Partially Null Curves of Constant Breadth in Semi-Riemannian Space

Melih Turgut (Corresponding author) Buca Educational Faculty, Dokuz Eylül University 35160, Buca-Izmir, Turkey E-mail: melih.turgut@gmail.com Abstract In this work, some characterizations of partially null curves of constant breadth in Semi-Riemannian space are presented.


Introduction
Curves of constant breadth were introduced by L. Euler, 1780.Ö. Köse (1984) wrote some geometric properties of plane curves of constant breadth.And, in another work Ö.Köse (1986) extended these properties to the Euclidean 3-Space 3 E .Moreover, M. Fujivara (1914) obtained a problem to determine whether there exist space curve of constant breadth or not, and he defined 'breadth' for space curves and obtained these curves on a surface of constant breadth.A. Magden and Ö. Köse (1997) studied this kind curves in four dimensional Euclidean space .4 E E. Cartan opened door of notion of null curves (for more details see C. Boyer, 1968).And, thereafter null curves were deeply studied by W.B. Bonnor (1969) in Minkowski space-time.In the same space, special null; Partially and Pseudo Null curves are defined by J. Walrave (1995).Additionally, M. Petrovic et. al. (2005) defined Frenet equations of pseudo null and partially null curves in .

E
In this paper, we extend the notion of curves of constant breadth to null curves in Semi-Riemannian space.Some characterizations are obtained by means of Frenet equations defined by M. Petrovic et. al.(2005).We used the method of Ö. Köse (1984).

Preliminary Notes
To meet the requirements in the next sections, here, the basic elements of the theory of curves in the space 4 2 E are briefly presented.A more complete elementary treatment can be found in B. O'Neill (1983).
Semi-Riemannian space 4 2 E is an Euclidean space 4  E provided with the standard flat metric given by are respectively space-like, time-like or null.Also, recall the norm of a vector v r is given by .
The velocity of the curve α r is given by .are, respectively, the tangent, the principal normal, the first binormal and the second binormal vector fields.Recall that a space-like curve with time-like principal normal N r and null first and second binormal is called a partially null curve in   E .These curves will be denoted by .
C The normal plane at every point P on the curve meets the curve at a single point Q other than P .We call the point Q the opposite point of P .We consider a partially null curve in the class Γ as in M. Fujivara (1914)   Then, using system (4), we easily have the following differential equations with respect to 1 m and 2 m as (5) These equations are characterizations for the curve .By the method of variation of parameters, the solution of (13) yields that c cannot be zero.

2 E
α r ′ Thus, a space-like or a time-like curve α r is said to be parametrized by arclength function s , if .two space-like vectors in 4 , then, there is unique real

4 2 E 2 E
M. Petrovic et.al. (2005).Let α r be a partially null curve in the space , 4 parametrized by arclength function s .Then for the curve α r the following Frenet equations are given by M. Petrovic et.al. (2005): second and third curvature of the curveα r , respectively.M. Petrovic et.al. (2005) gave a characterization about partially null curves with the following statement.Theorem 2.1 A partially null unit speed curve ) closed partially null curve in the space 4 2 . A simple closed curve of constant breadth having parallel tangents in opposite directions at opposite points can be represented with respect to Frenet frame by the equation , vector' of .C Differentiating both sides of (1) and considering Frenet equations, we have we call φ as the angle between the tangent of the curve C at point ϕ r with a given fixed direction and consider ,

r
If the distance between opposite points of C and * C is constant, then, due to null frame vectors, we can write that constant