Research Article
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Year 2023, Volume: 72 Issue: 2, 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, Volume: 72 Issue: 2, 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 Volume: 72 Issue: 2

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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