ON FOCAL SURFACES FORMED BY TIMELIKE NORMAL RECTILINEAR CONGRUENCE

Hakan Şimşek; Mustafa Özdemır

doi:10.1501/Commua1_0000000759

ON FOCAL SURFACES FORMED BY TIMELIKE NORMAL RECTILINEAR CONGRUENCE

Year 2016, Volume: 65 Issue: 2, 55 - 64, 01.08.2016

Hakan Şimşek Mustafa Özdemır

https://doi.org/10.1501/Commua1_0000000759

Keywords

Minkowski space, rectilinear congruence, spacelike surface, timelikefocal surfaces

References

B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press Inc., London, 1983.
B. J. Papantoniou, Investigation on the smooth surface S of Rthe normals of which, establish a rectilinear congruence such that AK1+ BK2= 0;Tensor, N. S. Vol. 47 (1988).
C. Weihuan, L. Haizhong, Spacelike Weingarten Surfaces in R3and the Sine-Gordon Equa- 1 tion, J ournal of Mathematical Analysis and Applications 214, 459-474 (1997).
D. J. Struik, Lectures on Classical Diğ erential Geometry, Second Edition Dover Publications, INC. New York (1961).
F. Tari, Caustics of surfaces in the Minkowski 3-space, Q. J. Math. 63(1), 189–209 (2012).
G. Tsagas, On the rectilinear congruences whose straight lines are tangent to one parameter family of curves on a surface, Tensor, N. S., 29 (1975), 287-294.
H. ¸Sim¸sek, M. Özdemir, The Sub-Parabolic Lines in the Minkowski 3-Space, Results in Math- ematics, 67, 417-430 (2015), Doi: 10.1007/s00025-014-0409-z.
Hou Zhong Hua, Li li, A kind of rectilinear congruences in the Minkowski 3-Space, Journal of Mathematical Research & Exposition Nov., 2008, Vol. 28, No. 4 pp. 911–918.
M. Özdemir, A. A. Ergin, Spacelike Darboux Curves in Minkowski 3-Space, Diğer. Geom. Dyn. Syst. 9, 131-137, 2007. Current address : Hakan ¸Sim¸sek, Department of Mathematics, Akdeniz University, Antalya, TURKEY E-mail address : hakansimsek@akdeniz.edu.tr Current address : Mustafa Özdemir, Department of Mathematics, Akdeniz University, Antalya, TURKEY E-mail address : mozdemir@akdeniz.edu.tr

Year 2016, Volume: 65 Issue: 2, 55 - 64, 01.08.2016

Hakan Şimşek Mustafa Özdemır

https://doi.org/10.1501/Commua1_0000000759

References

B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press Inc., London, 1983.
B. J. Papantoniou, Investigation on the smooth surface S of Rthe normals of which, establish a rectilinear congruence such that AK1+ BK2= 0;Tensor, N. S. Vol. 47 (1988).
C. Weihuan, L. Haizhong, Spacelike Weingarten Surfaces in R3and the Sine-Gordon Equa- 1 tion, J ournal of Mathematical Analysis and Applications 214, 459-474 (1997).
D. J. Struik, Lectures on Classical Diğ erential Geometry, Second Edition Dover Publications, INC. New York (1961).
F. Tari, Caustics of surfaces in the Minkowski 3-space, Q. J. Math. 63(1), 189–209 (2012).
G. Tsagas, On the rectilinear congruences whose straight lines are tangent to one parameter family of curves on a surface, Tensor, N. S., 29 (1975), 287-294.
H. ¸Sim¸sek, M. Özdemir, The Sub-Parabolic Lines in the Minkowski 3-Space, Results in Math- ematics, 67, 417-430 (2015), Doi: 10.1007/s00025-014-0409-z.
Hou Zhong Hua, Li li, A kind of rectilinear congruences in the Minkowski 3-Space, Journal of Mathematical Research & Exposition Nov., 2008, Vol. 28, No. 4 pp. 911–918.
M. Özdemir, A. A. Ergin, Spacelike Darboux Curves in Minkowski 3-Space, Diğer. Geom. Dyn. Syst. 9, 131-137, 2007. Current address : Hakan ¸Sim¸sek, Department of Mathematics, Akdeniz University, Antalya, TURKEY E-mail address : hakansimsek@akdeniz.edu.tr Current address : Mustafa Özdemir, Department of Mathematics, Akdeniz University, Antalya, TURKEY E-mail address : mozdemir@akdeniz.edu.tr

There are 9 citations in total.

Details

Primary Language	English
Journal Section	Research Articles
Authors	Hakan Şimşek This is me Mustafa Özdemır This is me
Publication Date	August 1, 2016
Published in Issue	Year 2016 Volume: 65 Issue: 2

Cite

APA	Şimşek, H., & Özdemır, M. (2016). ON FOCAL SURFACES FORMED BY TIMELIKE NORMAL RECTILINEAR CONGRUENCE. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 65(2), 55-64. https://doi.org/10.1501/Commua1_0000000759
AMA	Şimşek H, Özdemır M. ON FOCAL SURFACES FORMED BY TIMELIKE NORMAL RECTILINEAR CONGRUENCE. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. August 2016;65(2):55-64. doi:10.1501/Commua1_0000000759
Chicago	Şimşek, Hakan, and Mustafa Özdemır. “ON FOCAL SURFACES FORMED BY TIMELIKE NORMAL RECTILINEAR CONGRUENCE”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 65, no. 2 (August 2016): 55-64. https://doi.org/10.1501/Commua1_0000000759.
EndNote	Şimşek H, Özdemır M (August 1, 2016) ON FOCAL SURFACES FORMED BY TIMELIKE NORMAL RECTILINEAR CONGRUENCE. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 65 2 55–64.
IEEE	H. Şimşek and M. Özdemır, “ON FOCAL SURFACES FORMED BY TIMELIKE NORMAL RECTILINEAR CONGRUENCE”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 65, no. 2, pp. 55–64, 2016, doi: 10.1501/Commua1_0000000759.
ISNAD	Şimşek, Hakan - Özdemır, Mustafa. “ON FOCAL SURFACES FORMED BY TIMELIKE NORMAL RECTILINEAR CONGRUENCE”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 65/2 (August 2016), 55-64. https://doi.org/10.1501/Commua1_0000000759.
JAMA	Şimşek H, Özdemır M. ON FOCAL SURFACES FORMED BY TIMELIKE NORMAL RECTILINEAR CONGRUENCE. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2016;65:55–64.
MLA	Şimşek, Hakan and Mustafa Özdemır. “ON FOCAL SURFACES FORMED BY TIMELIKE NORMAL RECTILINEAR CONGRUENCE”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 65, no. 2, 2016, pp. 55-64, doi:10.1501/Commua1_0000000759.
Vancouver	Şimşek H, Özdemır M. ON FOCAL SURFACES FORMED BY TIMELIKE NORMAL RECTILINEAR CONGRUENCE. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2016;65(2):55-64.