CUMALİ EKİCİ, YASİNÜNLÜTÜRK, MURADİYEÇİMDİKER

ON ELLIPTIC LINEAR TIMELIKE PARALLEL WEINGARTEN SURFACES SATISFYING THE CONDITION 2a^rH^r + b^rK^r = c^r

In this study, firstly, we obtain the parallel Weingarten surfaceswhich satisfy the condition 2aHr+ brKr= cin Minkowski 3-space. Thenwe give some geometric properties of these kind of surfaces, such as theirGauss map Nrand conformal structures. By using the conformal structuresinduced by arψr− brN, we derive two fundamental elliptic partial differentialequations which involve the immersion and the Gauss map

Keywords:

Elliptic linearity, Gauss map Laplacian operator, conformal structure, parallel Weingarten surfaces,

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Beem, J. K., Ehrlich, P. E. and Easley, K. E., Global Lorentzian Geometry, Marcel Dekker, New York, 1996.
Beltrami, E., Risoluzione di un problema relativo alla teoria delle superficie gobbe, Ann. Mat. Pura Appl., 7 (1865), 139-150.
Chern, S. S., Some New Characterizations of the Euclidean Sphere. Duke Math. J., 12 (1945), 279-290.
Craig, T., Note on parallel surfaces, Journal Fur Die Reine und Angewandte Mathematikuke Math J (Crelle’s Journal), 94 (1883), 162-170.
C¸ ¨oken, A. C., C¸ iftci, ¨U. and Ekici, C., On parallel timelike ruled surfaces with timelike rulings, Kuwait Journal of Science & Engineering, 35 (2008), 21–31.
do Carmo, M. P., Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc., New Jersey, 1976.
Gray, A., Modern Differential Geometry of Curves and Surfaces, CRC Press, Inc., Florida, 1993.
Galvez, J. A., Martinez, A. and Milan, F., Linear Weingarten surfaces in R3, Monotsh. Math., 138 (2003), 133-144.
G¨org¨ul¨u, A. and C¸ ¨oken, A.C., The Dupin indicatrix for parallel pseudo-Euclidean hypersur- 1faces in pseudo-Euclidean space Rn
, Journ. Inst. Math. and Comp. Sci. (Math. Series), 7 (1994), no. 3, 221-225.
Hopf, H., Differential Geometry in the Large, Lect Notes Math 1000 Berlin Heidelberg Newyork: Springer 1983.
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.
Nizamoglu, S¸., Surfaces r´egl´ees parall`eles, Ege ¨Univ. Fen Fak. Derg., 9 (1986), (Ser. A), 37-48. [13] O’Neill, B., Semi Riemannian Geometry with Applications to Relativity, Academic Press, Inc. New York, 1983.
Park, K. R. and Kim, G. I., Offsets of ruled surfaces. J. Korean Computer Graphics Society; 4 (1998), 69-75.
Rosenberg, H. and Sa Earp, R., The Geometry of properly embedded special surfaces in R3; e. g., surfaces satisfying aH + bK = 1 where a and b are positive, Duke Math J., (1994) 73: 291-306.
Spiegel, M. R., Theory and Problems of Advanced Calculus SI (metric) Edition 1963, 1974 by Mc Graw-Hill, Inc. Singapore.
Willmore, T. J., Riemannian Geometry 1993 Published in the United States by Oxford Uni- versity Press Inc., Newyork.
¨Unl¨ut¨urk, Y., On parallel ruled Weingarten surfaces in 3-dimensional Minkowski space, (in Turkish), PhD thesis, Eskisehir Osmangazi University, Graduate School of Natural Sciences, Eskisehir, 2011.
Yoon, D. W., Polynomial translation surfaces of Weingarten types in Euclidean 3-space, Central European Journal of Mathematic, 8 (2010), no. 3, 430-436.
Eskis¸ehir Osmangazi University, Department of Mathematics-Computer, 26480, Eskis¸ehir- Turkey
E-mail address: cekici@ogu.edu.tr
Department of Mathematics, Kırklareli University, 39100, Kırklareli-Turkey
E-mail address: yasinunluturk@kirklareli.edu.tr
Department of Mathematics, Kırklareli University, 39100, Kırklareli-Turkey
E-mail address: muradiye.1001@hotmail.com