An Integral Characterization for the Curves on the Hyperbolic Sphere 30 H in the Minkowski Space-time

In this paper, we showed that a hyperbolic spherical curve in the Minkowski space-time 4 1 E can be given by Frenet-like equations of a space-like curve in the Minkowski 3-space 3 1 E . Thus, finding an integral characterization for a hyperbolic spherical 4 1 E curve is identical to finding it for a space-like 3 1 E curve. For a unit speed space-like curve (s) α in 3 1 E with '( ) s =T α , we showed that (s) α is a space-like Frenet curve with curvature ( ) s κ and torsion ( ) s τ if and only if there are constant vectors a and b such that ( ) { } 0 '( ) ( ) cosh ( ) ( ) ( ) ( ) T a b T ξ ξ κ ξ ξ δ δ κ δ δ − = + − − ∫ s s s e e s d , where 0 ( ) ( ) . s s d ξ τ δ δ = ∫ 10 Mustafa Kazaz et al. Mathematics Subject Classifications: 53C50, 55C40, 34A05


1
E can be given by Frenet-like equations of a space-like curve in the Minkowski 3-space 3   1 E .Thus, finding an integral characterization for a hyperbolic spherical 4   1 E -curve is identical to finding it for a space-like 3     In the Euclidean 3-space 3  E , the differential equation characterizing a spherical curve is given by 1 ( ) ( ) 0, ( ) where s is the arc length, ( ) 1 ( ) s s ρ κ = is the radius of curvature and ( ) s τ is the torsion of the curve [5].Note that, the equation ( 1) has a meaning if ( ) 0 s τ ≠ and ( ) 0. s κ ≠ In [10], Wong gave an explicit solution of this differential equation as 0 where a and b are arbitrary constants (see also [1]).On the other hand, Wong [9] gave a condition for a curve to be a spherical curve without any preconditions on ( ) s τ and ( ) s κ : A unit speed curve in the Euclidean 3-space 3   E is a spherical curve if and only if the curvature ( ) s κ is nowhere zero (then its torsion ( ) s τ is defined), and there exists a function which are the Frenet analogue equations for a unit-speed curve in the Euclidean plane 2 E .Similarly, Dannon [2] showed that spherical curves in the Euclidean 4-space 4  E can be given by Frenet analogue equations in

∫
. By using the same method, similar characterizations of time-like and space-like spherical curves lying on a Lorentzian sphere in 4   1 E have been given by the authors in [3].
On the other hand, the differential equation characterizing a hyperbolic spherical curve in Minkowski 3-space 3   1 τ ≠ [7].An explicit solution of this differential equation is also given in [7], namely, ( ) i if ( ) s α is a unit speed space-like curve with a space-like principal normal N , then α is a unit speed space-like curve with a time-like principal normal N , then , where a and b are arbitrary constants.We should note that there are no time-like and light-like (null) unit speed curves which lie on the hyperbolic sphere in Minkowski 3-space 3   1 It is also shown in [7] that a unit speed space-like curve in 3   1 E with a space-like principal normal N (resp., with a time-like principal normal N ) lies on a hyperbolic sphere if and only if there is a function Thus, by considering the Frenet equations in the Lorentzian space, we obtain the following important results: ( ) i The characteristic of a hyperbolic spherical curve in 4   1 E is the Frenet Pattern.Thus, the problem of getting an integral characterization to a hyperbolic spherical curve is nothing else but integration of Frenet equations.In particular, Lorentzian correspondence of the Breuer-Gottlieb characterization is isomorphic to the integral characterization of Frenet equations in the Lorentzian plane.( ) ii The methods used to obtain characterizations of hyperbolic spherical curves can be extended to include Frenet equations in 3   1 E .

Preliminaries
Minkowski space-time . Two vectors , v w in where { } , , , T T T T are mutually orthogonal vectors satisfying equations We can write these equations as . Now we can write

E Frenet Curves
Extending the methods used to obtain characterizations of hyperbolic spherical curves, we obtain the integral characterization of an where the variable is u .Then we can easily find a particular solution for ( )

1 E
curve.For a unit speed space-like curve (s) are constant vectors a and b such that

4 1 Eand 4 T 1 E
are said to be orthogonal if 0 ( Let us denote the moving Frenet frame along the space-like curve ( ) are the tangent, the principal normal, the first binormal and second binormal vector fields, respectively.In this Frenet frame,1  2 , T T and 3 T are space-like vectors, and 4 T is time-like vector (the fourth orthonormal time-like vector to 1 = T T , 2 = T N , 3 = T B ). Now, let ( ) s α be a space-like curve in the space-time 4 , parameterized Integral characterization for curves 13 by arc length function s .Then the Frenet formulae ( 4 T time-like) for ( )

1 E 1 E hyperbolic spherical curve equations have the structure of 3 1 E 1 E 1 E
hyperbolic sphere of radius r with center 0 x .■ Note that, the equations (3) are the Frenet formulae of a 3 -space-like curve with time-like 3 = T B .Thus 4 space-like Frenet curves.Therefore, finding an integral characterization for a 4 hyperbolic spherical curve is identical to finding it for a 3 space-like Frenet curve.4An Integral Characterizations of 3 1 Assume that the condition holds.Let us define the functions Then it is easy to check that T , N and B satisfy the Frenet equations.( ) ( ) i ii ⇒ : Assume that the equations hold.Then the above ( ) 3