ON CURVES IN THE LIGHTLIKE CONE 1. Introduction Lorentzian ...

H.B. ÖZTEKIN, M. ERGÜT: ON CURVES IN THE LIGHTLIKE CONE 223and < x, dx >= 0, dx(t) is spacelike. Then the induced arc length (or simply the arc length) sof the curve x(t) can be defined byds 2 =< dx(t), dx(t) > .If we take the arc length s of the curve x(t) as the parameter and denote x(s) = x(t(s)), thenx ′ (s) = dxdsis a spacelike unit tangent vector field of the curve x(s). Now we choose the vectorfield y(s), the spacelike normal space V n−1 of the curve x(s) such that they satisfy the followingconditions:< x(s), y(s) >= 1, < x(s), x(s) >=< y(s), y(s) >=< x ′ (s), y(s) >= 0,V n−1 = {span R {x, y, x ′ }} ⊥ , span R {x, y, x ′ , V n−1 } = E n+21 .Therefore, choosing the vector fields α 2 (s), α 3 (s), ..., α n (s) ∈ V n−1 suitably, we have thefollowing Frenet formulasx ′ (s) = α 1 (s)α ′ 1 (s) = κ 1(s)x(s) − y(s) + τ 1 (s)α 2 (s)α ′ 2 (s) = κ 2(s)x(s) − τ 1 (s)α 1 (s) + τ 2 (s)α 3 (s)α ′ 3 (s) = κ 3(s)x(s) − τ 2 (s)α 2 (s) + τ 3 (s)α 4 (s)...α ′ i (s) = κ i(s)x(s) − τ i−1 (s)α i−1 (s) + τ i (s)α i+1 (s)...α ′ n−1 (s) = κ n−1(s)x(s) − τ n−2 (s)α n−2 (s) + τ n−1 (s)α n (s)α n(s) ′ = κ n (s)x(s) − τ n−1 (s)α n−1 (s)y ′ ∑(s) = − n κ i (s)α i (s),i=1where α 2 (s), α 3 (s), ..., α n (s) ∈ V n−1 , < α i , α j >= δ ij , i, j = 1, 2, ..., n. The function κ 1 (s), ...,κ n (s), τ 1 (s), ..., τ n−1 (s) are called cone curvature functions of the curve x(s). The frame{x(s), y(s), α 1 (s), α 2 (s), ..., α n (s)}is called the asymptotic orthonormal frame on E n+21 along the curve x(s) in Q n+1 .3. Curves with constant cone curvature in the lightlike cone Q 2In this section, we consider curves in the lightlike cone Q 2 and define support function forcurves with constant cone curvature κ in the 2−dimensional lightlike cone and the evoluteinvolutecurves and then characterize curves which satisfy eigenvalue equations for the supportfunction in relation to the evolute-involute curves.Let x : I → Q 2 ⊂ E1 3 be a curve, then from (1), we havex ′ (s) = α(s),y ′ (s) = −κ(s)α(s),α ′ (s) = κ(s)x(s) − y(s),where s is an arc length parameter of the curve x(s) and x(s), y(s), α(s) satisfy< x, x >=< y, y >=< x, α >=< y, α >= 0,< x, y >=< α, α >= 1.For an arbitrary parameter t of the curve x(t), the cone curvature function κ is given byκ(t) = < dxdt , d2 x> 2 − < d2 x, d2 x>< dxdt 2dt 2 dt 2 dt , dxdt >2 < dxdt , dx. (3)dt >5(1)(2)