Pure and Applied Mathematics Journal 2015; 4(1-2): 24-27 Published online January 10, 2015 (http://www.sciencepublishinggroup.com/j/pamj) doi: 10.11648/j.pamj.s.2015040102.16 ISSN: 2326-9790 (Print); ISSN: 2326-9812 (Online) Constant Curvatures of Parallel Hypersurfaces in E 1 n+1 Lorentz Space Ayşe Yavuz 1 , F. Nejat Ekmekci 2 1 NecmettinErbakan University, Faculty of Education, Education of Mathematics, Konya, Turkey 2 Ankara University, Faculty of Sciences, Department of Mathematics,Ankara, Turkey Email address: ayasar@konya.edu.tr (A. Yavuz), ekmekci@science.ankara.edu.tr (F. N. Ekmekci) To cite this article: Ayşe Yavuz, F. Nejat Ekmekci. Constant Curvatures of Parallel Hypersurfaces in E 1 n+1 Lorentz Space. Pure and Applied Mathematics Journal. Special Issue: Applications of Geometry. Vol. 4, No. 1-2, 2015, pp.24-27.doi: 10.11648/j.pamj.s.2015040102.16 Abstract: In this paper generalized Gaussian and mean curvatures of a parallel hypersurface in   Euclidean space will be denoted respectively by   and   , and Generalized Gaussian and mean curvatures of a parallel hypersurface in ₁ⁿ⁺¹ Lorentz space will be denoted respectively by   and   .Generalized Gaussian curvature and mean curvatures,   and   ofaparallel hypersurface in   Euclidean space are givenin[2].Before nowwe studied relations between curvatures of a hypersurface in Lorentzian space and we introduced higher order Gaussian curvatures of hypersurfaces in Lorentzian space. In this paper, by considering our last studieson higher order Gaussian and mean curvatures, we calculate the generalized   and   ofaparallel hypersurface in ₁ⁿ⁺¹ Lorentz space and we prove theorems about generalized   and   ofa parallel hypersurface in ₁ⁿ⁺¹ Lorentz space. Keywords: Gaussian Curvatures, Mean Curvatures, Parallel Hypersurface, Higher Order Gaussian Curvatures 1. Introduction Suppose that  is an n-dimensional vector space over the real numbers for  = 1,2, . .. A symmetric bilinear form :  ×  →R is i) positive (resp.negative) definite if and only if     ≠0 implies (   ,   ) > 0 (!"#$. (, ) < 0)for all     in , ii) non-degenerate if and only if (   ,&)= 0 for all & in  implies that     =0  , and iii) indefinite if and only if there exist     and & in  with (   ,   )>0 and (&,&) < 0. A non-degenerate, symmetric bilinear form is called a scalar product. For an indefinite scalar product  on  , a vector     ≠0   is said to be(see [5], p. 4) a) spacelike if and only if (   ,   )>0, b) timelike if and only if (   ,   )<0, and c) null if and only if (   ,   )=0. 2. Basic Concepts Definition 1.1. Let ’ be a unit normal vector field on semi- Riemannianhypersurfaces (⊂(  . The (1,1) tensor field * on ( such that 〈*(), ,〉 = 〈..(, ,), ’〉/0!122, , ∈ ℵ(() is called the shape operator of (⊂(  derived from ’. (see [1], p. 107) Definition 1.2. Let ( and (  be two hypersurfaces in E₁ⁿ⁺¹ with unit normal vectors 5of (and5  of (  . 5=67 8 9 9 : ;  8< where each 7 8 is a = > function of M. If there exists a function/,from M to (  such that /: ( → (  ? → /(?) = ? + !5 A Then(  is called parallel hypersurfaceof (, where !∈B. (see[3])