Study on BCN and BAN Ruled Surfaces in $\mathbb{E}^{3}$

Hamdy N. Abd-Ellah, Abdelrahim Khalifa Omran


As a continuation to the study in [8, 12, 15, 17], we construct bi-conservative central normal (BCN) and bi-conservative asymptomatic normal (BAN) ruled surfaces in Euclidean 3-space $\mathbb{E}^{3}$. For such surfaces, local study is given and some examples are constructed using computer aided geometric design (CAGD). 


Bi-conservative surface; Geodesic Frenet trihedron; Shape operator

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Abdel-All, N. H. and Abd-Ellah, H. N., Critical values of deformed osculating hyperruled surfaces, Indian J. Pure Appl. Math. 32 (8) (2001), 1217-1228, 2001.

Abdel-All, N. H. and Abd-Ellah, H. N., Stability of closed hyperruled surfaces, Chaos, Solitons and Fractals 13 (2002), 1077–1092.

Abdel-All, N. H. and Hamdoon, F. M., Extension B-scroll surfaces in Lorentz 3-dimensional space, Rivista Di Math. 3 (6) (2000), 57–67.

Abdel-All, N. H. , Hamdoon, F. M. , Abd-Ellah, H. N. and Omran, A. K., Corresponding developable ruled surfaces in Euclidean 3-space E3, J. Math. Comput. Sci, 6 (3) (2016), 390–405.

Abdel-All, N. H., Abdel-Baky, R. A. and Hamdoon, F. M., Ruled surfaces with timelike rullings, Appl. Math. Comput. 147 (2004), 241–253.

Abd-Ellah, H. N., Translation L/W-surfaces in Euclidean 3-space E3, Journal of the Egyptian Mathematical Society 23 (2015), 513–517.

Caddeo, R., Montaldo, S., Oniciuc, C. and Piu, P., Surfaces in the three- dimensional space forms with divergence-free stress-bienergy tensor, Ann. Mat.

Pura Appl., 193 (4) (2014), 529–550.

Chen, B. Y. and Munteanu, M. I., Biharmonic ideal hypersurfaces in Euclidean spaces , Differ. Geom. Appl., 31 (2013), 1–16.

Chen, B. Y., Geometry of Submanifolds, Marcel Dekker, New York, USA, 1973.

Chen, B. Y., Some open problems and conjectures on submanifolds of finite type, Soochow Journal of Mathematics, 17 (2) (1991), 169–188.

Fu, Yu and li, Lan, A class of Weingarten surfaces in Euclidean 3-space , Hindawi Publishing Corporation Abstract and Applied Analysis, 2013.

Fu, Yu, On bi-conservative surfaces in Minkowski 3-space, Journal of Geometry and Physics 33 (2013), 71–79.

Gray, A., Modern Differential Geometry of Curves and Surfaces CRC Press, Boca Raton, FL, Tokyo, 1993.

Gray, A., Abbenda, E. and Salamon, S., Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition , Chapman, Hall/CRC, 2006.

Hamdoon, F. M. and Omran, A. K., Studying on a skew ruled surface by using the geodesic Frenet trihedron of its generator, Korean J. Math. 24 (4) (2016), 613–626.

Hasanis, T. and Vlachos, T., Hypersurfaces in E4 with harmonic mean curvature vector field, Mathematische Nachrichten 172 (1995), 145–169.

Montaldo, S., Oniciuc, C. and Ratto, A. , Bi-conservative surfaces, J. Geom. Anal., to appear.

Orbay, K. and Kasap, E., Mannheim offsets of ruled surfaces, Mathematical Proplems in Engineering, 2009.

Ravani, B. , and Ku, T. S., Bertrand offsets of ruled and developable surfaces, Computer-Aided Design 23 (2) (1991), 145–152.

Shifrin, T., Differential Geometry: A First Course in Curves and Surfaces, Preliminary Version, Spring, 2015.

Soliman, M. A., Abdel-All, Hussien, R. A. and Said, A. A., Geometric properties and invariants of Mannheim offsets of timelike ruled surface with timelike ruling, Mitteilungen Klosterneuburg J. 65 (2015), 285–299.

Yilmaz, T. and Murat, K. K., On the geometry of the first and second fundamental forms of canal surfaces, math. DG, 2011, ArXiv:1106.3177 v1.



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