Research Article
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Year 2023, , 331 - 339, 23.06.2023
https://doi.org/10.31801/cfsuasmas.1062426

Abstract

References

  • Arslan, K., Bayram, B. K., Bulca, B., Öztürk, G., Generalized rotation surfaces in $E^{4}$, Results Math., 61(3) (2012), 315–327. https://doi.org/10.1007/s00025-011-0103-3
  • Arslan, K., Deszcz, R., Yaprak, S¸., On Weyl pseudosymmetric hypersurfaces, Colloq. Math., 72(2) (1997), 353–361.
  • Arslan, K., Milousheva, V., Meridian surfaces of elliptic or hyperbolic type with pointwise 1-type Gauss map in Minkowski 4-space, Taiwanese J. Math., 20(2) (2016), 311–332. https://doi.org/10.11650/tjm.19.2015.5722
  • Arvanitoyeorgos, A. , Kaimakamis, G., Magid, M., Lorentz hypersurfaces in $E_{1}^{4}$ satisfying $\Delta H=\alpha H,$ Illinois J. Math., 53(2) (2009), 581–590. https://doi.org/10.1215/ijm/1266934794
  • Beneki, Chr. C., Kaimakamis, G., Papantoniou, B. J., Helicoidal surfaces in three-dimensional Minkowski space, J. Math. Anal. Appl., 275 (2002), 586–614. https://doi.org/10.1016/S0022-247X(02)00269-X
  • Chen, B. Y., Total Mean Curvature and Submanifolds of Finite Type, 2nd Ed., World Scientific, Singapore, 2014. https://doi.org/10.1142/9237
  • Cheng, Q. M., Wan, Q. R., Complete hypersurfaces of $R^{4}$ with constant mean curvature, Monatsh. Math., 118 (1994), 171–204. https://doi.org/10.1007/BF01301688
  • Cheng, S. Y., Yau, S. T., Hypersurfaces with constant scalar curvature, Math. Ann., 225 (1977), 195–204. https://doi.org/10.1007/BF01425237
  • Dillen, F., Fastenakels, J., Van der Veken, J., Rotation hypersurfaces of $S^{n}×R$ and $H^{n}×R,$ Note Mat., 29(1) (2009), 41–54. https://doi.org/10.1285/i15900932v29n1p41
  • Do Carmo, M. P., Dajczer, M., Rotation hypersurfaces in spaces of constant curvature, Trans. Am. Math. Soc., 277 (1983), 685–709. https://doi.org/10.1007/978-3-642-25588-517
  • Dursun, U., Hypersurfaces with pointwise 1-type Gauss map, Taiwanese J. Math., 11(5) (2007), 1407–1416. https://doi.org/10.11650/twjm/1500404873
  • Dursun, U., Turgay, N. C., Space-like surfaces in Minkowski space $E_{1}^{4}$ with pointwise 1-type Gauss map, Ukr. Math. J., 71(1) (2019), 64–80. https://doi.org/10.1007/s11253-019-01625-8
  • Ferrandez, A., Garay, O. J., Lucas, P., On a certain class of conformally flat Euclidean hypersurfaces, In Global Differential Geometry and Global Analysis, Lecture Notes in Mathematics, vol 1481. Springer, Heidelberg, Berlin, Germany, 1991, 48–54. https://doi.org/10.1007/BFb0083627
  • Ganchev, G., Milousheva, V., General rotational surfaces in the 4-dimensional Minkowski space, Turkish J. Math., 38 (2014), 883–895. https://doi.org/10.3906/mat-1312-10
  • Güler, E., Helical hypersurfaces in Minkowski geometry $E_{1}^{4}$, Symmetry, 12(8) (2020), 1–16. https://doi.org/10.3390/sym12081206
  • Güler, E., Fundamental form IV and curvature formulas of the hypersphere, Malaya J. Mat., 8(4) (2020), 2008–2011. https://doi.org/10.26637/MJM0804/0116
  • Güler, E., Rotational hypersurfaces satisfying $\Delta ^{I}R = AR$ in the four-dimensional Euclidean space, J. Polytech., 24(2) (2021), 517–520. https://doi.org/10.2339/politeknik.670333
  • Güler, E., Hacısalihoglu, H. H., Kim, Y.H., The Gauss map and the third Laplace–Beltrami operator of the rotational hypersurface in 4-space, Symmetry, 10(9) (2018), 1–12. https://doi.org/10.3390/sym10090398
  • Güler, E., Magid, M., Yaylı, Y., Laplace–Beltrami operator of a helicoidal hypersurface in four-space, J. Geom. Symmetry Phys., 41 (2016), 77–95. https://doi.org/10.7546/jgsp-41-2016-77-95
  • Güler, E., Turgay, N. C., Cheng–Yau operator and Gauss map of rotational hypersurfaces in 4-space, Mediterr. J. Math., 16(3) (2019), 1–16. https://doi.org/10.1007/s00009-019-1333-y
  • Hasanis, Th., Vlachos, Th., Hypersurfaces in $E^{4} with harmonic mean curvature vector field, Math. Nachr., 172 (1995), 145–169. https://doi.org/10.1002/mana.19951720112
  • Kim, Y. H., Turgay, N. C., Surfaces in $E^{4}$ with $L_{1}$-pointwise 1-type Gauss map, Bull. Korean Math. Soc., 50(3) (2013), 935–949. http://dx.doi.org/10.4134/BKMS.2013.50.3.935
  • Lawson, H. B., Lectures on Minimal Submanifolds, Vol. I., Second ed., Mathematics Lecture Series, 9. Publish or Perish, Wilmington, Del., 1980.
  • Magid, M., Scharlach, C., Vrancken, L., Affine umbilical surfaces in R4, Manuscripta Math., 88 (1995), 275–289. http://dx.doi.org/10.1007/BF02567823
  • Moore, C., Surfaces of rotation in a space of four dimensions, Ann. Math., 21 (1919), 81–93. https://doi.org/10.2307/2007223
  • Moore, C., Rotation surfaces of constant curvature in space of four dimensions, Bull. Amer. Math. Soc., 26 (1920), 454–460. https://doi.org/10.1090/S0002-9904-1920-03336-7
  • O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.
  • Takahashi, T., Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18 (1966), 380–385. https://doi.org/10.2969/jmsj/01840380
  • Turgay, N. C., Upadhyay, A., On biconservative hypersurfaces in 4-dimensional Riemannian space forms, Math. Nachr., 292(4) (2019), 905–921. https://doi.org/10.1002/mana.201700328

Timelike rotational hypersurfaces with timelike axis in Minkowski four-space

Year 2023, , 331 - 339, 23.06.2023
https://doi.org/10.31801/cfsuasmas.1062426

Abstract

We introduce the timelike rotational hypersurfaces $\textbf{x}$ with timelike axis in Minkowski 4-space $\mathbb{E}_1^{4}$. We obtain the equations for the curvatures of the hypersurface. Moreover, we present a theorem for the rotational hypersurfaces with timelike axis supplying $\Delta\textbf{x}=\mathcal{T}\textbf{x}$, where $\mathcal{T}$ is a 4x4 real matrix.

References

  • Arslan, K., Bayram, B. K., Bulca, B., Öztürk, G., Generalized rotation surfaces in $E^{4}$, Results Math., 61(3) (2012), 315–327. https://doi.org/10.1007/s00025-011-0103-3
  • Arslan, K., Deszcz, R., Yaprak, S¸., On Weyl pseudosymmetric hypersurfaces, Colloq. Math., 72(2) (1997), 353–361.
  • Arslan, K., Milousheva, V., Meridian surfaces of elliptic or hyperbolic type with pointwise 1-type Gauss map in Minkowski 4-space, Taiwanese J. Math., 20(2) (2016), 311–332. https://doi.org/10.11650/tjm.19.2015.5722
  • Arvanitoyeorgos, A. , Kaimakamis, G., Magid, M., Lorentz hypersurfaces in $E_{1}^{4}$ satisfying $\Delta H=\alpha H,$ Illinois J. Math., 53(2) (2009), 581–590. https://doi.org/10.1215/ijm/1266934794
  • Beneki, Chr. C., Kaimakamis, G., Papantoniou, B. J., Helicoidal surfaces in three-dimensional Minkowski space, J. Math. Anal. Appl., 275 (2002), 586–614. https://doi.org/10.1016/S0022-247X(02)00269-X
  • Chen, B. Y., Total Mean Curvature and Submanifolds of Finite Type, 2nd Ed., World Scientific, Singapore, 2014. https://doi.org/10.1142/9237
  • Cheng, Q. M., Wan, Q. R., Complete hypersurfaces of $R^{4}$ with constant mean curvature, Monatsh. Math., 118 (1994), 171–204. https://doi.org/10.1007/BF01301688
  • Cheng, S. Y., Yau, S. T., Hypersurfaces with constant scalar curvature, Math. Ann., 225 (1977), 195–204. https://doi.org/10.1007/BF01425237
  • Dillen, F., Fastenakels, J., Van der Veken, J., Rotation hypersurfaces of $S^{n}×R$ and $H^{n}×R,$ Note Mat., 29(1) (2009), 41–54. https://doi.org/10.1285/i15900932v29n1p41
  • Do Carmo, M. P., Dajczer, M., Rotation hypersurfaces in spaces of constant curvature, Trans. Am. Math. Soc., 277 (1983), 685–709. https://doi.org/10.1007/978-3-642-25588-517
  • Dursun, U., Hypersurfaces with pointwise 1-type Gauss map, Taiwanese J. Math., 11(5) (2007), 1407–1416. https://doi.org/10.11650/twjm/1500404873
  • Dursun, U., Turgay, N. C., Space-like surfaces in Minkowski space $E_{1}^{4}$ with pointwise 1-type Gauss map, Ukr. Math. J., 71(1) (2019), 64–80. https://doi.org/10.1007/s11253-019-01625-8
  • Ferrandez, A., Garay, O. J., Lucas, P., On a certain class of conformally flat Euclidean hypersurfaces, In Global Differential Geometry and Global Analysis, Lecture Notes in Mathematics, vol 1481. Springer, Heidelberg, Berlin, Germany, 1991, 48–54. https://doi.org/10.1007/BFb0083627
  • Ganchev, G., Milousheva, V., General rotational surfaces in the 4-dimensional Minkowski space, Turkish J. Math., 38 (2014), 883–895. https://doi.org/10.3906/mat-1312-10
  • Güler, E., Helical hypersurfaces in Minkowski geometry $E_{1}^{4}$, Symmetry, 12(8) (2020), 1–16. https://doi.org/10.3390/sym12081206
  • Güler, E., Fundamental form IV and curvature formulas of the hypersphere, Malaya J. Mat., 8(4) (2020), 2008–2011. https://doi.org/10.26637/MJM0804/0116
  • Güler, E., Rotational hypersurfaces satisfying $\Delta ^{I}R = AR$ in the four-dimensional Euclidean space, J. Polytech., 24(2) (2021), 517–520. https://doi.org/10.2339/politeknik.670333
  • Güler, E., Hacısalihoglu, H. H., Kim, Y.H., The Gauss map and the third Laplace–Beltrami operator of the rotational hypersurface in 4-space, Symmetry, 10(9) (2018), 1–12. https://doi.org/10.3390/sym10090398
  • Güler, E., Magid, M., Yaylı, Y., Laplace–Beltrami operator of a helicoidal hypersurface in four-space, J. Geom. Symmetry Phys., 41 (2016), 77–95. https://doi.org/10.7546/jgsp-41-2016-77-95
  • Güler, E., Turgay, N. C., Cheng–Yau operator and Gauss map of rotational hypersurfaces in 4-space, Mediterr. J. Math., 16(3) (2019), 1–16. https://doi.org/10.1007/s00009-019-1333-y
  • Hasanis, Th., Vlachos, Th., Hypersurfaces in $E^{4} with harmonic mean curvature vector field, Math. Nachr., 172 (1995), 145–169. https://doi.org/10.1002/mana.19951720112
  • Kim, Y. H., Turgay, N. C., Surfaces in $E^{4}$ with $L_{1}$-pointwise 1-type Gauss map, Bull. Korean Math. Soc., 50(3) (2013), 935–949. http://dx.doi.org/10.4134/BKMS.2013.50.3.935
  • Lawson, H. B., Lectures on Minimal Submanifolds, Vol. I., Second ed., Mathematics Lecture Series, 9. Publish or Perish, Wilmington, Del., 1980.
  • Magid, M., Scharlach, C., Vrancken, L., Affine umbilical surfaces in R4, Manuscripta Math., 88 (1995), 275–289. http://dx.doi.org/10.1007/BF02567823
  • Moore, C., Surfaces of rotation in a space of four dimensions, Ann. Math., 21 (1919), 81–93. https://doi.org/10.2307/2007223
  • Moore, C., Rotation surfaces of constant curvature in space of four dimensions, Bull. Amer. Math. Soc., 26 (1920), 454–460. https://doi.org/10.1090/S0002-9904-1920-03336-7
  • O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.
  • Takahashi, T., Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18 (1966), 380–385. https://doi.org/10.2969/jmsj/01840380
  • Turgay, N. C., Upadhyay, A., On biconservative hypersurfaces in 4-dimensional Riemannian space forms, Math. Nachr., 292(4) (2019), 905–921. https://doi.org/10.1002/mana.201700328
There are 29 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Erhan Güler 0000-0003-3264-6239

Publication Date June 23, 2023
Submission Date January 24, 2022
Acceptance Date November 27, 2022
Published in Issue Year 2023

Cite

APA Güler, E. (2023). Timelike rotational hypersurfaces with timelike axis in Minkowski four-space. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 72(2), 331-339. https://doi.org/10.31801/cfsuasmas.1062426
AMA Güler E. Timelike rotational hypersurfaces with timelike axis in Minkowski four-space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. June 2023;72(2):331-339. doi:10.31801/cfsuasmas.1062426
Chicago Güler, Erhan. “Timelike Rotational Hypersurfaces With Timelike Axis in Minkowski Four-Space”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72, no. 2 (June 2023): 331-39. https://doi.org/10.31801/cfsuasmas.1062426.
EndNote Güler E (June 1, 2023) Timelike rotational hypersurfaces with timelike axis in Minkowski four-space. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72 2 331–339.
IEEE E. Güler, “Timelike rotational hypersurfaces with timelike axis in Minkowski four-space”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 72, no. 2, pp. 331–339, 2023, doi: 10.31801/cfsuasmas.1062426.
ISNAD Güler, Erhan. “Timelike Rotational Hypersurfaces With Timelike Axis in Minkowski Four-Space”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72/2 (June 2023), 331-339. https://doi.org/10.31801/cfsuasmas.1062426.
JAMA Güler E. Timelike rotational hypersurfaces with timelike axis in Minkowski four-space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72:331–339.
MLA Güler, Erhan. “Timelike Rotational Hypersurfaces With Timelike Axis in Minkowski Four-Space”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 72, no. 2, 2023, pp. 331-9, doi:10.31801/cfsuasmas.1062426.
Vancouver Güler E. Timelike rotational hypersurfaces with timelike axis in Minkowski four-space. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72(2):331-9.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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