Monatsh. Math. 139, 235–245 (2003) DOI 10.1007/s00605-002-0526-8 Mappings of Ruled Surfaces in Simply Isotropic Space I 1 3 that Preserve the Generators By Z ˇ eljka Milin S ˇ ipus ˇ 1 and Blaz ˇenka Divjak 2 1 University of Zagreb, Zagreb, Croatia 2 University of Zagreb, Vara zdin, Croatia Received December 18, 2001; in revised form July 12, 2002 Published online April 4, 2003 # Springer-Verlag 2003 Abstract. In this paper we study some mappings of skew ruled surfaces in simply isotropic space which preserve the generators. We study isometries, conformal mappings and mappings which preserve the area. Furthermore, we study mappings of surfaces in I 1 3 which preserve the asymptotic lines. 2000 Mathematics Subject Classification: 53A35 Key words: Simply isotropic space, ruled surfaces, Minding isometry, generator-preserving mapping In this paper we study mappings of skew ruled surfaces in the simply isotropic space I 1 3 which also map the generators of one surface into the generators of another (i.e. they preserve the generators of ruled surfaces). The most important are isometries which are also generator-preserving. In Euclidean space these mappings were studied by F. Minding and are therefore called the Minding isom- etries. A differential geometric treatment of that problem goes to Kruppa [5]. As in Euclidean space, where the existence of a local isometry between a helicoid and a catenoid shows that the condition for the second surface to be ruled is not trivial, the same conclusion can be made in the space I 1 3 . Furthermore, we study some other mappings that preserve the generators of surfaces: the conformal mappings and the mappings that preserve the area. These problems in Euclidean space were studied by Brauner in [3] and [4]. Finally, we show that, contrary to the Euclidean case [2], there exist non-ruled surfaces which admit non-trivial isometries that preserve a family of asymptotic curves. 1. Minding Isometries of Ruled Surfaces in I 1 3 The simply isotropic space I 1 3 is a three-dimensional affine space whose group of motions B 1 6 is a six-parameter group given (in affine coordinates) by  x ¼ a þ x cos ’  y sin ’  y ¼ b þ x sin ’ þ y cos ’  z ¼ c þ c 1 x þ c 2 y þ z: