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

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.

Keywords:

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

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[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.