphere in R 3 . The parallelism P 72 contains exactly one regular spread, all other members of P 72 are piecewise regular spreads with two segments. However, P 72 is not topological.Mathematics Subject Classification (2010). 51H10, 51A15, 51A40, 51M30.

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“…For spreads with net generating regulizations and references to this subject, see [16], especially Remarks 2.8 and 2.9 are relevant exclusively for elliptic regulizations. For the real projective 3-space PG (3,~) examples of non-regular spreads admitting elliptic regulizations are given in [16,Sections 4.1 and 4.3].…”

Section: Introductionmentioning

confidence: 99%

“…[7, p.3]. This terminology is not ambiguous because Fix YZ = Aut 5r implies that Z is the only net generating regutization of Y (see [16,Theorem 2.7]). This terminology is not ambiguous because Fix YZ = Aut 5r implies that Z is the only net generating regutization of Y (see [16,Theorem 2.7]).…”

Section: Introductionmentioning

confidence: 99%

Spreads admitting elliptic regulizations

Riesinger¹

1997

J Geom

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SPREADS ADMITTING ELLIPTIC REGULIZATIONS Roll RiesingerThroughout this paper, the underlying projective space is 3-dimensional and Pappian.A spread 50 admits a reguttzatton Y., if Z is a collection of reguli contained in 5 0 and if each element of 50, except at most two lines, is contained either in exactly one regulus of Z or in all reguli of Z. Replacement of each regulus of Z by its complementary regulus (exceptional lines remain unchanged) produces the complementsry congruence 50~. of 50 with respect to E. If ~0~ is an elliptic linear congruence of lines, then we call Z an elliptic regulization. Applying a method due to Thas and Walker we construct topological spreads of PG(3,gR) which admit one elliptic and no further regulization. For each of these spreads we determine the group of automorphic collineations. Among others we obtain also spreads which are the complete intersection of a general linear complex of lines and of a cubic complex of lines.

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Spreads admitting net generating regulizations

Cited by 8 publications

References 19 publications

Flocks of topological circle planes

Flocks of topological circle planes

Parallelisms of $${{\rm PG}(3, {\mathbb R})}$$ composed of non-regular spreads

Spreads admitting elliptic regulizations

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