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A characterization of the projective transformation in Minkowski 3-space

We consider transformations preserving asymptotic directions of surfaces in Minkowski 3-space and show that a transformation preserves the asymptotic directions of a surface if only if it is the projective one. Therefore, we obtain a characterization of the projective transformation. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Turk J Math (2014) 38: 896 – 904 ¨ ITAK ˙ c TUB ⃝ doi:10.3906/mat-1402-12 Research Article A characterization of the projective transformation in Minkowski 3-space ¨ ∗ Yasemin ALAGOZ ˙ Department of Mathematics, Faculty of Arts and Sciences, Yıldız Technical University, Istanbul, Turkey Received: 04.02.2014 • Accepted: 01.05.2014 • Published Online: 01.07.2014 • Printed: 31.07.2014 Abstract: We consider transformations preserving asymptotic directions of surfaces in Minkowski 3-space and show that a transformation preserves the asymptotic directions of a surface if only if it is the projective one. Therefore, we obtain a characterization of the projective transformation. Key words: Minkowski space, asymptotic direction, projective transformation 1. Introduction The projective transformation has been studied by many researchers in the Euclidean space. They characterize some properties of this transformation as follows. A transformation is the projective one if and only if it transforms a straight line to the other straight line [5]. In 3-dimensional Euclidean space, the projective transformation transforms an infinitesimally rigid surface to the other infinitesimally rigid surface, that is, it preserves the infinitesimal rigidity [8, 6, p.355 ]. The projective transformation also preserves the asymptotic lines of surfaces [3, p.202 ]. The transformations preserving asymptotic directions of hypersurfaces in the Euclidean space were considered by Alag¨oz and Soyu¸cok in [2]. Moreover, they gave a characterization of the projective transformation in [1]. In this study, we investigate the properties of transformation preserving asymptotic directions of surfaces in Minkowski 3-space. We also show that a transformation preserves the asymptotic directions of a Minkowski surface if and only if it is the projective one. 2. Preliminaries Let E13 be a Minkowski 3-space with the scalar product A.B =