1-TYPE AND BIHARMONIC FRENET CURVES IN LORENTZIAN 3-SPACE *

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1 Iranian Journal of Science & Technology, Transaction A, ol., No. A Printed in the Islamic Republic of Iran, 009 Shiraz University -TYPE AND BIHARMONIC FRENET CURES IN LORENTZIAN -SPACE * H. KOCAYIGIT ** AND H. H. HACISALIHOGLU Celal Bayar University, Faculty of Science and Arts Department of Mathematics Muradiye Campus, Muradiye, Manisa, Turkey Ankara University, Faculty of Science, Department of Mathematics, Tandogan, Ankara, Turkey huseyin.kocayigit@bayar.edu.tr Abstract -type and biharmonic curves by using Laplace operator in Lorentzian -space are studied and some theorems and characterizations are given for these curves. Keywords -type curve, biharmonic curve, helix, degenerate helices. INTRODUCTION n Chen and Ishikawa [] classified biharmonic curves in semi-euclidean space E v. They showed that every biharmonic curve lies in a -dimensional totally geodesic subspace, thus, it suffices to classify biharmonic curves in a semi-euclidean -space. Inoguchi [] pointed out that every biharmonic Frenet curve in Monkowski -space E is a helix whose curvature and torsion satisfy. In this paper we shall give the characterizations of -type and biharmonic curves in semi-euclidean -space in terms of curvature and torsion from a different point. Firstly, we point out the general differential equation characterizing Frenet curves (both non-null and null) in a Lorentz -space. Moreover, we classify the special curves from the differential equations. In the final section, we give some theorems, corollaries and propositions.. PRELIMINARIES Let ( M, g ) be a time-oriented Lorentz -manifold. Let : I M be a unit speed curve. Namely the velocity vector field satisfies g(, ). The constant is called the causal character of. A unit speed curve is said to be spacelike or timelike if its causal character is or -, respectively, a unit speed curve is said to be a geodesic if 0, where is the Levi-Civita connection of ( M, g ). A unit speed curve is said to be a Frenet curve if g(, ) 0. Like Euclidean geometry, every Frenet curve on ( M, g ) admits a Frenet frame field along. Here, a Frenet frame filed,, is an orthonormal frame filed along such that () s and,, satisfies the following Frenet- Serret formula [, 4, 5] () Received by the editor June 7, 006 and in final revised form January 0, 00 Corresponding author

2 60 H. H. Hacisalihoglu / H. Kocayigit The functions 0 and are called the curvature and torsion, respectively, the vector fields,, are called tangent vector field, principle normal vector field and binormal vector field of, respectively. The constants and defined by g (, ), i, () i i i are called second causal character and third causal character of, respectively. Note that. () As in the case of Riemannian geometry, a Frenet curve is a geodesic if and only if 0. A Frenet curve with constant curvature and zero torsion is called a pseudo circle. A circular helix is a Frenet curve whose curvature and torsion are constants. Pseudo circles are regarded as degenerate helices. Helices, which are not circles, are frequently called proper helices. The mean curvature vector field of a unit speed curve is. If is a Frenet curve, then is given by. (4) To close this section, we recall the notion of biharmonicity for unit speed curves. Let () s be a unit speed curve on the Lorentz -manifold ( M, g ) defined on an interval I. Denote by *TM the vector bundle over I obtained by pulling back the tangent bundle TM : * TM : T M s. (5) Iranian Journal of Science & Technology, Trans A, olume, Number A Spring 009 si The Laplace operator action on the space *TM explicitly by []. ( ) of all smooth sections of *TM is given. (6) Definition.. []. A unit speed curve : I M on a Lorentz -manifold M is said to be biharmonic if 0. If M is the semi-euclidean -space, then is biharmonic if and only if 0 Definition.. [6]. A unit speed curve : I M on a Lorentz -manifold M is said to be -type if. Definition.. [7, 8]. A unit speed curve : I M on a Lorentz -manifold M is said to be a general helix, which means that its is constant but and are not.. -TYPE AND BIHARMONIC CURES IN LORENTZIAN -MANIFOLDS Chen and Ishikawa classified biharmonic curves in a semi-euclidean -space. In particular, they showed that in a Euclidean -space, there are no proper biharmonic curves (i.e., biharmonic curves which are not harmonic). On the other hand, in an indefinite semi-euclidean -space, proper biharmonic curves exist. Inoguchi pointed out that every biharmonic Frenet curve in Minkowski -space E is a helix whose curvature and torsion satisfy. Here, we recall their classification theorems. Theorem.. []. Let be a spacelike curve in indefinite semi-euclidean -space biharmonic if and only if is congruent to one of the following: E v. Then is

3 -type and biharmonic frenet curves in lorentzian -space 6 () a spacelike line; () a spacelike curve () s ( as bs, as bs, s) in E, where a and b are constants such that a b 0 ; () a spacelike curve () s ( as /6, as /, a s /6 s) in E, where a is a nonzero constant; (4) a spacelike curve () s ( as /6, as /, a s /6 s) in E, where a is a nonzero constant. Theorem.. []. Let : I M be a Frenet curve on a Lorentz -manifold ( M, g ). Denote by the Laplace operator acting on *TM. Then satisfies if and only if is a helix (including a geodesic). In this case the eigenvalue is ( ). Corollary.. []. Let be a Frenet curve on a Lorentz -manifold ( M, g ). Then is a non-geodesic biharmonic curve if and only if it is one of the following: () is a spacelike helix with a spacelike principal normal such that ; () is a timelike helix such that. Note that no biharmonic spacelike curve on M exists with spacelike principal normals. Corollary.4. []. Let be a Frenet curve on ( M, g ). Then is a helix if and only if 0 (7) for some constant. In this case equals ( ). Note that Ikawa obtained Corollary.4 for timelike curves [9]. Thus, here Inoguchi gave an analytic meaning of (7). Since he treated both spacelike and timelike curves in corollary.4, he got a generalization of [9]. In the case where M is the Minkowski -space E, it is known that helices with 0 are cubic curves, and one can explicitly give the formula of such helices [0]. Moreover, it is easy to check that such spacelike helices are congruent to the curves given in Theorem.. Now, we rephrase the classification due to Chen and Ishikawa. Since case (4) in Theorem. is the image of a timelike helix satisfying a under the following anti-isometry from E onto E : E u v w w u v we may restrict our attention to curves in Minkowski -space (,, ) (,, ), (8) E. Proposition.5. []. Let be a unit speed curve in Minkowski -space only if is congruent to one of the following: () a spacelike or timelike line; () a spacelike curve such that g(, ) 0 given by s as bs as bs s () (,, ), E. Then is biharmonic if and where a and b are constants such that a b 0 ; () a spacelike helix with a spacelike principal normal vector field satisfying a, Spring 009 Iranian Journal of Science & Technology, Trans A, olume, Number A

4 6 H. H. Hacisalihoglu / H. Kocayigit () as, as, as s, 6 6 s (4) a timelike helix satisfying a. as as as () s,, 6 s A GENERAL CHARACTERIZATION FOR A NON-NULL FRENET CURE AND SOME RESULTS Theorem 4.. Let be a unit speed non-null Frenet curve in a Lorentz -space. Then satisfies the following differential equation where Proof: By () we get, (9) 0,,.. (0) Since, we have, by (0), that. () Since we have that, we get. () Now combining (0) with () we may write. () Taking the covariant derivative of () and considering () we obtain the equation (9). Corollary 4.. Let be a unit speed non-null Frenet curve in a Lorentz -space. () The diffential equation characterizing a timelike curve is Iranian Journal of Science & Technology, Trans A, olume, Number A Spring 009

5 -type and biharmonic frenet curves in lorentzian -space 0. 6 (4) () The differential equation characterizing a spacelike curve whose principal normal is timelike is 0. (5) () The differential equation characterizing a spacelike curve whose binormal is timelike is 0. (6) Proof: (). (9) gives us (4) since we have,, because the curve is timelike. (). (9) gives us (5) since we have,, because the curve is spacelike and has a principal normal that is timelike. (). (9) gives us (6) since we have,, because the curve is spacelike and has a binormal that is timelike. Theorem 4.. Let be a unit speed non-null Frenet curve in Lorentz -space. Then is a general helix if and only if, (7) where and. Proof: Suppose that is a general helix, i.e., is constant, in other words. If we replace the value in (9), then we get (7). Conversely, assume that (7) holds. We show that the curve is a general helix. To obtain (9) from (7), must be constant. Thus is a general helix. 0 Spring 009 Iranian Journal of Science & Technology, Trans A, olume, Number A

6 64 H. H. Hacisalihoglu / H. Kocayigit Corollary 4.4. The equation (7) is a general form of the equation (7). Proof: If is a circular helix, then and are constants. By Theorem 4. we get (7) as a special case. Corollary 4.5. Let be a unit speed non-null Frenet curve in Lorentz -space. () The differential equation characterizing a timelike general helix is 0 (8) () The differential equation characterizing a spacelike general helix whose principal normal is timelike is 0 (9) () The differential equation characterizing a spacelike curve whose binormal is timelike is 0. (0) Proof: (). (7) gives us (8) since we have,, because the curve is timelike. (). (7) gives us (9) since we have,, because the curve is spacelike and has a principal normal that is timelike. (). (7) gives us (0) since we have,, because the curve is spacelike and has a binormal that is timelike. Note that K. Ilarslan provided these results in another way []. Theorem 4.6. Let be a unit speed non-null Frenet curve in Lorentz -space. Then is a general helix if and only if where 0, (),. Proof: According to (7), (4) and (6) we have (). Sufficiency is clear. Iranian Journal of Science & Technology, Trans A, olume, Number A Spring 009

7 -type and biharmonic frenet curves in lorentzian -space 65 Corollary 4.7. Let be a unit speed non-null Frenet curve in Lorentz -space. Then is a general helix if and only if the equation (4.) satisfies one of the following conditions: () is a timelike general helix such that and ; () is a spacelike general helix whose principal normal is timelike such that and ; () is a spacelike general helix whose binormal is timelike such that and. The proof is obvious. Corollary 4.8. In the special case that is a circular helix in a Lorentz -space, the equation (4.) gives us the equation ( ) 0. () Proof: Since is a circular helix, then and are constants. Then, by (9) and (), we get () since 0. Corollary 4.9. Let be a unit speed curve in Lorentz -space. is a circular helix if and only if the equation () becomes one of the following equations: () the differential equation characterizing a timelike circular helix is ( ) 0; () () the differential equation characterizing a spacelike circular helix whose principal normal is timelike is ( ) 0; (4) () the differential equation characterizing a spacelike circular helix whose binormal is timelike is ( ) 0. (5) The proof is clear. Note that Ikawa, Inoguchi and Ilarslan obtained the equations (), (4) and (5) from another way. As a result, we can say that -type (which is a circular helix) and biharmonic non-null curves (which are timelike helix and spacelike helix) can be studied by only one general differential equation (9) which is for a Frenet curve. 5. A GENERAL CHARACTERIZATION FOR A NULL FRENET CURE AND SOME RESULTS The following theorem gives a general characterization for a Frenet null curve in Lorenz -space. Theorem 5.. Let be a Frenet null curve in Lorentz -space. Then satisfies the following differential equation Spring 009 Iranian Journal of Science & Technology, Trans A, olume, Number A

8 66 H. H. Hacisalihoglu / H. Kocayigit, (6) 0 where Proof: By a Cartan frame,,, ( ),. of we mean a family of vector field (), s (), s (), s along the curve satisfying the following conditions: () s, g(, ) g(, ) 0, g (, ), g (, ) g (, ) 0, g (, ), , (7) where and are the curvature and torsion of, respectively [9]. Here and are null vectors and is a unit spacelike vector. (7) gives us (6). Theorem 5.. Let : I M be a null Frenet curve in Lorentz -space. Denote the Laplace operator by. Then satisfies if and only if is a circular null helix. In this case the eigenvalue of the operator is. (8) Proof: Suppose satisfies. Let us denote by the mean curvature vector field of and by the Laplace-Beltrami operator of. Then the mean curvature vector field is defined by and the Laplacian operator along is given by [, 6]. By (7), (9) and (0) we have and From the equations (), () and tr( ) (, ) (9) (0) ( ) ( ) (), we get. () 0, 0 () Iranian Journal of Science & Technology, Trans A, olume, Number A Spring 009

9 -type and biharmonic frenet curves in lorentzian -space 67 and. (4) According to (), is a circular null helix. Since is a circular null helix, the equation (4) reduces to. Conversely, suppose that is a circular null helix and that. If we replace (4) and () in (), we obtain. In this case the curve is -type. Corollary 5.. Let be a Frenet null curve in Lorentz -space. Then is a biharmonic null curve if and only if is a pseudo null circle which is a degenerate null helix. Proof: Since the curve is a Frenet curve in which 0, 0 0. This completes the proof. Corollary 5.4. Let be a Frenet null curve in Lorentz -space. Then is a general null helix if and only if where, (5) 0 and. Proof: Suppose that is a general null helix, i.e., is constant but and are not constants, in other words,. If we write the value in (6), then we get (5), must be constant. Thus is a general null helix. Note that Ilarslan obtained Corollary 5.4 in a different way []. Corollary 5.5. Let be a Frenet null curve in Lorentz -space. Then is a general null helix if and only if where 0, (6) and. Proof: From (5), (9) and (0), the proof is clear as the same method in Theorem 4.6. Note that this corollary is a general case of Theorem 5.. Corollary 5.6. For a null-frenet curve in Lorentz -space the differential equation characterizes that is a circular null helix. 0 (7) Spring 009 Iranian Journal of Science & Technology, Trans A, olume, Number A

10 68 H. H. Hacisalihoglu / H. Kocayigit Proof: By (6), we easily get (7) since and are constants. Ilarslan obtained Corollary 5.6 in a different way []. -type and biharmonic curves for a Frenet null curve can also be studied easily by Equation (6) REFERENCES. Chen, B. Y. & Ishikawa, S. (99). Biharmonic surface in pseudo-euclidean spaces. Mem. Fac. Sci. Kyushu Univ. A 45, Inoguchi, J. (00). Biharmonic curves in Minkowski -space. International J. Math. Math. Sci., Ferrandez, A., Lucas, P. & Merono, M., A. (998). Biharmonic Hopf cylinders. Rocky Mountain J. 8, Izumiya, S. & Takiyama, A. (997). A time-like surface in Minkowski -space which contain pseudocircles, Proc. Edinburgh Math. Soc. 40, Izumiya, S. & Takiyama, A. (999). A time-like surface in Minkowski -space which contain light-like lines. J. Geom. 64, Chen, B. Y. (99). On the total curvature of immersed, I: Submanifolds of finite type and their applications. Bull. Ins. Math. Acad. Sinica, n 7. Hacisalihoglu, H. H. & Ozturk, R. (00). On the characterization of inclined curves in E, I. Tensor, N., S., 64, n 8. Hacisalihoglu, H. H. & Ozturk, R. (00). On the characterization of inclined curves in E, II. Tensor, N., S., 64, Ikawa, T. (985). On curves and submanifolds in an indefinite Riemannian manifolds. Tsukuba J. Math. 9, Kobayashi, O. (98). Maximal surfaces in the -dimensional Minkowski space L. Tokyo J. Math. 6, Ilarslan, K. (00). Some special curves on non-euclidean manifolds. Ph. D. thesis, University of Ankara. Iranian Journal of Science & Technology, Trans A, olume, Number A Spring 009