Research Article
BibTex RIS Cite
Year 2021, Volume: 9 Issue: 1, 132 - 136, 28.04.2021

Abstract

References

  • [1] K. Arslan, B. Bulca, V. Milousheva, Meridian Surfaces in E4. with Pointwise 1-type Gauss Map, Bull. Korean Math. Soc. 51 (2014) 911–922.
  • [2] K. Arslan, R. Deszcz, S. Yaprak, On Weyl Pseudosymmetric Hypersurfaces, Colloq. Math. 72(2) (1997) 353 361.
  • [3] B.Y. Chen, Geometry of Submanifolds. Pure and Applied Mathematics, No. 22. Marcel Dekker, Inc., New York, 1973.
  • [4] U. Dierkes, S. Hildebrandt, F. Sauvigny, Minimal Surfaces, Springer-Verlag, Berlin, Heidelberg. 2nd ed. 2010.
  • [5] L.P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Dover Publications, N.Y. 1909.
  • [6] G. Ganchev, V. Milousheva, On the Theory of Surfaces in the Four-Dimensional Euclidean Space, Kodai Math. J. 31 (2008) 183-198.
  • [7] G. Ganchev, V. Milousheva, An Invariant Theory of Surfaces in the Four-Dimensional Euclidean or Minkowski Space, Pliska Stud. Math. Bulgar. 21 (2012) 177 200.
  • [8] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed., CRC Press, Boca Raton, FL. 1998.
  • [9] E. Guler, H.H. Hacısalihoglu, 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.
  • [10] E. Guler, O. Kisi, Weierstrass Representation, Degree and Classes of the Surfaces in the Four Dimensional Euclidean Space, Celal Bayar Un. J. Sci., 13-1 (2017) 155-163.
  • [11] E. Guler, O. Kisi, C. Konaxis, Implicit Equation of the Henneberg-Type Minimal Surface in the Four Dimensional Euclidean Space, Mathematics Sp. Iss. : Comp. Alg. Sci. Comp. 6(12) (2018) 1 10.
  • [12] E. Guler, M. Magid, Y. Yaylı, Laplace Beltrami Operator of a Helicoidal Hypersurface in Four Space, J. Geom. Sym. Phys. 41 (2016) 77–95.
  • [13] E. Guler, N.C. Turgay, Cheng-Yau operator and Gauss map of rotational hypersurfaces in 4-space, Mediterr. J. Math. 16(3) (2019) 1–16.
  • [14] L. Henneberg, Uber Salche Minimalflache, Welche Eine Vorgeschriebene Ebene Curve Sur Geodatishen¨ Line Haben, Doctoral Dissertation, Eid- genossisches Polythechikum, Zurich 1875.
  • [15] L. Henneberg, Uber Diejenige Minimalflache,¨ Welche Die Neil’sche Paralee Zur Ebenen Geodatischen¨ Line Hat, Vierteljschr Natuforsch, Ges. Zurich,¨ 21 (1876) 66 70.
  • [16] L. Henneberg, Bestimmung Der Neidrigsten Classenzahl Der Algebraischen Minimalflachen¨, Annali di Matem. Pura Appl. 9 (1878) 54 57.
  • [17] D.A. Hoffman, R. Osserman, The Geometry of the Generalized Gauss Map, Memoirs of the AMS, 1980.
  • [18] F. Kahraman Aksoyak, Y. Yaylı, Boost Invariant Surfaces with Pointwise 1-type Gauss Map in Minkowski 4-Space E41; Bull.,Korean Math. Soc. 51 (2014) 1863 1874.
  • [19] F. Kahraman Aksoyak, Y. Yaylı, General Rotational surfaces with Pointwise 1-type Gauss Map in Pseudo-Euclidean Space E42, Indian J. Pure Appl. Math. 46 (2015) 107–118.
  • [20] C. Moore, Surfaces of Rotation in a Space of Four Dimensions, The Annals of Math., 2nd Ser., 21(2) (1919) 81 93.
  • [21] J.C.C. Nitsche, Lectures on Minimal Surfaces. Vol. 1. Introduction, fundamentals, geometry and basic boundary value problems, Cambridge Un Press, Cambridge. 1989.
  • [22] M.E.G.G. de Oliveira, Some New Examples of Nonorientable Minimal Surfaces, Proc. Amer. Math. Soc. 98-4 (1986) 629 636.
  • [23] R. Osserman, A Survey of Minimal Surfaces, Van Nostrand Reinhold Co., New York-London-Melbourne. 1969.
  • [24] K. Weierstrass, Untersuchungen Uber Die Flachen, Deren Mittlere Krummung¨ Uberall Gleich Null Ist., Monatsber d Berliner Akad. (1866) 612 625.
  • [25] K. Weierstrass, Mathematische Werke, Vol. 3, Mayer & Muller, Berlin, 1903.

Weingarten Map of the Henneberg-Type Minimal Surfaces in 4-Space

Year 2021, Volume: 9 Issue: 1, 132 - 136, 28.04.2021

Abstract

We consider a two parameter family of Henneberg-type minimal surfaces $ \mathfrak{H}_{m,n}$ using the Weierstrass representation in the four dimensional Euclidean space $\mathbb{E}^{4}$. An invariant linear map of Weingarten type in the tangent space of the Henneberg-type minimal surface $\mathfrak{H}_{4,2}$ which generates two invariants $\kappa $ and $\varkappa $, is characterized by $ \varkappa^{2}=\kappa $ in $\mathbb{E}^{4}$.

References

  • [1] K. Arslan, B. Bulca, V. Milousheva, Meridian Surfaces in E4. with Pointwise 1-type Gauss Map, Bull. Korean Math. Soc. 51 (2014) 911–922.
  • [2] K. Arslan, R. Deszcz, S. Yaprak, On Weyl Pseudosymmetric Hypersurfaces, Colloq. Math. 72(2) (1997) 353 361.
  • [3] B.Y. Chen, Geometry of Submanifolds. Pure and Applied Mathematics, No. 22. Marcel Dekker, Inc., New York, 1973.
  • [4] U. Dierkes, S. Hildebrandt, F. Sauvigny, Minimal Surfaces, Springer-Verlag, Berlin, Heidelberg. 2nd ed. 2010.
  • [5] L.P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Dover Publications, N.Y. 1909.
  • [6] G. Ganchev, V. Milousheva, On the Theory of Surfaces in the Four-Dimensional Euclidean Space, Kodai Math. J. 31 (2008) 183-198.
  • [7] G. Ganchev, V. Milousheva, An Invariant Theory of Surfaces in the Four-Dimensional Euclidean or Minkowski Space, Pliska Stud. Math. Bulgar. 21 (2012) 177 200.
  • [8] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed., CRC Press, Boca Raton, FL. 1998.
  • [9] E. Guler, H.H. Hacısalihoglu, 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.
  • [10] E. Guler, O. Kisi, Weierstrass Representation, Degree and Classes of the Surfaces in the Four Dimensional Euclidean Space, Celal Bayar Un. J. Sci., 13-1 (2017) 155-163.
  • [11] E. Guler, O. Kisi, C. Konaxis, Implicit Equation of the Henneberg-Type Minimal Surface in the Four Dimensional Euclidean Space, Mathematics Sp. Iss. : Comp. Alg. Sci. Comp. 6(12) (2018) 1 10.
  • [12] E. Guler, M. Magid, Y. Yaylı, Laplace Beltrami Operator of a Helicoidal Hypersurface in Four Space, J. Geom. Sym. Phys. 41 (2016) 77–95.
  • [13] E. Guler, N.C. Turgay, Cheng-Yau operator and Gauss map of rotational hypersurfaces in 4-space, Mediterr. J. Math. 16(3) (2019) 1–16.
  • [14] L. Henneberg, Uber Salche Minimalflache, Welche Eine Vorgeschriebene Ebene Curve Sur Geodatishen¨ Line Haben, Doctoral Dissertation, Eid- genossisches Polythechikum, Zurich 1875.
  • [15] L. Henneberg, Uber Diejenige Minimalflache,¨ Welche Die Neil’sche Paralee Zur Ebenen Geodatischen¨ Line Hat, Vierteljschr Natuforsch, Ges. Zurich,¨ 21 (1876) 66 70.
  • [16] L. Henneberg, Bestimmung Der Neidrigsten Classenzahl Der Algebraischen Minimalflachen¨, Annali di Matem. Pura Appl. 9 (1878) 54 57.
  • [17] D.A. Hoffman, R. Osserman, The Geometry of the Generalized Gauss Map, Memoirs of the AMS, 1980.
  • [18] F. Kahraman Aksoyak, Y. Yaylı, Boost Invariant Surfaces with Pointwise 1-type Gauss Map in Minkowski 4-Space E41; Bull.,Korean Math. Soc. 51 (2014) 1863 1874.
  • [19] F. Kahraman Aksoyak, Y. Yaylı, General Rotational surfaces with Pointwise 1-type Gauss Map in Pseudo-Euclidean Space E42, Indian J. Pure Appl. Math. 46 (2015) 107–118.
  • [20] C. Moore, Surfaces of Rotation in a Space of Four Dimensions, The Annals of Math., 2nd Ser., 21(2) (1919) 81 93.
  • [21] J.C.C. Nitsche, Lectures on Minimal Surfaces. Vol. 1. Introduction, fundamentals, geometry and basic boundary value problems, Cambridge Un Press, Cambridge. 1989.
  • [22] M.E.G.G. de Oliveira, Some New Examples of Nonorientable Minimal Surfaces, Proc. Amer. Math. Soc. 98-4 (1986) 629 636.
  • [23] R. Osserman, A Survey of Minimal Surfaces, Van Nostrand Reinhold Co., New York-London-Melbourne. 1969.
  • [24] K. Weierstrass, Untersuchungen Uber Die Flachen, Deren Mittlere Krummung¨ Uberall Gleich Null Ist., Monatsber d Berliner Akad. (1866) 612 625.
  • [25] K. Weierstrass, Mathematische Werke, Vol. 3, Mayer & Muller, Berlin, 1903.
There are 25 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Erhan Güler 0000-0003-3264-6239

Ömer Kişi 0000-0001-6844-3092

Publication Date April 28, 2021
Submission Date July 3, 2020
Acceptance Date April 21, 2021
Published in Issue Year 2021 Volume: 9 Issue: 1

Cite

APA Güler, E., & Kişi, Ö. (2021). Weingarten Map of the Henneberg-Type Minimal Surfaces in 4-Space. Konuralp Journal of Mathematics, 9(1), 132-136.
AMA Güler E, Kişi Ö. Weingarten Map of the Henneberg-Type Minimal Surfaces in 4-Space. Konuralp J. Math. April 2021;9(1):132-136.
Chicago Güler, Erhan, and Ömer Kişi. “Weingarten Map of the Henneberg-Type Minimal Surfaces in 4-Space”. Konuralp Journal of Mathematics 9, no. 1 (April 2021): 132-36.
EndNote Güler E, Kişi Ö (April 1, 2021) Weingarten Map of the Henneberg-Type Minimal Surfaces in 4-Space. Konuralp Journal of Mathematics 9 1 132–136.
IEEE E. Güler and Ö. Kişi, “Weingarten Map of the Henneberg-Type Minimal Surfaces in 4-Space”, Konuralp J. Math., vol. 9, no. 1, pp. 132–136, 2021.
ISNAD Güler, Erhan - Kişi, Ömer. “Weingarten Map of the Henneberg-Type Minimal Surfaces in 4-Space”. Konuralp Journal of Mathematics 9/1 (April 2021), 132-136.
JAMA Güler E, Kişi Ö. Weingarten Map of the Henneberg-Type Minimal Surfaces in 4-Space. Konuralp J. Math. 2021;9:132–136.
MLA Güler, Erhan and Ömer Kişi. “Weingarten Map of the Henneberg-Type Minimal Surfaces in 4-Space”. Konuralp Journal of Mathematics, vol. 9, no. 1, 2021, pp. 132-6.
Vancouver Güler E, Kişi Ö. Weingarten Map of the Henneberg-Type Minimal Surfaces in 4-Space. Konuralp J. Math. 2021;9(1):132-6.
Creative Commons License
The published articles in KJM are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.