Innovations in Incidence Geometry

Isometries and collineations of the Cayley surface

Johannes Gmainer and Hans Havlicek

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Let F be Cayley’s ruled cubic surface in a projective three-space over any commutative field K . We determine all collineations fixing F , as a set, and all cubic forms defining F . For both problems the cases | K | = 2 , 3 turn out to be exceptional. On the other hand, if | K | 4 then the set of simple points of F can be endowed with a non-symmetric distance function. We describe the corresponding circles, and we establish that each isometry extends to a unique projective collineation of the ambient space.

Article information

Innov. Incidence Geom., Volume 2, Number 1 (2005), 109-127.

Received: 23 November 2004
Accepted: 17 March 2005
First available in Project Euclid: 28 February 2019

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Zentralblatt MATH identifier

Primary: 51B15: Laguerre geometries 51N25: Analytic geometry with other transformation groups 51N35: Questions of classical algebraic geometry [See also 14Nxx]

Cayley surface non-symmetric distance isometry


Gmainer, Johannes; Havlicek, Hans. Isometries and collineations of the Cayley surface. Innov. Incidence Geom. 2 (2005), no. 1, 109--127. doi:10.2140/iig.2005.2.109.

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