On Parallel Surfaces of Timelike Ruled Weingarten Surfaces

[1] Beem, J. K., Ehrlich, P. E. and Easley, K. E., Global Lorentzian geometry, Marcel Dekker, New
York, 1996.
[2] Beltrami, E., "Risoluzione di un problem a relativeo alla teoria delle superficie gobbe", Ann. Mat.
Pura Appl., 7, pp. 139-150, 1865.
[3] Brunt, B., "Weingarten surfaces design and application of curves and surfaces", Fisher, R., (Ed.),
Mathematics of surfaces V, Oxford Univ. Press., 1994.
[4] Brunt, B. and Grant, K., "Potential applications of Weingarten surfaces in CAGD. I: Weingarten
surfaces and surface shape investigation", Comput. Aided Geom. Des.,13, pp. 569-582, 1996.
[5] Coken, A. C., Ciftci, U. and Ekici, C., "On parallel timelike ruled surfaces with timelike rulings",
Kuwait Journal of Science & Engineering, 35, pp. 21-31, 2008.
[6] Craig, T., "Note on parallel surfaces", Journal für die Reine und Angewandte Mathematik
(Crelle's journal), 94, pp. 162-170, 1883.
[7] Dillen, F. and Kühnel, W., "Ruled Weingarten surfaces in Minkowski 3-space", Manuscripta
Math.,98, pp. 307-320, 1999.
[8] Dillen, F. and Sodsiri, W., "Ruled surfaces of Weingarten type in Minkowski 3-space", J. Geom.,
83, pp. 10-21, 2005.
[9] Dillen, F. and Sodsiri, W., "Ruled surfaces of Weingarten type in Minkowski 3-space-II", J.
Geom., 84, pp. 37-44, 2005.
[10] Dini, U., "Sullesuperficiegobbenellequaliunodei due raggidi curvature principale e una funzione
dellaltro", Ann. Mat. Pura Appl., 7, pp. 205-210, 1865.
[11] Gorgulu, A. and Coken, C., "The Dupin indicatrix for parallel pseudo-Euclidean hyper surfaces in
pseudo-Euclidean space in semi-Euclidean space Rⁿ".Journ. Inst. Math. and Comp. Sci. (Math
Series), 7(3), pp. 221-225, 1994.
[12] Gray, A., Modern differential geometry of curves and surfaces, CRC Press, Inc., 1993.
[13] Hou, Z. H. and Ji, F., "Helicoidal surfaces with H²=K in Minkowski 3-space", J. Math. Anal.
Appl., 325, pp. 101-113, 2007.
[14] Kim, M. H. and Yoon, D. W., "Weingarten quadric surfaces in a Euclidean 3-space", Turk. J.
Math.,34, pp. 1-7, 2010.
[15] Koch, R., "Die Weingarten-regelflächen", J. Geom., 47, pp. 77-85, 1993.
[16] Kuhnel1 :Kühnel, W., "Ruled W-surfaces", Arch. Math. (Basel), 62, pp. 475-480, 1994.
[17] 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.
[18] O'Neill, B., Semi Riemannian geometry with applications to relativity, Academic Press, Inc. New
York, 1983.
[19] Schucking, E. L. and Wang, J. Z., "The two-dimensional Ivor", In Gravitation and Geometry, a
Volume in Honor of I. Robinson, W. Rindler and A. Trautman, eds., Naples: Bibliopolis, 1987.
[20] Stamou, G., "Regelflachenvom Weingarten-typ", Colloq. Math.,79, pp. 77-84, 1999.
[21] Turgut, A., On spacelike and timelike ruled surfaces in 3-dimensional Minkowski space, (in
Turkish), unpublished Ph. D thesis, Gazi University, Graduate School of Natural Sciences,
Ankara, 1995.
[22] Unluturk, Y., On parallel ruled Weingarten surfaces in 3-dimensional Minkowski space, (in
Turkish), unpublished Ph. D thesis, Eskisehir Osmangazi University, Graduate School of Natural
Sciences, Eskisehir, 2011.
[23] Unluturk, Y., and Ozusaglam, E., "On Parallel Surfaces in Minkowski 3-Space" in 1st
International Conference on Mathematical Sciences and Applications (IECMSA 2012),
September 3-7, Prishtine University, Prishtine, Kosovo.
[24] Weingarten, J., "Über eine klasse auf einander abwickelbarer flächen", J. ReineAngew. Math., 59,
pp. 382-393, 1861.
[25] Weingarten, J., "Über eine flächen, derer normalen eine gegebene fläche-berühren", Journal für
die Reineund Angewandte Mathematik, 62, pp. 61-63, 1863.
[26] D. W. Yoon, "Some properties of parallel surfaces in Euclidean 3-spaces", Honam Mathematical
J., 30(4), pp, 637-644, 2008.