ON PARALLEL SURFACES IN MINKOWSKI 3-SPACE

Year 2013, Volume 3, Issue 2, 214 - 222, 01.12.2013

Abstract

In this paper, we study on some properties of parallel surfaces in Minkowski 3-space. The results given in this paper were given in Euclidean space by [7, 8]. By using these two former studies, we show these properties in Minkowski 3-space. Also we give the relation among the fundamental forms of parallel surfaces in Minkowski 3-space. Finally we show that how a curve which is geodesic on M become again a geodesic on parallel surface Mr by the normal map in Minkowski 3-space.

References

• [1] C¸ ¨oken, A. C¸., C¸iftci, U. and Ekici, C., 2008, On parallel timelike ruled surfaces with timelike rulings, ¨ Kuwait Journal of Science & Engineering, 35, 21–31.
• [2] Craig, T., Note on parallel surfaces, Journal f¨ur die Reine und Angewandte Mathematik (Crelle’s journal), 94, 162–170., 1883.
• [3] Eisenhart, L.P., 1909, A treatise on the differential geometry of curves and surfaces, Ginn and Company, Boston: New York.
• [4] G¨org¨ul¨u, A. and C¸ ¨oken, C., The dupin indicatrix for parallel pseudo-Euclidean hypersurfaces in pseudo-Euclidean space in semi-Euclidean space R n 1 . Journ. Inst. Math. and Comp. Sci. (Math Series), 7(3), 221-225., 1994.
• [5] A. G¨org¨ul¨u and A. C. C¸ ¨oken, The Euler theorem for parallel pseudo-Euclidean hypersurfaces in pseudo-Euclidean space E n+1 1 , Journ. Inst. Math. and Comp. Sci. (Math. Series) Vol:6, No.2 (1993), 161-165.
• [6] Gray, A., Modern differential geometry of curves and surfaces, CRC Press, Inc., 1993.
• [7] Hacısaliho˘glu, H. H., 2000, Diferensiyel geometri, Cilt I-II, Ankara Universitesi, Fen Fak¨ultesi ¨ Yayınları.
• [8] Hicks, Noel J., Notes on Differential Geometry, Van Nostrant Reinhold Company, London, 1974.
• [9] K¨uhnel, W., 2002, Differential geometry, curves-surfaces-manifolds, American Mathematical Society.
• [10] Lopez, R., Differential geometry of curves and surfaces in Lorentz-Minkowski space, Mini-Course taught at the Instituto de Matematica e Estatistica (IME-USP), University of Sao Paulo, Brasil, 2008
• [11] Nizamoglu, S¸., 1986, Surfaces r´egl´ees paralleles, Ege Univ. Fen Fak. Derg., 9 (Ser. A), 37-48. ¨ [12] O’Neill, B., Semi Riemannian geometry with applications to relativity, Academic Press, Inc. New York, 1983.
• [13] Patriciu, A. M., 2010, On some 1,3H3-helicoidal surfaces and their parallel surfaces at a certain distance in 3-dimensional Minkowski space, Annals of the University of Craiova, Maths. and Compt.Science Series, V 37 (4), 93-98.
Year 2013, Volume 3, Issue 2, 214 - 222, 01.12.2013

References

• [1] C¸ ¨oken, A. C¸., C¸iftci, U. and Ekici, C., 2008, On parallel timelike ruled surfaces with timelike rulings, ¨ Kuwait Journal of Science & Engineering, 35, 21–31.
• [2] Craig, T., Note on parallel surfaces, Journal f¨ur die Reine und Angewandte Mathematik (Crelle’s journal), 94, 162–170., 1883.
• [3] Eisenhart, L.P., 1909, A treatise on the differential geometry of curves and surfaces, Ginn and Company, Boston: New York.
• [4] G¨org¨ul¨u, A. and C¸ ¨oken, C., The dupin indicatrix for parallel pseudo-Euclidean hypersurfaces in pseudo-Euclidean space in semi-Euclidean space R n 1 . Journ. Inst. Math. and Comp. Sci. (Math Series), 7(3), 221-225., 1994.
• [5] A. G¨org¨ul¨u and A. C. C¸ ¨oken, The Euler theorem for parallel pseudo-Euclidean hypersurfaces in pseudo-Euclidean space E n+1 1 , Journ. Inst. Math. and Comp. Sci. (Math. Series) Vol:6, No.2 (1993), 161-165.
• [6] Gray, A., Modern differential geometry of curves and surfaces, CRC Press, Inc., 1993.
• [7] Hacısaliho˘glu, H. H., 2000, Diferensiyel geometri, Cilt I-II, Ankara Universitesi, Fen Fak¨ultesi ¨ Yayınları.
• [8] Hicks, Noel J., Notes on Differential Geometry, Van Nostrant Reinhold Company, London, 1974.
• [9] K¨uhnel, W., 2002, Differential geometry, curves-surfaces-manifolds, American Mathematical Society.
• [10] Lopez, R., Differential geometry of curves and surfaces in Lorentz-Minkowski space, Mini-Course taught at the Instituto de Matematica e Estatistica (IME-USP), University of Sao Paulo, Brasil, 2008
• [11] Nizamoglu, S¸., 1986, Surfaces r´egl´ees paralleles, Ege Univ. Fen Fak. Derg., 9 (Ser. A), 37-48. ¨ [12] O’Neill, B., Semi Riemannian geometry with applications to relativity, Academic Press, Inc. New York, 1983.
• [13] Patriciu, A. M., 2010, On some 1,3H3-helicoidal surfaces and their parallel surfaces at a certain distance in 3-dimensional Minkowski space, Annals of the University of Craiova, Maths. and Compt.Science Series, V 37 (4), 93-98.

Details

Primary Language English Research Article Yasin UNLUTURK This is me Department of Mathematics, Kırklareli University, 39060, Kırklareli, TURKEY Erdal OZUSAGLAM This is me Department of Mathematics, Aksaray University, 68100, Aksaray, TURKEY. December 1, 2013 Year 2013, Volume 3, Issue 2

Cite

 Bibtex @ { twmsjaem761412, journal = {TWMS Journal of Applied and Engineering Mathematics}, issn = {2146-1147}, eissn = {2587-1013}, address = {Işık University ŞİLE KAMPÜSÜ Meşrutiyet Mahallesi, Üniversite Sokak No:2 Şile / İstanbul}, publisher = {Turkic World Mathematical Society}, year = {2013}, volume = {3}, number = {2}, pages = {214 - 222}, title = {ON PARALLEL SURFACES IN MINKOWSKI 3-SPACE}, key = {cite}, author = {Unluturk, Yasin and Ozusaglam, Erdal} }