Constant Curvatures of Parallel Hypersurfaces in E1n+1Lorentz Space

Ayşe Yavuz

doi:10.11648/j.pamj.s.2015040102.16

Abstract

In this paper generalized Gaussian and mean curvatures of a parallel hypersurface in E^(n+1) Euclidean space will be denoted respectively by K ̅ and H ̅, and Generalized Gaussian and mean curvatures of a parallel hypersurface in E₁ⁿ⁺¹ Lorentz space will be denoted respectively by K ̿ and H ̿.Generalized Gaussian curvature and mean curvatures, K ̅and H ̅ofaparallel hypersurface in E^(n+1)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 K ̿and H ̿ofaparallel hypersurface in E₁ⁿ⁺¹ Lorentz space and we prove theorems about generalized K ̿and H ̿ ofa parallel hypersurface in E₁ⁿ⁺¹ Lorentz space.

Highlights

IntroductionSuppose that is an n-dimensional vector space over the real numbers for = 1,2,
Suppose that is an n-dimensional vector space over the real numbers for = 1,2, . ..A symmetric bilinear form : × →R is i) positive 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
We know that the generalized Gaussian curvature of a parallel hypersurface is given by def(g,h1,h2,...,he)

Summary

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 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

Basic Concepts

Generalized Theorems