Tubular Hypersurfaces According to Extended Darboux Frame Field of First Kind in E4

Yıl 2022, Cilt: 7 Sayı: 2, 75 - 84, 30.09.2022

Mustafa Altın , Ahmet Kazan

Öz

In this chapter, we study tubular hypersurfaces according to one of the extended Darboux frame field in Euclidean 4-space. We obtain the Gaussian and mean curvatures of tubular hypersurfaces according to extended Darboux frame field of first kind and give some results for them. Also, we prove a theorem about linear Weingarten tubular hypersurface and construct an example.

Anahtar Kelimeler

Tubular hypersurface, Extended Darboux frame field of First Kind, Linear Weingarten hypersurface

Kaynakça

• [1] H.S. Abdel-Aziz and M.K. Saad; Computation of Smarandache curves according to Darboux frame in Minkowski 3-space, Journal of the Egyptian Mathematical Society, 25, (2017), 382-390.
• [2] M. Akyigit, K. Eren and H.H. Kosal; Tubular Surfaces with Modified Orthogonal Frame in Euclidean 3-Space, Honam Mathematical J., 43(3), (2021), 453–463.
• [3] M. Altın, A. Kazan and H.B. Karadag; Monge Hypersurfaces in Euclidean 4-Space with Density, Journal of Polytechnic, 23(1), (2020), 207-214.
• [4] M. Altın, A. Kazan, and D.W. Yoon; 2-Ruled hypersurfaces in Euclidean 4-space, Journal of Geometry and Physics, 166, (2021), 1-13.
• [5] M. Altın and A. Kazan; Rotational Hypersurfaces in Lorentz-Minkowski 4-Space, Hacettepe Journal of Mathematics and Statistics, (2021), 1-25.
• [6] S. Aslan and Y. Yaylı; Canal Surfaces with Quaternions, Adv. Appl. Clifford Algebr., 26, (2016), 31-38.
• [7] M.E. Aydın and I. Mihai; On certain surfaces in the isotropic 4-space, Mathematical Communications, 22(1), (2017), 41-51.
• [8] R.L. Bishop; There is more than one way to frame a curve, Amer. Math. Monthly, 82(3), (1975), 246-251.
• [9] M.P.D. Carmo; Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ, 1976.
• [10] F. Casorati; Mesure de la courbure des surfaces suivant l’id´ee commune, Acta Mathematica, 14, (1890), 95-110.
• [11] F. Dogan and Y. Yaylı; ˘ Tubes with Darboux Frame, Int. J. Contemp.Math. Sci., 7(16), (2012), 751-758.
• [12] M. Düldül, B.U. Düldül, N. Kuruoğlu and E. Özdamar; Extension of the Darboux frame into Euclidean 4-space and its invariants, Turk J Math., 41, (2017), 1628-1639.
• [13] R. Garcia, J. Llibre and J. Sotomayor; Lines of Principal Curvature on Canal Surfaces in R3, An. Acad. Brasil. Cienc., 78(3), (2006), 405-415.
• [14] H. Gluck; Higher curvatures of curves in Euclidean space, Amer Math Monthly, 73, (1966), 699-704.
• [15] E. Guler, Helical Hypersurfaces in Minkowski Geometry E14, Symmetry, 12, (2020), 1206.
• [16] E. Guler, H.H. Hacısalihoğlu and Y.H. Kim, The Gauss map and the third Laplace-Beltrami operator of the rotational hypersurface in 4-Space, Symmetry, 10(9), (2018), 1-11.
• [17] A. Gray; Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd edn. CRC Press, Boca Raton, 1999.
• [18] E. Hartman; Geometry and Algorithms for Computer Aided Design, Dept. of Math. Darmstadt Univ. of Technology, 2003.
• [19] S. Izumiya and M .Takahashi; On caustics of submanifolds and canal hypersurfaces in Euclidean space, Topology Appl., 159, (2012), 501-508.
• [20] M.K. Karacan, H. Es and Y. Yaylı; Singular Points of Tubular Surfaces in Minkowski 3-Space, Sarajevo J. Math., 2(14), (2006), 73-82.
• [21] M.K. Karacan and Y. Tuncer; Tubular Surfaces of Weingarten Types in Galilean and Pseudo-Galilean, Bull. Math. Anal. Appl., 5(2), (2013), 87-100.
• [22] M.K. Karacan, D.W. Yoon and Y. Tuncer; Tubular Surfaces of Weingarten Types in Minkowski 3-Space, Gen. Math. Notes, 22(1), (2014), 44-56.
• [23] A.Kazan, M. Altın and D.W. Yoon; Geometric Characterizations of Canal Hypersurfaces in Euclidean Spaces, arXiv:2111.04448v1 [math.DG], (2021).
• [24] A. Kazan and H.B. Karadağ; Magnetic Curves According to Bishop Frame and Type-2 Bishop Frame in Euclidean 3-Space, British Journal of Mathematics & Computer Science, 22(4), (2017), 1-18.
• [25] S. Kızıltuğ, M. Dede and C. Ekici; Tubular Surfaces with Darboux Frame in Galilean 3-Space, Facta Universitatis Ser. Math. Inform., 34(2), (2019), 253-260.
• [26] Y.H. Kim, H. Liu and J. Qian; Some Characterizations of Canal Surfaces, Bull. Korean Math. Soc., 53(2), (2016), 461-477.
• [27] S.N. Krivoshapko and C.A.B. Hyeng; Classification of Cyclic Surfaces and Geometrical Research of Canal Surfaces, International Journal of Research and Reviews in Applied Sciences, 12(3), (2012), 360-374.
• [28] Z. Küçükarslan Yüzbaşı and D.W. Yoon; Tubular Surfaces with Galilean Darboux Frame in G3, Journal of Mathematical Physics, Analysis, Geometry, 15(2), (2019), 278-287.
• [29] J.M. Lee; Riemannian Manifolds-An Introduction to Curvature, Springer-Verlag New York, Inc, 1997.
• [30] T. Maekawa, N.M. Patrikalakis, T. Sakkalis and G. Yu; Analysis and Applications of Pipe Surfaces, Comput. Aided Geom. Design, 15, (1998), 437-458.
• [31] B. O’Neil; Semi-Riemannian Geometry with Applications to Relativity, Academic Press, London, 1983.
• [32] M. Peternell and H. Pottmann; Computing Rational Parametrizations of Canal Surfaces, J. Symbolic Comput., 23, (1997), 255-266.
• [33] J.S. Ro and D.W. Yoon; Tubes of Weingarten Type in a Euclidean 3-Space, Journal of the Chungcheong Mathematical Society, 22(3), (2009), 359-366.
• [34] A. Uçum and K. İlarslan; New Types of Canal Surfaces in Minkowski 3-Space, Adv. Appl. Clifford Algebr., 26, (2016), 449-468.
• [35] Z. Xu, R. Feng and J-G. Sun; Analytic and Algebraic Properties of Canal Surfaces, J. Comput. Appl. Math., 195, (2006), 220-228.