TAILIEUCHUNG - 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 Turk J Math (2014) 38: 896 – 904 ¨ ITAK ˙ c TUB ⃝ doi: 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: • Accepted: • Published Online: • Printed: 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, ]. The projective transformation also preserves the asymptotic lines of surfaces [3, ]. 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 =

• Toán học
• Minkowski space
• Asymptotic direction
• Projective transformation
• Projective transformation
• Minkowski 3 space
• Null curves
• Transversal vector bundle
• Minkowski 4 space
• Distinguished Frenet frame
• Surfaces in the 4 dimensional Minkowski space
• General rotational surfaces
• Minimal surfaces
• Flat surfaces
• Surfaces with flat normal connection
• Generalized bertrand curves
• Bertrand curve
• Minkowski space time
• Frenet planes
• Second binormal vector fields
• Timelike minimal surface
• Björling problem
• Helicoidal motions
• Lorentz–Minkowski space
• Vector field
• Formation of crystals
• Multi dimensional space
• The Lorenz quasitransformation
• Bindering organic in Minkovsky space
• Null curve
• Frenet frame
• Cartan curvature
• Curvature functions