Araştırma Makalesi
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Simple Rotational Surfaces in Euclidean 4-Space with Generalized 1-Type Gauss Map

Yıl 2025, Cilt: 18 Sayı: 1, 48 - 59, 24.04.2025

Nurettin Cenk Turgay , Mustafa Sağdiç , Erol Kılıç

Öz

In this paper, we consider simple rotational surfaces in the Euclidean 4-space $\mathbb E^4$ with the profile curve contained in a 2-plane. In terms of having generalized 1-type Gauss map, we obtain some classification results of minimal surfaces, flat simple rotational surfaces and simple rotational surfaces with constant Gaussian curvature.

Anahtar Kelimeler

Submanifolds with finite type Gauss map surfaces with finite type Gauss ma rotational surfaces with generalized 1-type Gauss map

Kaynakça

  • Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. US Government printing office. Washington (2006).
  • Arslan, K., Bayram, B. K., Bulca, B., Kim, Y. H., Murathan, C., Öztürk, G.: Rotational embeddings in $\mathbb{E}^{4}$ with pointwise 1-type Gauss map, Turk. J. Math., 35 (3), 493–499 (2011).
  • Baikoussis, C., Chen, B. Y., Verstraelen, L.: Surfaces with finite type Gauss map, Geometry and Topology of submanifolds, 4, 214–216 (1992).
  • Bektaş, B., Canfes, E. Ö., Dursun, U.: On rotational surfaces in pseudo-Euclidean space $\mathbb{E}^{4}_t$ with pointwise 1-type Gauss map, Acta Universitatis Apulensis, 45, 43–59 (2016).
  • Chen, B. Y.: Total Mean Curvature and Submanifold of Finite Type. Academic Press. London (1984).
  • Chen, B. Y., Choi,M., Kim, Y. H.: Surfaces of revolution with pointwise 1-type Gauss map, J. Korean Math. Soc., 42 (4), 447–455 (2005).
  • Chen, B. Y., Piccinni, P.: Submanifolds with finite type Gauss map, Bull. Austral. Math. Soc., 35 (2), 161–186 (1987).
  • Choi, M. K., Kim, D. S., Kim, Y. H.: Helicoidal surfaces with pointwise 1-type Gauss map. J. Korean Math. Soc. 46 (1), 215–223 (2009).
  • Choi, M. K., Kim, Y. H.: Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map, Bulletin of the Korean Mathematical Society, 38 (4), 753–761 (2001).
  • Cole, F. N.: On rotations in space of four dimensions, Amer. J. Math., 12 (2), 191–210 (1890).
  • Dursun, U., Arsan, G. G.: Surfaces in the Euclidean space $\mathbb{E}^{4}$ with pointwise 1-type Gauss map, Hacettepe Journal of Mathematics and Statistics, 40 (5), 617–625 (2011).
  • Dursun, U., Bektaş, B.: Spacelike Rotational Surfaces of Elliptic, Hyperbolic and Parabolic Types in Minkowski Space $\mathbb{E}^{4}_1$ with Pointwise 1-Type Gauss Map, Mathematical Physics, Analysis and Geometry, 17, 247–263 (2014).
  • Dursun, U., Turgay, N. C.: General rotational surfaces in Euclidean space $\mathbb{E}^{4}$ with pointwise 1-type Gauss map, Mathematical Communications, 17 (1), 71–81 (2012).
  • Dursun, U., Yeğin, R.: Hyperbolic submanifolds with finite type hyperbolic Gauss map, International Journal of Mathematics, 26 (2), 1550014 (2015).
  • Moore, C. L. E.: Surfaces of rotation in a space of four dimensions, Annals. of Math., 21 (2), 81–93 (1919).
  • Ki, U. H., Kim, D. S., Kim, Y. H., Roh, Y. M.: Surfaces of revolution with pointwise 1-type Gauss map in Minkowski 3-space, Taiwanese J. Math., 13 (1), 317–338 (2009).
  • Kim, Y. H., Yoon, D. W.: Classification of rotation surfaces in pseudo-Euclidean space, J. Korean Math. Soc., 41 (2), 379–396 (2004).
  • Kim, Y. H., Yoon, D. W.: Ruled surfaces with pointwise 1-type Gauss map, J. Geom. Phys. 34 (3-4), 191–205 (2000).
  • Niang, A.: Rotation surfaces with 1-type Gauss map, Bull. Korean Math. Soc., 42 (1), 23–27 (2005).
  • Qian, J. H., Su, M. F., Kim, Y. H.: Canal surfaces with generalized 1-type Gauss map, Rev. Union Mat. Argent, 62, 199–211 (2021).
  • Şen, R. Y., Dursun, U.: On Submanifolds with 2-Type Pseudo-Hyperbolic Gauss Map in Pseudo-Hyperbolic Space, Mediterranean Journal of Mathematics, 14, 1–20 (2017).
  • Turgay, N. C.: Euclıd ve Yarı-Euclıd Uzaylarının Noktasal 1-tipinden Gauss Tasvirine Sahip Alt Manifoldları. Ph.D. thesis. ˙Istanbul Technical University (2013).
  • Yeğin, R., Dursun, U.: On submanifolds of pseudo-hyperbolic space with 1-type pseudo-hyperbolic Gauss map, J. Math. Phys. Anal. 12 (4), 315–337 (2016).
  • Yoon, D. W.: Rotation surfaces with finite type Gauss Map in $\mathbb{E}^{4}$, Indian J. Pure. Appl. Math., 32 (5), 1803–1808 (2001).
  • Yoon, D. W.: Some properties of the Clifford Torus as rotation surface, Indian J. Pure. Appl. Math., 34 (3), 907–915 (2004).
  • Yoon, D. W., Kim, D. S., Kim, Y. H., Lee, J. W.: Hypersurfaces with generalized 1-type Gauss maps, Mathematics, 6 (8), 130 (2018)

Yıl 2025, Cilt: 18 Sayı: 1, 48 - 59, 24.04.2025

Nurettin Cenk Turgay , Mustafa Sağdiç , Erol Kılıç

Öz

Kaynakça

  • Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. US Government printing office. Washington (2006).
  • Arslan, K., Bayram, B. K., Bulca, B., Kim, Y. H., Murathan, C., Öztürk, G.: Rotational embeddings in $\mathbb{E}^{4}$ with pointwise 1-type Gauss map, Turk. J. Math., 35 (3), 493–499 (2011).
  • Baikoussis, C., Chen, B. Y., Verstraelen, L.: Surfaces with finite type Gauss map, Geometry and Topology of submanifolds, 4, 214–216 (1992).
  • Bektaş, B., Canfes, E. Ö., Dursun, U.: On rotational surfaces in pseudo-Euclidean space $\mathbb{E}^{4}_t$ with pointwise 1-type Gauss map, Acta Universitatis Apulensis, 45, 43–59 (2016).
  • Chen, B. Y.: Total Mean Curvature and Submanifold of Finite Type. Academic Press. London (1984).
  • Chen, B. Y., Choi,M., Kim, Y. H.: Surfaces of revolution with pointwise 1-type Gauss map, J. Korean Math. Soc., 42 (4), 447–455 (2005).
  • Chen, B. Y., Piccinni, P.: Submanifolds with finite type Gauss map, Bull. Austral. Math. Soc., 35 (2), 161–186 (1987).
  • Choi, M. K., Kim, D. S., Kim, Y. H.: Helicoidal surfaces with pointwise 1-type Gauss map. J. Korean Math. Soc. 46 (1), 215–223 (2009).
  • Choi, M. K., Kim, Y. H.: Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map, Bulletin of the Korean Mathematical Society, 38 (4), 753–761 (2001).
  • Cole, F. N.: On rotations in space of four dimensions, Amer. J. Math., 12 (2), 191–210 (1890).
  • Dursun, U., Arsan, G. G.: Surfaces in the Euclidean space $\mathbb{E}^{4}$ with pointwise 1-type Gauss map, Hacettepe Journal of Mathematics and Statistics, 40 (5), 617–625 (2011).
  • Dursun, U., Bektaş, B.: Spacelike Rotational Surfaces of Elliptic, Hyperbolic and Parabolic Types in Minkowski Space $\mathbb{E}^{4}_1$ with Pointwise 1-Type Gauss Map, Mathematical Physics, Analysis and Geometry, 17, 247–263 (2014).
  • Dursun, U., Turgay, N. C.: General rotational surfaces in Euclidean space $\mathbb{E}^{4}$ with pointwise 1-type Gauss map, Mathematical Communications, 17 (1), 71–81 (2012).
  • Dursun, U., Yeğin, R.: Hyperbolic submanifolds with finite type hyperbolic Gauss map, International Journal of Mathematics, 26 (2), 1550014 (2015).
  • Moore, C. L. E.: Surfaces of rotation in a space of four dimensions, Annals. of Math., 21 (2), 81–93 (1919).
  • Ki, U. H., Kim, D. S., Kim, Y. H., Roh, Y. M.: Surfaces of revolution with pointwise 1-type Gauss map in Minkowski 3-space, Taiwanese J. Math., 13 (1), 317–338 (2009).
  • Kim, Y. H., Yoon, D. W.: Classification of rotation surfaces in pseudo-Euclidean space, J. Korean Math. Soc., 41 (2), 379–396 (2004).
  • Kim, Y. H., Yoon, D. W.: Ruled surfaces with pointwise 1-type Gauss map, J. Geom. Phys. 34 (3-4), 191–205 (2000).
  • Niang, A.: Rotation surfaces with 1-type Gauss map, Bull. Korean Math. Soc., 42 (1), 23–27 (2005).
  • Qian, J. H., Su, M. F., Kim, Y. H.: Canal surfaces with generalized 1-type Gauss map, Rev. Union Mat. Argent, 62, 199–211 (2021).
  • Şen, R. Y., Dursun, U.: On Submanifolds with 2-Type Pseudo-Hyperbolic Gauss Map in Pseudo-Hyperbolic Space, Mediterranean Journal of Mathematics, 14, 1–20 (2017).
  • Turgay, N. C.: Euclıd ve Yarı-Euclıd Uzaylarının Noktasal 1-tipinden Gauss Tasvirine Sahip Alt Manifoldları. Ph.D. thesis. ˙Istanbul Technical University (2013).
  • Yeğin, R., Dursun, U.: On submanifolds of pseudo-hyperbolic space with 1-type pseudo-hyperbolic Gauss map, J. Math. Phys. Anal. 12 (4), 315–337 (2016).
  • Yoon, D. W.: Rotation surfaces with finite type Gauss Map in $\mathbb{E}^{4}$, Indian J. Pure. Appl. Math., 32 (5), 1803–1808 (2001).
  • Yoon, D. W.: Some properties of the Clifford Torus as rotation surface, Indian J. Pure. Appl. Math., 34 (3), 907–915 (2004).
  • Yoon, D. W., Kim, D. S., Kim, Y. H., Lee, J. W.: Hypersurfaces with generalized 1-type Gauss maps, Mathematics, 6 (8), 130 (2018)
Toplam 26 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Cebirsel ve Diferansiyel Geometri
Bölüm Araştırma Makalesi
Yazarlar

Nurettin Cenk Turgay 0000-0002-0171-3876

Mustafa Sağdiç 0000-0002-0993-6281

Erol Kılıç 0000-0001-7536-0404

Erken Görünüm Tarihi 20 Nisan 2025
Yayımlanma Tarihi 24 Nisan 2025
Gönderilme Tarihi 12 Mayıs 2024
Kabul Tarihi 22 Kasım 2024
Yayımlandığı Sayı Yıl 2025 Cilt: 18 Sayı: 1

Kaynak Göster

APA Turgay, N. C., Sağdiç, M., & Kılıç, E. (2025). Simple Rotational Surfaces in Euclidean 4-Space with Generalized 1-Type Gauss Map. International Electronic Journal of Geometry, 18(1), 48-59.
AMA Turgay NC, Sağdiç M, Kılıç E. Simple Rotational Surfaces in Euclidean 4-Space with Generalized 1-Type Gauss Map. Int. Electron. J. Geom. Nisan 2025;18(1):48-59.
Chicago Turgay, Nurettin Cenk, Mustafa Sağdiç, ve Erol Kılıç. “Simple Rotational Surfaces in Euclidean 4-Space With Generalized 1-Type Gauss Map”. International Electronic Journal of Geometry 18, sy. 1 (Nisan 2025): 48-59.
EndNote Turgay NC, Sağdiç M, Kılıç E (01 Nisan 2025) Simple Rotational Surfaces in Euclidean 4-Space with Generalized 1-Type Gauss Map. International Electronic Journal of Geometry 18 1 48–59.
IEEE N. C. Turgay, M. Sağdiç, ve E. Kılıç, “Simple Rotational Surfaces in Euclidean 4-Space with Generalized 1-Type Gauss Map”, Int. Electron. J. Geom., c. 18, sy. 1, ss. 48–59, 2025.
ISNAD Turgay, Nurettin Cenk vd. “Simple Rotational Surfaces in Euclidean 4-Space With Generalized 1-Type Gauss Map”. International Electronic Journal of Geometry 18/1 (Nisan 2025), 48-59.
JAMA Turgay NC, Sağdiç M, Kılıç E. Simple Rotational Surfaces in Euclidean 4-Space with Generalized 1-Type Gauss Map. Int. Electron. J. Geom. 2025;18:48–59.
MLA Turgay, Nurettin Cenk vd. “Simple Rotational Surfaces in Euclidean 4-Space With Generalized 1-Type Gauss Map”. International Electronic Journal of Geometry, c. 18, sy. 1, 2025, ss. 48-59.
Vancouver Turgay NC, Sağdiç M, Kılıç E. Simple Rotational Surfaces in Euclidean 4-Space with Generalized 1-Type Gauss Map. Int. Electron. J. Geom. 2025;18(1):48-59.
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