Abstract
In this paper, a study of time-like ruled surfaces in Minkowski 3-space is investigated by strictly connected time-like straight line moving with Darboux's frame along a differentiable space-like curve. By using the striction curve and the distribution parameter of time-like ruled surfaces, some theorems related to the geodesic curvature and the second fundamental form tensor are obtained.
Highlights
Let R3 be endowed with Lorentzian inner product of X and Y is defined by⟨X, Y ⟩ = x1y1 + x2y2 − x3y3, for X, Y ∈ R3. (R3, ⟨, ⟩) is called Minkowski 3-space denoted by R13
We study some characteristic properties of time-like ruled surfaces related to the geodesic curvature and the second fundamental form tensor by the means of the information given above
We firstly study on a time-like ruled surface swept out by a time-like straight line X moving along a differentiable space-like curve α
Summary
Let R3 be endowed with Lorentzian inner product of X and Y is defined by. (R3, ⟨, ⟩) is called Minkowski 3-space denoted by R13. Lorentzian vector product of X and Y is defined by. The vectors X ∈ R13 are called a space-like, time-like and null (light-like) vector if ⟨X, X⟩ > 0 or X = 0,. For X ∈ R13, the norm of X is defined by ∥X∥ = |⟨X, X⟩| and X is said to be a unit vector if ∥X∥ = 1, (see [8]). For any t ∈ I, the curve α is said to be a space-like, time-like and null curve if the velocity vector α′(t) is a space-like, time-like or null vector, respectively, (see [8])
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