Rotational spreads and rotational parallelisms and oriented parallelisms of PG(3,R) View Full Text


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Article Info

DATE

2019-04

AUTHORS

Rainer Löwen

ABSTRACT

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 SO3R-action (only the Clifford parallelism admits a larger group (Löwen in Innov Incid Geom. arXiv:1702.03328), and it turns out surprisingly that there are far more oriented parallelisms of this kind than ordinary parallelisms. More specifically, Betten and Riesinger (Aequ Math 81:227–250, 2011) construct ordinary parallelisms by applying SO3R to rotational Betten spreads. We show that these are the only ordinary parallelisms compatible with this group action, but also the ‘acentric’ rotational spreads considered by them yield oriented parallelisms. The automorphism group of the resulting (oriented or non-oriented) parallelisms is always SO3R, no matter how large the automorphism group of the non-regular spread is. The isomorphism type of the parallelism depends not only on the isomorphism type of the spread used, but also on the rotation group applied to it. We also study the rotational Betten spreads used in this construction and their automorphisms. More... »

PAGES

11

References to SciGraph publications

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s00022-018-0466-7

DOI

http://dx.doi.org/10.1007/s00022-018-0466-7

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https://app.dimensions.ai/details/publication/pub.1111100346


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