TAILIEUCHUNG - Classification of metallic shaped hypersurfaces in real space forms

We define the notion of a metallic shaped hypersurface and give the full classification of metallic shaped hypersurfaces in real space forms. We deduce that every metallic shaped hypersurface in real space forms is a semisymmetric hypersurface. | Turk J Math (2015) 39: 784 – 794 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Classification of metallic shaped hypersurfaces in real space forms ¨ ¨ ∗, Nihal YILMAZ OZG ¨ ¨ Cihan OZG UR UR Department of Mathematics, Balıkesir University, C ¸ a˘ gı¸s, Balıkesir, Turkey Received: • Accepted/Published Online: • Printed: Abstract: We define the notion of a metallic shaped hypersurface and give the full classification of metallic shaped hypersurfaces in real space forms. We deduce that every metallic shaped hypersurface in real space forms is a semisymmetric hypersurface. Key words: Hypersurface, real space form, metallic means family, pseudosymmetric hypersurface, semisymmetric hypersurface 1. Introduction The generalized secondary Fibonacci sequence (see [5]) is given by the relation G(n + 1) = pG(n) + qG(n − 1), n ≥ 1, where G(0) = a, G(1) = b , p and q are real numbers. If p = q = 1 , then we obtain secondary Fibonacci sequence. If the limit G(n + 1) x = lim n→∞ G(n) exists then it is a root of the equation x2 − px − q = 0; () see [4]. Let p and q be two integers. The positive solution of equation () is called a member of the metallic means family (briefly MMF) [4]. The positive solution of the above equation is σp,q = p+ √ p2 + 4q . 2 These numbers are called (p, q) -metallic numbers [4]. For the special values of p and q, we have the following (see [5]): i) For p = q = 1 we obtain σG = √ 1+ 5 2 , which is the golden mean; √ ii) For p = 2 and q = 1 we obtain σAg = 1 + 2, which is the silver mean; iii) For p = 3 and q = 1 we obtain σBr = ∗Correspondence: √ 3+ 13 2 , which is the bronze mean; cozgur@ 2010 AMS Mathematics Subject Classification: 53C40, 53C42, 53A07. 784 ¨ ¨ and YILMAZ OZG ¨ ¨ OZG UR UR/Turk J Math iv) For p = 1 and q = 2 we obtain σCu = 2 , which is the copper mean; v) For p = 1 and q = 3 we

• Toán học
• Real space form
• Metallic means family
• Pseudosymmetric hypersurface
• Semisymmetric hypersurface
• Metallic shaped hypersurfaces