SOME CHARACTERIZATIONS FOR SPACELIKE INCLINED CURVES

In this paper by establishing the Frenet frame {T,N,B1, B2} for a spacelike curve we give some characterizations for the spacelike inclined curves and B2-slant helices in R4 2.


Introduction
In the classical di¤erential geometry inclined curves and slant helices are well known.A general helix or an iclined curve in E 3  1 de…ned as a curve whose tangent lines make a constant angle with a …xed direction called the axis of the helix.A helix curve is characterized by the fact that the ratio k1 k2 is constant along the curve, where k 1 and k 2 denote the …rst curvature and the second curvature(torsion), respectively.Analogue to that A. Magden has given a characterization for a curve x(s) to be a helix in Euclidean 4-space E 4 .He characterizes a helix i¤ the function 2 is constant where k 1 , k 2 and k 3 are …rst, second and third curvatures of Euclidean curve x(s), respectively and they are not zero anywhere [2].Similar characterizations of timelike helices in Minkowski 4-space E 4 1 were given by H. Kocayigit and M. Onder [6].
S. Yilmaz and M. Turgut presented necessary and su¢ cient conditions to be inclined for spacelike and timelike curves in terms of Frenet equations in Minkowski spacetime E 4  1 [12].A. T. Ali and R. Lopez studied the generalized timelike helices in Minkowski 4-space and gave some characterizations for these curves [3].
M. Onder, H. Kocayigit and M. Kazaz gave the di¤erential equations characterizing the spacelike helices and also gave the integral characterizations for these curves in E 4  1 [7].
Izumiya and Takeuchi have introduced the concept of slant helix by considering that the normal lines make a constant angle with a …xed direction.They characterized a slant helix if and only if the function is constant [10].
A. T. Ali and R. Lopez gave di¤erent characterizations of slant helices in terms of their curvature functions [4].Kula and Yayli investigated spherical images, the tangent indicatrix and the binormal indicatrix of a slant helix and they obtained that the spherical images are spherical helices [9].
M. Onder, H. Kocayigit and M. Kazaz gave the characterizations of spacelike B 2 -slant helix by means of curvatures of the spacelike curve in Minkowski 4-space.Moreover they gave the integral characterizations of the spacelike B 2 -slant helix [8].
In this study we investigate the conditions for spacelike curves to be inclined or B 2 -slant helix in R 4  2 and we give some characterizations and theorems for these curves.

Preliminaries
The Semi-Euclidean space R 4  2 is the standart vector space equipped with an inde…nite ‡at metric h; i given by where (x 1 ; x 2 ; x 3 ; x 4 ) is a rectangular coordinate system of R 4 2 .A vector v in R 4 2 is called a spacelike, timelike or null(lightlike) if respectively hold hv; vi > 0, hv; vi < 0 or hv; vi = 0 and v 6 = 0 = (0; 0; 0; 0).The norm of a vector v is given by kvk = p jhv; vij.Two vectors v and w are said to be orthogonal if hv; wi = 0.An arbitrary curve : I !R 4 2 can locally be spacelike, timelike or null if respectively all of its velocity vectors 0 (s) are spacelike, timelike or null.
Let a and b be two spacelike vectors in R 4 2 .Then there is unique real number 0 < < , called angel between a and b, such that ha; bi =k a k : k b k :cos .Let fT (s); N (s); B 1 (s); B 2 (s)g be the moving Frenet frame along the curve (s) in R 4  2 .Then T; N; B 1 ; B 2 are the tangent, the principal normal, the …rst binormal and the second binormal …elds respectively and let r T T is spacelike.
Let be a spacelike curve in R 4 2 , parametrized by arclength function of s.The following cases occur for the spacelike curve .Let the vector N is spacelike, B 1 and B 2 be timelike.In this case there exists only one Frenet frame fT; N; B 1 ; B 2 g for which (s) is a spacelike curve with Frenet equations where T , N , B 1 and B 2 are mutually orthogonal vectors satisfying the equations Recall that the functions k 1 = k 1 (s), k 2 = k 2 (s) and k 3 = k 3 (s) are called the …rst, the second and the third curvature of the spacelike curve (s), respectively and we will assume throughout this work that all the three curvatures satisfy k i (s) 6 = 0, 1 i 3.

Some Characterizations for Spacelike Inclined Curves and
Let (s) be a non-geodesic spacelike curve in R 4 2 and let fT; N; B 1 ; B 2 g denotes the Frenet frame of the curve (s).A spacelike curve in R 4  2 is said to be an inclined curve if its tangent vector forms a constant angle with a constant vector U .From the de…nition of the inclined curve we can write where U is a spacelike constant vector.Di¤erentiating both sides of this equations we have k 1 N:U = 0 (5) Thus we arrive N ?U .Considering this we can compose U as where u i , 1 i 3 are arbitrary functions.Di¤erentiating (6) and considering Frenet equations, we have From (7) we …nd the equations 8 > > < > > : By using the equations above we have and From the equation u 0 Di¤erentiating u 2 we have By a direct computation we have the di¤erential equation By using exchange variable t = R s 0 k 3 (s)ds in (13) we …nd The general solution of ( 14) is where m 1 ; m 2 2 R. Replacing variable t = R s 0 k 3 (s)ds in (15) we have Considering equation ( 16) and ( 9) we have From the equations above we …nd and By taking or Conversely, let us consider vector given by Di¤erentiating vector U and considering di¤erential equation of ( 21) we obtain Thus U is a constant vector and so the curve (s) is an inclined curve in R 4 2 .Thus we have the following theorem.
Theorem 1.Let = (s) be a spacelike curve in R 4  2 .is an inclined curve if and only if Proof.It is obvious from the computations above.
Corollary 2. Let = (s) be a spacelike curve in R 4 2 .is an inclined curve if and only if Proof.If we di¤erentiate the equation (24) respect to s we …nd the equation (25).Now let us solve the equation (25) respect to k1 k2 .If we use exchange variable So we arrive where W 1 and W 2 are real numbers.Now we will give a di¤erent characterization for inclined curves.Let be an inclined curve in R 4  2 .By di¤erentiating (24) with respect to s we get and hence If we de…ne a function f (s) as By using ( 28) and (31) we have Conversely, consider the function and assume that f 0 (s) = k1k3 k2 .We compute As consequence of above computations 3 that is the function ) 02 is constant.Therefore we have the following theorem.
where k 1 , k 2 and k 3 are the curvatures of .
Proof.The proof can be completed from the computations above.Now let (s) be a spacelike curve in R 4  2 and let fT; N; B 1 ; B 2 g denotes the Frenet frame of the curve (s).We call (s) as spacelike B 2 -slant helix if its second binormal vector makes a constant angle with a …xed direction in a vector U .From the de…nition of the B 2 -slant helix we can write where U is a spacelike constant vector.Di¤erentiating both sides of this equations we have Since k 3 6 = 0 we arrive B 1 ?U .Considering this we can compose U as where u i , 1 i 3 are arbitrary functions.Di¤erentiating (38) and considering Frenet equations, we have From (39) we …nd the equations 8 > > < > > : By using the equations above we have and From the equation u 0 Di¤erentiating By a direct computation we have the di¤erential equation By using exchange variable t = R s 0 k 1 (s)ds in (45) we …nd The general solution of (46) is where m 1 ; m 2 2 R. Replacing variable t = R s 0 k 1 (s)ds in (47) we have Considering equation (48) we have From the equations above we …nd and By taking or Conversely, let us consider vector given by Di¤erentiating vector U and considering di¤erential equation of (53) we obtain Thus U is a constant vector and so the curve (s) is a spacelike B 2 slant helix in R 4 2 .As a result we can give the following theorem.Theorem 4. Let = (s) be a spacelike curve in R 4  2 .is a spacelike B 2 slant helix if and only if Proof.The proof can easily seen from the computations above. Proof.
So we arrive where L 1 and L 2 are real numbers.Now we will give a di¤erent characterization for B 2 -slant helices.Let be a spacelike B 2 -slant helix in R 4  2 .By di¤erentiaing (56) with respect to s we get and hence If we de…ne a function f (s) as From f (s)f 0 (s) = ( k3 k2 )( k3 k2 ) 0 and f 00 (s) = k 0 1 ( k3 k2 ) k 1 ( k3 k2 ) 0 we obtain f 0 (s)f 00 (s) = k 1 k 0 1 ( As a consequence of above computations that is the function ( k3(s) k2(s) ) 2 + 1 k 2 1 (s) f( k3(s) k2(s) ) 02 is constant.Therefore we have the following theorem.Proof.It is obvious from the above computations.

Theorem 3 .
Let be a unit speed spacelike curve in R4  2 .Then is an inclined curve if and only if the function f

Theorem 6 .
Let be a unit speed spacelike curve in R 4 2 .Then is a B 2 -slant helix if and only if the function f (s) = 1 k1(s) ( k3 k2 ) 0 = L 1 sin R s 0 k 1 (s)ds L 2 cos R s 0 k 1 (s)ds satis…es f 0 (s) = k1k3 k2 ,where k 1 , k 2 and k 3 are the curvatures of .
If we di¤erentiate the equation (56) respect to s we have the equation (57).