1Bartın University, Faculty of Sciences, Department of Mathematics, 74100 Bartın, Turkey eguler@bartin.edu.tr, ergler@gmail.com
2Faculty of Mathematics and Engineering Sciences, Vari, Attiki, Greece gmiamis@gmail.com
3Department of Mathematics, Wellesley College, Wellesley, MA 02458, United States mmagid@wellesley.edu
Abstract
Helicoidal hypersurfaces in the four dimensional Minkowski space are defined. There are three types, depending on the axis of rotation. Equations for the Gaussian and mean curvature are derived and many examples of the various types of hypersurfaces are given. A theorem classifying the helicoids with timelike axes and ΔH=AH is obtained.
Keywords: Helicoidal hypersurface; Laplace-Beltrami operator; Gaussian curvature;
mean curvature; Minkowski 4-space.
References
[1] A. Arvanitoyeorgos, G. Kaimakamis and M. Magid, Lorentz hypersurfaces in E14 satisfying ΔH=αH, Illinois J. Math., 53(2): 581-590, 2009.
[2] Chr.C. Beneki, G. Kaimakamis, and B.J. Papantoniou, Helicoidal surfaces in three- dimensional Minkowski space, J. Math. Anal. Appl., 275: 586-614, 2002.
[3] E. Bour, Théorie de la déformation des surfaces, J. de l.Êcole Imperiale Polytech., 22(39):
1-148, 1862.
[4] B.Y. Chen, Total mean curvature and submanifolds of finite type, World Scientific, Singapore, 1984.
[5] Q.M. Cheng and Q.R. Wan, Complete hypersurfaces of R4 with constant mean curvature, Monatsh. Math., 118(3-4): 171-204, 1994.
[6] M. Choi and Y.H. Kim, Characterization of the helicoid as ruled surfaces with pointwise 1- type Gauss map, Bull. Korean Math. Soc., 38: 753-761, 2001.
[7] F., Dillen, J. Pas and L. Verstraelen, On surfaces of finite type in Euclidean 3-space, Kodai Math. J., 13: 10-21, 1990.
[8] M. Do Carmo and M. Dajczer, Helicoidal surfaces with constant mean curvature, Tohoku Math. J., 34: 351-367, 1982.
[9] A. Ferrandez, O.J. Garay and P. Lucas, On a certain class of conformally at Euclidean hypersurfaces, Proc. of the Conf. in Global Analysis and Global Differential Geometry, Berlin, 1990.
[10] G. Ganchev and V. Milousheva, General rotational surfaces in the 4-dimensional Minkowski space, Turkish J. Math., 38: 883-895, 2014.
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[11] E. Güler, Bour's theorem and lightlike profile curve, Yokohama Math. J., 54(1): 55-77, 2007.
[12] E. Güler, M. Magid and Y. Yaylı, Laplace-Beltrami Operator of a Helicoidal Hypersurface in Four-Space, J. Geom. Symmetry Phys., 41: 77-95, 2016.
[13] H.B. Lawson, Lectures on minimal submanifolds, Vol. 1, Rio de Janeiro, 1973.
[14] M. Magid, C. Scharlach and L. Vrancken, Affine umbilical surfaces in R4, Manuscripta Math., 88: 275-289, 1995.
[15] C. Moore, Surfaces of rotation in a space of four dimensions, Ann. Math., 21 81-93, 1919.
[16] C. Moore, Rotation surfaces of constant curvature in space of four dimensions, Bull. Amer.
Math. Soc., 26: 454-460, 1920.
[17] M. Moruz and M.I. Munteanu, Minimal translation hypersurfaces in E4, J. Math. Anal.
Appl., 439: 798-812, 2016.
[18] C. Scharlach, Affine geometry of surfaces and hypersurfaces in R4, Symposium on the Differential Geometry of Submanifolds, France, 251-256, 2007.
[19] B. Senoussi and M. Bekkar, Helicoidal surfaces with ∆ =Jr Ar in 3-dimensional Euclidean space, Stud. Univ. Babeş-Bolyai Math., 60(3): 437-448, 2015.
[20] T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18:
380-385, 1966.
[21] L. Verstraelen, J. Walrave and S. Yaprak, The minimal translation surfaces in Euclidean space, Soochow J. Math., 20(1): 77-82, 1994.
[22] Th. Vlachos, Hypersurfaces in E4 with harmonic mean curvature vector field, Math.
Nachr., 172: 145-169, 1995.
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Surface Growth Kinematics in Galilean Space
Zehra Özdemir1, Gül Tuğ2, İsmail Gök3 and F. Nejat Ekmekci41Amasya University, Department of Mathematics, Amasya/TURKEY zehra.ozdemir@amasya.edu.tr
2Karadeniz Teknik University, Department of Mathematics, Trabzon/TURKEY gguner@ktu.edu.tr
3Ankara University, Department of Mathematics, Ankara/TURKEY igok@science.ankara.edu.tr
4Ankara University, Department of Mathematics, Ankara/TURKEY ekmekci@science.ankara.edu.tr
Abstract
In this study, we investigate the mathematical framework to model the kinematics of the surface growth of objects such as some crustacean creatures in Galilean 3-Space. For this, the growth kinematics of the different species of these creatures is obtained by applying a method with a system of differential equations. Using the analytical solutions of this system, various surface examples, including some seashells are provided and the shapes of these surfaces are illustrated.
Keywords: Alternative moving frame; Accretive growth; Darboux vector; General helix.
References
[1] P.J. Davis., The Schwarz Function and its Applications, Carius Monograph 17, Mathematical Association of America, 1974.
[2] C. Illert, Formulation and solution of the classical problem, I Seashell geometry, Nuovo Cimento, 9(7): 791-814, 1987.
[3] C. Illert, Formulation and solution of the classical problem, II Tubular three dimensional surfaces, Nuovo Cimento, 11: 761-780, 1989.
[4] D.E. Moulton and A. Goriely, Mechanical growth and morphogenesis of seashells, J. Theor.
Biol., 311: 69–79, 2012.
[5] D.E. Moulton, A. Goriely and R. Chirat, Surface growth kinematics via local curve evolution, J. Math. Biol., 68: 81–108, 2014.
[6] R. Skalak, D. Farrow and A. Hoger, Kinematics of surface growth, J. Math. Biol., 35(8):
869–907, 1997.
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Möbius-Type Hypersurface in 4-Space
Erhan Güler1 and H. Hilmi Hacısalihoğlu21Bartın University, Faculty of Sciences, Department of Mathematics, 74100 Bartın, Turkey eguler@bartin.edu.tr, ergler@gmail.com
2Şeyh Edebali University (Emeritus), Faculty of Sciences, Department of Mathematics 11230 Bilecik, Turkey
h.hacisalihoglu@bilecik.edu.tr
Abstract
Möbius-type hypersurface in the four dimensional Euclidean space is defined. Equations for the Gaussian and mean curvature are derived and some examples of hypersurface are given.
Keywords: Möbius-type hypersurface; Gaussian curvature; mean curvature; Euclidean 4-space.
References
[1] W.W.R. Ball and H.S.M. Coxeter, Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 127-128, 1987.
[2] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, New York: North Holland, p. 243, 1976.
[3] J. Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, New York: Penguin, 2004.
[4] C.T.J. Dodson and P.E., Parker, A User's Guide to Algebraic Topology, Dordrecht, Netherlands: Kluwer, pp. 121 and 284, 1997.
[5] M. Gardner, The Sixth Book of Mathematical Games from Scientific American, Chicago, IL: University of Chicago Press, p. 10, 1984.
[6] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC Press, Florida, 1998.
[7] H.H. Hacısalihoğlu, İki ve Üç Boyutlu Uzaylarda Dönüşümler ve Geometriler, Ankara Üniversitesi, Ankara, 1998.
[8] M. Henle, A Combinatorial Introduction to Topology, New York: Dover, p. 110, 1994.
[9] J.A.H. Hunter and J.S. Madachy, Mathematical Diversions, New York: Dover, pp. 41-45, 1975.
[10] J.B. Listing, Vorstudien zur Topologie, Göttinger Studien, Pt. 10, 1848. Facsimile Publisher 2016.
[11] J. S. Madachy, Madachy's Mathematical Recreations, New York: Dover, p. 7, 1979.
[12] S. Montel and A. Ross, Curves and Surfaces, AMS, Real Sociedad Matematica Espanola, 2005.
[13] A.F. Möbius, Werke, Vol. 2. p. 519, 1858.
[14] J.R. Munkres, Topology, Prentice Hall Inc., USA, 2000.
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[15] T. Pappas, "The Moebius Strip & the Klein Bottle," "A Twist to the Moebius Strip," "The 'Double' Moebius Strip." The Joy of Mathematics, San Carlos, CA: Wide World Publ./Tetra, p.
207, 1989.
[16] C.A. Pickover, The Möbius Strip: Dr. August Mobius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology, New York: Thunder's Mouth Press, 2006.
[17] A. Sabuncuoğlu, Diferensiyel Geometri, Nobel Yayın, Ankara, 2001.
[18] H. Steinhaus, Mathematical Snapshots, 3rd ed. New York: Dover, pp. 269-274, 1999.
[19] Wagon, S. "Rotating Circles to Produce a Torus or Möbius Strip." §7.4 in Mathematica in Action. New York: W. H. Freeman, pp. 229-232, 1991.
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