Null curves in Minkowski 3-space

Jun İchi Inoguchi; Sungwook Lee

Research Article

Year 2008, Volume: 1 Issue: 2, 40 - 83, 30.11.2008

Jun İchi Inoguchi , Sungwook Lee

Abstract

References

[1] Alias, L. J., Ferrandez, A., Lucas, P. and Merono, M. A., On the Gauss map of B-scroll, Tsukuba J. Math. 22 (1998), 371-377.
[2] Arroyo, J., Barros, M. and Garay, O. J., A characterization of helices and Cornu spirals in real space forms, Bull. Austral. Math. Soc. 56 (1997), 37-49.
[3] Balgetir, H., Bektas, M. Ergüt, M., On a characterization of null helices, Bull. Inst. Math. Acad. Sinica 29 (2001), 71-78.
[4] Balgetir, H., Bektas, M. and Inoguchi, J., Null Bertrand curves and their characterizations, Note Mat. 23 (2004), no. 1, 7-13.
[5] Barros, M., General helices and a theorem of Lancret, Proc. Amer. Math. Soc. 125 (1997), 1503-1509.
[6] Bektas, M., Balgetir, H. and Ergut, M., Inclined curves of null curves in the 3-dimensional Lorentzian manifold and their characterization, J. Inst. Math. Comput. Sci. Math. Ser. 12 (1999), 117-120.
[7] Bonnor, W. B., Null curves in a Minkowski space-time, Tensor (N. S.) 20 (1969), 229-242.
[8] Calini, A. and Ivey, T., Backlund transformations and knots of constant torsion, J. Knot Theory Ramifications 7 (1988), no. 6, 719-746.
[9] Chen, B. Y., When does the position vector of a space curve always lie in its rectifying plane ?, Amer. Math. Monthly 110 (2003), No. 2, 147-152.
[10] Cho, J. T., Inoguchi, J. and Lee, J. E., Slant curves in Sasakian space forms, Bull. Austral. Math. Soc. 74 (2006), no. 3, 359-367.
[11] Cho, J. T., Inoguchi, J. and Lee, J. E., Biharmonic curves in 3-dimensional Sasakian space forms, Ann. Mat. pura Appl., 186 (2007), 685-701.
[12] Cho, J. T. and Lee, J. E., Slant curves in a contact pseudo-Hermitian 3-manifold, preprint, 2006.
[13] Choi, S. M., On the Gauss map of ruled surfaces in a Minkowski 3-space, Tsukuba J. Math. 19 (1995) 285-304.
[14] Choi, S. M., Ki, U.H. and Suh, Y. J., On the Gauss map of null scrolls, Tsukuba J. Math. 22 (1998) 272-279 .
[15] Dajczer, M. and Nomizu, K., On flat surfaces in S31 and H31 , in: Manifolds and Lie Groups{ papers in honor of Yozo Matsushima (J. Hano et al eds.), Progress in Math. 14 (1981), Birkhauser, Boston, pp. 71-108.
[16] Dillen, F. and Künel, W., Ruled Weingarten surfaces in Minkowski 3-space, Manuscripta Math. 98 (1999), 307-320.
[17] Dillen, F., Van de Woestyne, I., Verstraelen, L. and Walrave, J., Ruled surfaces of finite type in 3-dimensional Minkowski space, Results Math. 27 (1995), 250-255.
[18] Dorfmeister, J., Inoguchi, J. and Toda, M.,Weierstra¼-type representation of timelike surfaces with constant mean curvature, Contem. Math. 308 (2002), Amer. Math. Soc., pp. 77-99.
[19] Duggal, K. L. and Bejancu, A., Lightlike Submanifolds of semi-Riemannian Manifolds and Applications, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1996.
[20] Eisenhart, L. P., A Treatise on the Di®erential Geometry of Curves and Surfaces, Ginn and Company, 1909. Reprinted as a Dover Phoenix Editions, 2004.
[21] Ferrandez, A., Riemannian versus Lorentzian submanifolds, some open problems, in: Proc. Workshop on Recent Topics in Di®erential Geometry, Santiago de Compostera, Depto. Geom. y Topologia, Univ. Santiago de Compostera, 89 (1998), 109-130.
[22] Ferrandez, A., Gimenez, A. and Lucas, P., Null helices in Lorentzian space forms, Internat. J. Modern Phys. A 16 (2001), 4845-4863.
[23] Ferrandez, A., Gimenez, A. and Lucas, P., Null generalized helices in Lorentz-Minkowski spaces, J. Phys. A: Math. Gen. 35 (2002), no. 39, 8243-8251.
[24] Ferrandez, A., Gimenez, A. and Lucas, P., Geometry of lightlike submanifolds in Lorentzian space forms, in: Proc. del Congreso Geometria de Lorentz. Benalmadena 2001, Publ. RSME (2003). (http://www.um.es/docencia/plucas/)
[25] Ferrandez, A. and Lucas, P. On the Gauss map of B-scrolls in 3-dimensional Lorentzian space forms, Czechoslovak Math. J. 50 (125) (2000), 699-704.
[26] Fujioka, A. and Inoguchi, J., Timelike Bonnet surfaces in Lorentzian space forms, Di®er. Geom. Appl. 18 (2003) 103-111.
[27] Fujioka, A. and Inoguchi, J., Timelike surfaces with harmonic inverse mean curvature, in: Surveys on Differential Geometry and Integrable Systems, Advanced Studies in Pure Math., Math. Soc. Japan, to appear.
[28] Grant, J. D. E. and Musso, E., Coisotropic variational problems, J. Geom. Phys. 50 (2004), 303-338.
[29] Graves, L. K., Codimension one isometric immersions between Lorentz spaces, Trans. Amer. Math. Soc. 252 (1979), 367-392.
[30] Guggenheimer, H. W., Differential Geometry, General Publishing Company, 1963, Dover Edition, 1977.
[31] Hartman, P. and Nirenberg, L., On spherical image maps whose Jacobian do not change sign, Amer. J. Math. 81 (1959), 901-920.
[32] Honda, K. and Inoguchi, J., Deformation of Cartan framed null curves preserving the torsion, Differ. Geom. Dyn. Syst. 5 (2003), 31-37. (http://vectron.mathem.pub.ro/dgds/v5n1/d51.htm)
[33] Honda, K. and Inoguchi, J., Cesµaro's method for Cartan framed null curves, preprint.
[34] Hong, J. Q., Timelike surfaces with mean curvature one in anti de Sitter 3-space, Kodai Math. J. 17 (1994), 341-350.
[35] Ikawa, T., On curves and submanifolds in an indefinite Riemannian manifold, Tsukuba J. Math. 9 (1985), 353-371.
[36] Inoguchi, J., Timelike surfaces of constant mean curvature in Minkowski 3-space, Tokyo J. Math. 21 (1998), 141-152.
[37] Inoguchi, J. and Lee, S., Lightlike surfaces in Minkowski 3-space, preprint.
[38] Inoguchi, J. and Toda, M., Timelike minimal surfaces via loop groups, Acta Appl. Math. 63 (2004), 313-355.
[39] S. Izumiya and A. Takiyama, A time-like surface in Minkowski 3-space which contain light-like lines, J. Geom. 64 (1999), 95-101.
[40] Kim, D. S. and Kim, Y. H., B-scrolls with nondiagonalizable shape operators, Rocky Moun- tain J. Math. 33 (2003), 175-190.
[41] Kim, Y. H. and Yoon, D. W., Ruled surfaces with pointwise 1-type Gauss map, J. Geom. Phys. 34 (2000), 191-205.
[42] Kim, Y. H. and Yoon, D. W., Classi¯cation of ruled surfaces in Minkowski 3-space, J. Geom. Phys., 49 (2004), 89-100.
[43] Kobayashi, O, Maximal surfaces in the three-dimensional Minkowski space L3, Tokyo J. Math. 6 (1983), 297-309.
[44] Lancret, M. A., Memoire sur les courbes µa double courbure, Memoires presentes µa l'Institut 1 (1806), 416-454.
[45] Lee, S., Timelike surfaces of constant mean curvature §1 in anti-de Sitter 3-space H31 (¡1),Ann. Global. Anal. Geom. 29 (2006), no. 4, 355-401.
[46] Magid, M. A., Timelike surfaces in Lorentz 3-space with prescribed mean curvature and Gauss map, Hokkaido Math. J. 20 (1991), 447-464.
[47] Massey,W. S., Surfaces of Gaussian curvature zero in Euclidean 3-space, Tohoku Math. J. 14 (1962), 73-79.
[48] McNertney,L. V., One-parameter families of surfaces with constant curvature in Lorentz 3- space, Ph. D. Thesis, Brown Univ., 1980.
[49] Milnor, T. K., Harmonic maps and classical surface theory in Minkowski 3-space, Trans. Amer. Math. Soc. 280 (1983), 161-185.
[50] Mira, P. and Pastor, J. A., Helicoidal maximal surfaces in Lorentz-Minkowski space, Monatsh. Math. 140 (2003), 315-334.
[51] Musso, E. and Nicolodi, L., Closed trajectories of a particle model on null curves in anti-de Sitter 3-space, Classical Quantum Gravity 24 (2007), 5401-5411.
[52] Musso, E. and Nicolodi, L., Reduction for constrained variational problems on 3D null curves, preprint, math.DG. 0710.0483.
[53] Nomizu, K. and Sasaki, T., A±ne Dfferential Geometry. Geometry of A±ne Immersions, Cambridge Tracts in Math. 111, Cambridge Univ. Press, 1994.
[54] Nutbourne, A. W. and Martin, R. R., Differential Geometry Applied to the Design of Curves and Surfaces, Ellis Horwood, Chichester, UK, 1988.
[55] O'Neill, B., Elementary Di®erential Geometry, Academic Press, 1966.
[56] O'Neill, B., Semi-Riemannian Geometry with Application to Relativity, Academic Press, 1983.
[57] Otsuki, T., Differential Geometry (in Japanese), Asakura, 1961.
[58] Nassar, H. A.A. and Fathi, M. H., On an extension of the B-scroll surface in Lorentz 3-space R31 , Riv. Mat. Univ. Parma (6) 3 (2000), 57-67 (2001).
[59] Petrovic-Torga¸sev, M. and ¸Sucurovic, E., Some characterizations of the Lorentzian spherical timelike and null curves, Math. Vesnik 53 (2001), 21-27. (http://www.emis.de/journals/MV/0112/3.html )
[60] Pogolerov, A. W., Continuous maps of a bounded variations, Dokl. Acad. Nauk SSSR 111 1956 757-759.
[61] Pogolerov, A. W., An extension of Gauss' theorem on the spherical representation of surfaces of bounded exterior curvature, Dokl. Acad. Nauk SSSR 111 1956 945-947.
[62] Sahin, B., Kilic, E. and Güneş, R., Null helices in R31, Differ. Geom. Dyn. Syst. 3 (2001), 31-36. (http://vectron.mathem.pub.ro/dgds/v3n2/v3n2.htm)
[63] Schief, W. K., On the integrability of Bertrand curves and Razzaboni surfaces, J. Geom. Phys. 45 (2003), 130-150.
[64] Scofield, P. D., Curves of constant precession, Amer. Math. Monthly 102 (1995), No. 6, 531-537.
[65] Spivak, M., A Comprehensive Introduction to Di®erential Geometry, Second Edition, Vol. 4, Publish or Perish, Wilmington, DE, 1979.
[66] Struik, D. J., Lectures on Classical Di®erential Geometry, Addison-Wesley Press Inc., Cam- bridge, Mass., 1950, Reprint of the second edition, Dover, New York, 1988.
[67] Van de Woestijne, I., Minimal surfaces of the 3-dimensional Minkowski space, in: Geometry and Topology of Submanifolds II (M. Boyom et al eds.), World Scienti¯c Publ., Teaneck, NJ, 1990, pp. 344-369.
[68] Weinstein, T., An Introduction to Lorentz Surfaces, de Gruyter Exposition in Math. vol. 22, Walter de Gruyter, Berlin.
[69] Wong, Y. C., A global formulation of the condition for a curve to be lie in a sphere, Monatsch. Math. 67 (1963), 363-365.
[70] Wong, Y. C., On an explicit characterization of spherical curves, Proc. Amer. Math. Soc. 34 (1972), 239{242. Erratum: 38(1973), 668.

Null curves in Minkowski 3-space

Year 2008, Volume: 1 Issue: 2, 40 - 83, 30.11.2008

Jun İchi Inoguchi , Sungwook Lee

Abstract

Keywords

Null curves, timelike minimal surfaces, timelike CMC-1 surfaces

References

[1] Alias, L. J., Ferrandez, A., Lucas, P. and Merono, M. A., On the Gauss map of B-scroll, Tsukuba J. Math. 22 (1998), 371-377.
[2] Arroyo, J., Barros, M. and Garay, O. J., A characterization of helices and Cornu spirals in real space forms, Bull. Austral. Math. Soc. 56 (1997), 37-49.
[3] Balgetir, H., Bektas, M. Ergüt, M., On a characterization of null helices, Bull. Inst. Math. Acad. Sinica 29 (2001), 71-78.
[4] Balgetir, H., Bektas, M. and Inoguchi, J., Null Bertrand curves and their characterizations, Note Mat. 23 (2004), no. 1, 7-13.
[5] Barros, M., General helices and a theorem of Lancret, Proc. Amer. Math. Soc. 125 (1997), 1503-1509.
[6] Bektas, M., Balgetir, H. and Ergut, M., Inclined curves of null curves in the 3-dimensional Lorentzian manifold and their characterization, J. Inst. Math. Comput. Sci. Math. Ser. 12 (1999), 117-120.
[7] Bonnor, W. B., Null curves in a Minkowski space-time, Tensor (N. S.) 20 (1969), 229-242.
[8] Calini, A. and Ivey, T., Backlund transformations and knots of constant torsion, J. Knot Theory Ramifications 7 (1988), no. 6, 719-746.
[9] Chen, B. Y., When does the position vector of a space curve always lie in its rectifying plane ?, Amer. Math. Monthly 110 (2003), No. 2, 147-152.
[10] Cho, J. T., Inoguchi, J. and Lee, J. E., Slant curves in Sasakian space forms, Bull. Austral. Math. Soc. 74 (2006), no. 3, 359-367.
[11] Cho, J. T., Inoguchi, J. and Lee, J. E., Biharmonic curves in 3-dimensional Sasakian space forms, Ann. Mat. pura Appl., 186 (2007), 685-701.
[12] Cho, J. T. and Lee, J. E., Slant curves in a contact pseudo-Hermitian 3-manifold, preprint, 2006.
[13] Choi, S. M., On the Gauss map of ruled surfaces in a Minkowski 3-space, Tsukuba J. Math. 19 (1995) 285-304.
[14] Choi, S. M., Ki, U.H. and Suh, Y. J., On the Gauss map of null scrolls, Tsukuba J. Math. 22 (1998) 272-279 .
[15] Dajczer, M. and Nomizu, K., On flat surfaces in S31 and H31 , in: Manifolds and Lie Groups{ papers in honor of Yozo Matsushima (J. Hano et al eds.), Progress in Math. 14 (1981), Birkhauser, Boston, pp. 71-108.
[16] Dillen, F. and Künel, W., Ruled Weingarten surfaces in Minkowski 3-space, Manuscripta Math. 98 (1999), 307-320.
[17] Dillen, F., Van de Woestyne, I., Verstraelen, L. and Walrave, J., Ruled surfaces of finite type in 3-dimensional Minkowski space, Results Math. 27 (1995), 250-255.
[18] Dorfmeister, J., Inoguchi, J. and Toda, M.,Weierstra¼-type representation of timelike surfaces with constant mean curvature, Contem. Math. 308 (2002), Amer. Math. Soc., pp. 77-99.
[19] Duggal, K. L. and Bejancu, A., Lightlike Submanifolds of semi-Riemannian Manifolds and Applications, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1996.
[20] Eisenhart, L. P., A Treatise on the Di®erential Geometry of Curves and Surfaces, Ginn and Company, 1909. Reprinted as a Dover Phoenix Editions, 2004.
[21] Ferrandez, A., Riemannian versus Lorentzian submanifolds, some open problems, in: Proc. Workshop on Recent Topics in Di®erential Geometry, Santiago de Compostera, Depto. Geom. y Topologia, Univ. Santiago de Compostera, 89 (1998), 109-130.
[22] Ferrandez, A., Gimenez, A. and Lucas, P., Null helices in Lorentzian space forms, Internat. J. Modern Phys. A 16 (2001), 4845-4863.
[23] Ferrandez, A., Gimenez, A. and Lucas, P., Null generalized helices in Lorentz-Minkowski spaces, J. Phys. A: Math. Gen. 35 (2002), no. 39, 8243-8251.
[24] Ferrandez, A., Gimenez, A. and Lucas, P., Geometry of lightlike submanifolds in Lorentzian space forms, in: Proc. del Congreso Geometria de Lorentz. Benalmadena 2001, Publ. RSME (2003). (http://www.um.es/docencia/plucas/)
[25] Ferrandez, A. and Lucas, P. On the Gauss map of B-scrolls in 3-dimensional Lorentzian space forms, Czechoslovak Math. J. 50 (125) (2000), 699-704.
[26] Fujioka, A. and Inoguchi, J., Timelike Bonnet surfaces in Lorentzian space forms, Di®er. Geom. Appl. 18 (2003) 103-111.
[27] Fujioka, A. and Inoguchi, J., Timelike surfaces with harmonic inverse mean curvature, in: Surveys on Differential Geometry and Integrable Systems, Advanced Studies in Pure Math., Math. Soc. Japan, to appear.
[28] Grant, J. D. E. and Musso, E., Coisotropic variational problems, J. Geom. Phys. 50 (2004), 303-338.
[29] Graves, L. K., Codimension one isometric immersions between Lorentz spaces, Trans. Amer. Math. Soc. 252 (1979), 367-392.
[30] Guggenheimer, H. W., Differential Geometry, General Publishing Company, 1963, Dover Edition, 1977.
[31] Hartman, P. and Nirenberg, L., On spherical image maps whose Jacobian do not change sign, Amer. J. Math. 81 (1959), 901-920.
[32] Honda, K. and Inoguchi, J., Deformation of Cartan framed null curves preserving the torsion, Differ. Geom. Dyn. Syst. 5 (2003), 31-37. (http://vectron.mathem.pub.ro/dgds/v5n1/d51.htm)
[33] Honda, K. and Inoguchi, J., Cesµaro's method for Cartan framed null curves, preprint.
[34] Hong, J. Q., Timelike surfaces with mean curvature one in anti de Sitter 3-space, Kodai Math. J. 17 (1994), 341-350.
[35] Ikawa, T., On curves and submanifolds in an indefinite Riemannian manifold, Tsukuba J. Math. 9 (1985), 353-371.
[36] Inoguchi, J., Timelike surfaces of constant mean curvature in Minkowski 3-space, Tokyo J. Math. 21 (1998), 141-152.
[37] Inoguchi, J. and Lee, S., Lightlike surfaces in Minkowski 3-space, preprint.
[38] Inoguchi, J. and Toda, M., Timelike minimal surfaces via loop groups, Acta Appl. Math. 63 (2004), 313-355.
[39] S. Izumiya and A. Takiyama, A time-like surface in Minkowski 3-space which contain light-like lines, J. Geom. 64 (1999), 95-101.
[40] Kim, D. S. and Kim, Y. H., B-scrolls with nondiagonalizable shape operators, Rocky Moun- tain J. Math. 33 (2003), 175-190.
[41] Kim, Y. H. and Yoon, D. W., Ruled surfaces with pointwise 1-type Gauss map, J. Geom. Phys. 34 (2000), 191-205.
[42] Kim, Y. H. and Yoon, D. W., Classi¯cation of ruled surfaces in Minkowski 3-space, J. Geom. Phys., 49 (2004), 89-100.
[43] Kobayashi, O, Maximal surfaces in the three-dimensional Minkowski space L3, Tokyo J. Math. 6 (1983), 297-309.
[44] Lancret, M. A., Memoire sur les courbes µa double courbure, Memoires presentes µa l'Institut 1 (1806), 416-454.
[45] Lee, S., Timelike surfaces of constant mean curvature §1 in anti-de Sitter 3-space H31 (¡1),Ann. Global. Anal. Geom. 29 (2006), no. 4, 355-401.
[46] Magid, M. A., Timelike surfaces in Lorentz 3-space with prescribed mean curvature and Gauss map, Hokkaido Math. J. 20 (1991), 447-464.
[47] Massey,W. S., Surfaces of Gaussian curvature zero in Euclidean 3-space, Tohoku Math. J. 14 (1962), 73-79.
[48] McNertney,L. V., One-parameter families of surfaces with constant curvature in Lorentz 3- space, Ph. D. Thesis, Brown Univ., 1980.
[49] Milnor, T. K., Harmonic maps and classical surface theory in Minkowski 3-space, Trans. Amer. Math. Soc. 280 (1983), 161-185.
[50] Mira, P. and Pastor, J. A., Helicoidal maximal surfaces in Lorentz-Minkowski space, Monatsh. Math. 140 (2003), 315-334.
[51] Musso, E. and Nicolodi, L., Closed trajectories of a particle model on null curves in anti-de Sitter 3-space, Classical Quantum Gravity 24 (2007), 5401-5411.
[52] Musso, E. and Nicolodi, L., Reduction for constrained variational problems on 3D null curves, preprint, math.DG. 0710.0483.
[53] Nomizu, K. and Sasaki, T., A±ne Dfferential Geometry. Geometry of A±ne Immersions, Cambridge Tracts in Math. 111, Cambridge Univ. Press, 1994.
[54] Nutbourne, A. W. and Martin, R. R., Differential Geometry Applied to the Design of Curves and Surfaces, Ellis Horwood, Chichester, UK, 1988.
[55] O'Neill, B., Elementary Di®erential Geometry, Academic Press, 1966.
[56] O'Neill, B., Semi-Riemannian Geometry with Application to Relativity, Academic Press, 1983.
[57] Otsuki, T., Differential Geometry (in Japanese), Asakura, 1961.
[58] Nassar, H. A.A. and Fathi, M. H., On an extension of the B-scroll surface in Lorentz 3-space R31 , Riv. Mat. Univ. Parma (6) 3 (2000), 57-67 (2001).
[59] Petrovic-Torga¸sev, M. and ¸Sucurovic, E., Some characterizations of the Lorentzian spherical timelike and null curves, Math. Vesnik 53 (2001), 21-27. (http://www.emis.de/journals/MV/0112/3.html )
[60] Pogolerov, A. W., Continuous maps of a bounded variations, Dokl. Acad. Nauk SSSR 111 1956 757-759.
[61] Pogolerov, A. W., An extension of Gauss' theorem on the spherical representation of surfaces of bounded exterior curvature, Dokl. Acad. Nauk SSSR 111 1956 945-947.
[62] Sahin, B., Kilic, E. and Güneş, R., Null helices in R31, Differ. Geom. Dyn. Syst. 3 (2001), 31-36. (http://vectron.mathem.pub.ro/dgds/v3n2/v3n2.htm)
[63] Schief, W. K., On the integrability of Bertrand curves and Razzaboni surfaces, J. Geom. Phys. 45 (2003), 130-150.
[64] Scofield, P. D., Curves of constant precession, Amer. Math. Monthly 102 (1995), No. 6, 531-537.
[65] Spivak, M., A Comprehensive Introduction to Di®erential Geometry, Second Edition, Vol. 4, Publish or Perish, Wilmington, DE, 1979.
[66] Struik, D. J., Lectures on Classical Di®erential Geometry, Addison-Wesley Press Inc., Cam- bridge, Mass., 1950, Reprint of the second edition, Dover, New York, 1988.
[67] Van de Woestijne, I., Minimal surfaces of the 3-dimensional Minkowski space, in: Geometry and Topology of Submanifolds II (M. Boyom et al eds.), World Scienti¯c Publ., Teaneck, NJ, 1990, pp. 344-369.
[68] Weinstein, T., An Introduction to Lorentz Surfaces, de Gruyter Exposition in Math. vol. 22, Walter de Gruyter, Berlin.
[69] Wong, Y. C., A global formulation of the condition for a curve to be lie in a sphere, Monatsch. Math. 67 (1963), 363-365.
[70] Wong, Y. C., On an explicit characterization of spherical curves, Proc. Amer. Math. Soc. 34 (1972), 239{242. Erratum: 38(1973), 668.

There are 70 citations in total.

Details

Primary Language	English
Journal Section	Research Article
Authors	Jun İchi Inoguchi Sungwook Lee This is me
Publication Date	November 30, 2008
Published in Issue	Year 2008 Volume: 1 Issue: 2

Cite

APA	Inoguchi, J. İ., & Lee, S. (2008). Null curves in Minkowski 3-space. International Electronic Journal of Geometry, 1(2), 40-83.
AMA	Inoguchi Jİ, Lee S. Null curves in Minkowski 3-space. Int. Electron. J. Geom. November 2008;1(2):40-83.
Chicago	Inoguchi, Jun İchi, and Sungwook Lee. “Null Curves in Minkowski 3-Space”. International Electronic Journal of Geometry 1, no. 2 (November 2008): 40-83.
EndNote	Inoguchi Jİ, Lee S (November 1, 2008) Null curves in Minkowski 3-space. International Electronic Journal of Geometry 1 2 40–83.
IEEE	J. İ. Inoguchi and S. Lee, “Null curves in Minkowski 3-space”, Int. Electron. J. Geom., vol. 1, no. 2, pp. 40–83, 2008.
ISNAD	Inoguchi, Jun İchi - Lee, Sungwook. “Null Curves in Minkowski 3-Space”. International Electronic Journal of Geometry 1/2 (November 2008), 40-83.
JAMA	Inoguchi Jİ, Lee S. Null curves in Minkowski 3-space. Int. Electron. J. Geom. 2008;1:40–83.
MLA	Inoguchi, Jun İchi and Sungwook Lee. “Null Curves in Minkowski 3-Space”. International Electronic Journal of Geometry, vol. 1, no. 2, 2008, pp. 40-83.
Vancouver	Inoguchi Jİ, Lee S. Null curves in Minkowski 3-space. Int. Electron. J. Geom. 2008;1(2):40-83.

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