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Year 2021, Volume: 13 Issue: 2, 270 - 281, 31.12.2021

Ayla Erdur Kara Muhittin Evren Aydın Mahmut Ergüt

https://doi.org/10.47000/tjmcs.882976

Abstract

References

  • [1] Agashe, N. S., Chafle, M.R., A semi-symmetric non-metric connection on a Riemannian manifold, Indian J. Pure Appl. Math., 23(6)(1992), 399–409.
  • [2] Agashe, N. S., Chafle, M.R., On submanifolds of a Riemannian manifold with a semi-symmetric non-metric connection, Tensor, 55(2)(1994), 120–130.
  • [3] Akyol, M. A., Beyendi, S., Riemannian submersion endowed with a semi-symmetric non-metric connection, Konuralp J. Math. 6(1)(2018), 188–193.
  • [4] Aydin, M.E., Erdur, A., Ergut, M., Singular minimal translation graphs in Euclidean spaces, J. Korean Math. Soc., 58(1)(2021), 109–122.
  • [5] Aydin, M.E., Mihai, A., Translation hypersurfaces and Tzitzeica Translation hyper-surfaces of the Euclidean space, Proc. Rom. Acad. Ser. A, Math. Phys. Tech. Sci. Inf. Sci., 16(4)(2015), 477–483.
  • [6] Böhme, R., Hildebrant, S., Taush, E., The two-dimensional analogue of the catenary, Pac. J. Math., 88(2)(1980), 247–278.
  • [7] Chaubey, S.K., Yildiz, A., Riemannian manifolds admitting a new type of semisymmetric nonmetric connection, Turk. J. Math., 43(4)(2019), 1887–1904.
  • [8] Darboux, J.G., Theorie Generale des Surfaces, Livre I, Gauthier-Villars, Paris, 1914.
  • [9] De, U.C., Barman, A., On a type of semisymmetric metric connection on a Riemannian manifold, Publ. Inst. Math., Nouv. Ser. 98(112)(2015), 211–218.
  • [10] Dierkes, U., A Bernstein result for enery minimizing hypersurfaces, Cal. Var. Part. Differ. Equ., 1(1)(1993), 37-54.
  • [11] Dierkes, U., Singular minimal surfaces, Geometric Analysis and Nonlinear Partial Differential Equations, Springer, Berlin, Heidelberg (2003), 176-193.
  • [12] Dillen, F., Van de Woestyne, I., Verstraelen, L., Walrave, J.T., The surface of Scherk in E3: A special case in the class of minimal surfaces defined as the sum of two curves, Bull. Inst. Math. Acad. Sin. (N.S.) 26(4)(1998), 257–267.
  • [13] Dillen, F., Verstraelen, L., Zafindratafa, G., A generalization of the translation surfaces of Scherk, Diff. Geom. in honor of Radu Rosca (KUL) (1991), 107–109.
  • [14] Dillen, F., Goemans, W., Van de Woestyne, I., Translation surfaces of Weingarten type in 3-space, Bull. Transilv. Univ. Bras¸ov, Ser. III 1(50)(2008), 109–122.
  • [15] Dogru, Y., On some properties of submanifolds of a Riemannian manifold endowed with a semi-symmetric non-metric connection, An. Şt. Univ. Ovidius Constanta, 19(3)(2011), 85–100.
  • [16] Gil, J.B., The catenary (almost) everywhere, Boletin de la AMV XII(2)(2005), 251–258.
  • [17] Goemans, W., Van de Woestyne, I., Translation and homothetical lightlike hypersurfaces of semi-Euclidean space, Kuwait J. Sci. Eng. 38(2)(2011), 35–42.
  • [18] Gozutok, A., Esin, E., Tangent bundle of hypersurface with semi symmetric metric connection, Int. J. Contemp. Math. Sci. 7(6)(2012), 279–289.
  • [19] Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC Press, 2nd Ed., 1998.
  • [20] Hasanis, T., Translation surfaces with non-zero constant mean curvature in Euclidean space, J. Geom. 110(20)(2019).
  • [21] Hasanis, T., Lopez, R., Translation surfaces in Euclidean space with constant Gaussian curvature, Arxiv 8 Sept 2018: https://arxiv.org/abs/1809.02758v1.
  • [22] Hasanis, T., Lopez, R., Classification and construction of minimal translation surfaces in Euclidean space, Result Math. 75(1)(2020), 1–22.
  • [23] Hayden, H. A., Subspaces of a space with torsion, Proc. London Math. Soc. S2-34(1)(1932), 27–50.
  • [24] Imai, T., Notes on semi-symmetric metric connections, Tensor, 24(1972), 293–296.
  • [25] Inoguchi, J., Lopez, R., Munteanu, M.I., Minimal translation surfaces in the Heisenberg group Nil3, Geom. Dedicata, 161(2012), 221–231.
  • [26] Jung, S.D., Liu, H., Liu, Y., Weingarten affine translation surfaces in Euclidean 3-space, Results Math. 72(4)(2017),1839–1848.
  • [27] Lima, B.P., Santos, N. L., Sousa, P.A., Generalized translation hypersurfaces in Euclidean space, J. Math. Anal. Appl. 470(2)(2019), 1129–1135.
  • [28] Liu, H., Translation surfaces with constant mean curvature in 3- dimensional spaces, J. Geom. 64(1999), 141–149.
  • [29] Liu, H., Jung, S.D., Affine translation surfaces with constant mean curvature in Euclidean 3-space, J. Geom. 108(2017), 423–428.
  • [30] Liu, H., Yu, Y., Affine translation surfaces in Euclidean 3-space, Proc. Japan Acad. 89(A)(2013), 111–113.
  • [31] Lopez, R., Minimal translation surfaces in hyperbolic space, Beitr. Algebra Geom. 52(1)(2011), 105–112.
  • [32] Lopez, R., Munteanu, M.I., Minimal translation surfaces in $Sol_3$, J. Math. Soc. Japan 64(3)(2012), 985–1003.
  • [33] Lopez, R., Moruz, M., Translation and homothetical surfaces in Euclidean space with constant curvature, J. Korean Math. Soc. 52(3)(2015), 523–535.
  • [34] Lopez, R., Constant Mean Curvature Surfaces with Boundary, Springer–Verlag, Berlin, 2013.
  • [35] Lopez, R., Seperation of variables in equations of mean curvature type , Poc. R. Soc. Edinb. 146(5)(2016), 1017–1035.
  • [36] Lopez, R., Perdomo, O., Minimal translation surfaces in Euclidean space, J. Geom. Anal. 27(4)(2017), 2926–2937.
  • [37] Lopez, R., Invariant singular minimal surfaces, Ann. Glob. Anal. Geom. 53(4)(2018), 521–541.
  • [38] Lopez, R., The Dirichlet problem for the $\alpha$–singular minimal surface equation, Arch. Math. 112(2)(2019), 213–222.
  • [39] Lopez, R., The two–dimensional analogue of the Lorentzian catenary and the Dirichlet problem, Pacific J. Math. 305(2)(2020), 693–719.
  • [40] Moruz, M., Munteanu, M.I., Minimal translation hypersurfaces in E4, J. Math. Anal. Appl. 439(2)(2016), 798–812.
  • [41] Munteanu, M.I., Palmas, O., Ruiz–Hernandez, G., Minimal translation hypersurfaces in Euclidean spaces, Mediterran. J. Math. 13(5)(2016), 2659–2676.
  • [42] Murathan, C., Ozgür, C., Riemannian manifolds with a semi–symmetric metric connection satisfying some semisymmetry conditions, Proc. Est. Acad. Sci. 57(4)(2008), 210–216.
  • [43] Nakao, Z., Submanifolds of a Riemannian manifold with semisymmetric metric connections, Proc. Amer. Math. Soc. 54(1)(1976), 261–266.
  • [44] Ozen Zengin, F., Altay Demirbag, S., Uysal, S. A., Some vector fields on a Riemannian manifold with semi-symmetric metric connection, Bull. Iranian Math. Soc. 38(2)(2012), 479–490.
  • [45] Ozgur, C., On submanifolds of a Riemannian manifold with a semi-symmetric non-metric connection, Kuwait J. Sci. Eng. 37(2)(2010), 17–30.
  • [46] Panayotounakos, D.E., Zarmpoutis, T.I., Construction of exact parametric or closed form solutions of some unsolvable classes of nonlinear ODEs (Abel’s nonlinear ODEs of the first kind and relative degenerate equations), Int. J. Math. Math. Sci. (2011), p. 387429.
  • [47] Polyanin, A.D., Zaitsev, V.F., Handbook of Exact Solutions for Ordinary Dfferential Equations, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2nd Ed., 2003.
  • [48] Scherk, H.F., Bemerkungen über die kleinste Flache innerhalb gegebener Grenzen, J. Reine Angew. Math. 13(1835), 185–208.
  • [49] Sekerci, G.A., Sevinc, S., Coken, A.C., On the translation hypersurfaces with Gauss map G satisfying $\triangle G = AG$, Miscolc Math. Notes, 20(2)(2019), 1215–1225.
  • [50] Seo, K., Translation hypersurfaces with constant curvature in space form, Osaka J. Math. 50(3)(2013), 631–641.
  • [51] Sun, H., On affine translation surfaces of constant mean curvature, Kumamoto J. Math. 13(2000), 49–57.
  • [52] Unal, I., On submanifolds of N(k)–quasi Einstein manifolds with a type of semi–symmetric metric connection, Univers. J. Math. Appl. 3(4)(2020), 167–172.
  • [53] Verstraelen, L., Walrave, J., Yaprak, S., The minimal translation surfaces in Euclidean space, Soochow J. Math. 20(1)(1994), 77–82.
  • [54] Wang, Y., Minimal translation surfaces with respect to semi–symmetric connections in $\mathbb{R}^{3}$ and $\mathbb{R}_{1}^{3}$, Bull. Korean Math. Soc. 58(4)(2021), 959–972.
  • [55] Yang, D., Fu, Y.,On affine translation surfaces in affine space, J. Math. Anal. Appl. 440(2)(2016), 437–450.
  • [56] Yang, D., Zhang, J., Fu, Y., A note on minimal translation graphs in Euclidean space, Mathematics 7(10)(2019).
  • [57] Yano, K., On semi–symmetric metric connection, Rev. Roumaine Math. Pures Appl. 15(1970), 1579–1586.
  • [58] Yano, K., Kon, M., Structures on Manifolds, Series in Pure Math., World Scientific, 1984.
  • [59] Yoon, D.W., On the Gauss map of translation surfaces in Minkowski 3-space, Taiwanese J. Math. 6(2002), 389–398.
  • [60] Yoon, D.W., Lee, C.W., Karacan, M.K., Some translation surfaces in the 3-dimensional Heisenberg group, Bull. Korean Math. Soc. 50(4)(2013), 1329–1343.
  • [61] Yucesan, A., Yasar, E., Non–degenerate hypersurfaces of a semi–Riemannian manifold with a semi–symmetric non–metric connection, Math. Rep. (Bucur) 14(64) (2012), no. 2, 209–219.

Singular Minimal Translation Surfaces in Euclidean Spaces Endowed with Semi-symmetric Connections

Year 2021, Volume: 13 Issue: 2, 270 - 281, 31.12.2021

Ayla Erdur Kara Muhittin Evren Aydın Mahmut Ergüt

https://doi.org/10.47000/tjmcs.882976

Abstract

In this paper, we study and classify singular minimal translation surfaces in a Euclidean space of dimension $3$ endowed with a certain semi-symmetric (non-)metric connection.

Keywords

Singular minimal surface translation surface $\alpha -$ catenary semi-symmetric connection

References

  • [1] Agashe, N. S., Chafle, M.R., A semi-symmetric non-metric connection on a Riemannian manifold, Indian J. Pure Appl. Math., 23(6)(1992), 399–409.
  • [2] Agashe, N. S., Chafle, M.R., On submanifolds of a Riemannian manifold with a semi-symmetric non-metric connection, Tensor, 55(2)(1994), 120–130.
  • [3] Akyol, M. A., Beyendi, S., Riemannian submersion endowed with a semi-symmetric non-metric connection, Konuralp J. Math. 6(1)(2018), 188–193.
  • [4] Aydin, M.E., Erdur, A., Ergut, M., Singular minimal translation graphs in Euclidean spaces, J. Korean Math. Soc., 58(1)(2021), 109–122.
  • [5] Aydin, M.E., Mihai, A., Translation hypersurfaces and Tzitzeica Translation hyper-surfaces of the Euclidean space, Proc. Rom. Acad. Ser. A, Math. Phys. Tech. Sci. Inf. Sci., 16(4)(2015), 477–483.
  • [6] Böhme, R., Hildebrant, S., Taush, E., The two-dimensional analogue of the catenary, Pac. J. Math., 88(2)(1980), 247–278.
  • [7] Chaubey, S.K., Yildiz, A., Riemannian manifolds admitting a new type of semisymmetric nonmetric connection, Turk. J. Math., 43(4)(2019), 1887–1904.
  • [8] Darboux, J.G., Theorie Generale des Surfaces, Livre I, Gauthier-Villars, Paris, 1914.
  • [9] De, U.C., Barman, A., On a type of semisymmetric metric connection on a Riemannian manifold, Publ. Inst. Math., Nouv. Ser. 98(112)(2015), 211–218.
  • [10] Dierkes, U., A Bernstein result for enery minimizing hypersurfaces, Cal. Var. Part. Differ. Equ., 1(1)(1993), 37-54.
  • [11] Dierkes, U., Singular minimal surfaces, Geometric Analysis and Nonlinear Partial Differential Equations, Springer, Berlin, Heidelberg (2003), 176-193.
  • [12] Dillen, F., Van de Woestyne, I., Verstraelen, L., Walrave, J.T., The surface of Scherk in E3: A special case in the class of minimal surfaces defined as the sum of two curves, Bull. Inst. Math. Acad. Sin. (N.S.) 26(4)(1998), 257–267.
  • [13] Dillen, F., Verstraelen, L., Zafindratafa, G., A generalization of the translation surfaces of Scherk, Diff. Geom. in honor of Radu Rosca (KUL) (1991), 107–109.
  • [14] Dillen, F., Goemans, W., Van de Woestyne, I., Translation surfaces of Weingarten type in 3-space, Bull. Transilv. Univ. Bras¸ov, Ser. III 1(50)(2008), 109–122.
  • [15] Dogru, Y., On some properties of submanifolds of a Riemannian manifold endowed with a semi-symmetric non-metric connection, An. Şt. Univ. Ovidius Constanta, 19(3)(2011), 85–100.
  • [16] Gil, J.B., The catenary (almost) everywhere, Boletin de la AMV XII(2)(2005), 251–258.
  • [17] Goemans, W., Van de Woestyne, I., Translation and homothetical lightlike hypersurfaces of semi-Euclidean space, Kuwait J. Sci. Eng. 38(2)(2011), 35–42.
  • [18] Gozutok, A., Esin, E., Tangent bundle of hypersurface with semi symmetric metric connection, Int. J. Contemp. Math. Sci. 7(6)(2012), 279–289.
  • [19] Gray, A., Modern Differential Geometry of Curves and Surfaces with Mathematica, CRC Press, 2nd Ed., 1998.
  • [20] Hasanis, T., Translation surfaces with non-zero constant mean curvature in Euclidean space, J. Geom. 110(20)(2019).
  • [21] Hasanis, T., Lopez, R., Translation surfaces in Euclidean space with constant Gaussian curvature, Arxiv 8 Sept 2018: https://arxiv.org/abs/1809.02758v1.
  • [22] Hasanis, T., Lopez, R., Classification and construction of minimal translation surfaces in Euclidean space, Result Math. 75(1)(2020), 1–22.
  • [23] Hayden, H. A., Subspaces of a space with torsion, Proc. London Math. Soc. S2-34(1)(1932), 27–50.
  • [24] Imai, T., Notes on semi-symmetric metric connections, Tensor, 24(1972), 293–296.
  • [25] Inoguchi, J., Lopez, R., Munteanu, M.I., Minimal translation surfaces in the Heisenberg group Nil3, Geom. Dedicata, 161(2012), 221–231.
  • [26] Jung, S.D., Liu, H., Liu, Y., Weingarten affine translation surfaces in Euclidean 3-space, Results Math. 72(4)(2017),1839–1848.
  • [27] Lima, B.P., Santos, N. L., Sousa, P.A., Generalized translation hypersurfaces in Euclidean space, J. Math. Anal. Appl. 470(2)(2019), 1129–1135.
  • [28] Liu, H., Translation surfaces with constant mean curvature in 3- dimensional spaces, J. Geom. 64(1999), 141–149.
  • [29] Liu, H., Jung, S.D., Affine translation surfaces with constant mean curvature in Euclidean 3-space, J. Geom. 108(2017), 423–428.
  • [30] Liu, H., Yu, Y., Affine translation surfaces in Euclidean 3-space, Proc. Japan Acad. 89(A)(2013), 111–113.
  • [31] Lopez, R., Minimal translation surfaces in hyperbolic space, Beitr. Algebra Geom. 52(1)(2011), 105–112.
  • [32] Lopez, R., Munteanu, M.I., Minimal translation surfaces in $Sol_3$, J. Math. Soc. Japan 64(3)(2012), 985–1003.
  • [33] Lopez, R., Moruz, M., Translation and homothetical surfaces in Euclidean space with constant curvature, J. Korean Math. Soc. 52(3)(2015), 523–535.
  • [34] Lopez, R., Constant Mean Curvature Surfaces with Boundary, Springer–Verlag, Berlin, 2013.
  • [35] Lopez, R., Seperation of variables in equations of mean curvature type , Poc. R. Soc. Edinb. 146(5)(2016), 1017–1035.
  • [36] Lopez, R., Perdomo, O., Minimal translation surfaces in Euclidean space, J. Geom. Anal. 27(4)(2017), 2926–2937.
  • [37] Lopez, R., Invariant singular minimal surfaces, Ann. Glob. Anal. Geom. 53(4)(2018), 521–541.
  • [38] Lopez, R., The Dirichlet problem for the $\alpha$–singular minimal surface equation, Arch. Math. 112(2)(2019), 213–222.
  • [39] Lopez, R., The two–dimensional analogue of the Lorentzian catenary and the Dirichlet problem, Pacific J. Math. 305(2)(2020), 693–719.
  • [40] Moruz, M., Munteanu, M.I., Minimal translation hypersurfaces in E4, J. Math. Anal. Appl. 439(2)(2016), 798–812.
  • [41] Munteanu, M.I., Palmas, O., Ruiz–Hernandez, G., Minimal translation hypersurfaces in Euclidean spaces, Mediterran. J. Math. 13(5)(2016), 2659–2676.
  • [42] Murathan, C., Ozgür, C., Riemannian manifolds with a semi–symmetric metric connection satisfying some semisymmetry conditions, Proc. Est. Acad. Sci. 57(4)(2008), 210–216.
  • [43] Nakao, Z., Submanifolds of a Riemannian manifold with semisymmetric metric connections, Proc. Amer. Math. Soc. 54(1)(1976), 261–266.
  • [44] Ozen Zengin, F., Altay Demirbag, S., Uysal, S. A., Some vector fields on a Riemannian manifold with semi-symmetric metric connection, Bull. Iranian Math. Soc. 38(2)(2012), 479–490.
  • [45] Ozgur, C., On submanifolds of a Riemannian manifold with a semi-symmetric non-metric connection, Kuwait J. Sci. Eng. 37(2)(2010), 17–30.
  • [46] Panayotounakos, D.E., Zarmpoutis, T.I., Construction of exact parametric or closed form solutions of some unsolvable classes of nonlinear ODEs (Abel’s nonlinear ODEs of the first kind and relative degenerate equations), Int. J. Math. Math. Sci. (2011), p. 387429.
  • [47] Polyanin, A.D., Zaitsev, V.F., Handbook of Exact Solutions for Ordinary Dfferential Equations, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2nd Ed., 2003.
  • [48] Scherk, H.F., Bemerkungen über die kleinste Flache innerhalb gegebener Grenzen, J. Reine Angew. Math. 13(1835), 185–208.
  • [49] Sekerci, G.A., Sevinc, S., Coken, A.C., On the translation hypersurfaces with Gauss map G satisfying $\triangle G = AG$, Miscolc Math. Notes, 20(2)(2019), 1215–1225.
  • [50] Seo, K., Translation hypersurfaces with constant curvature in space form, Osaka J. Math. 50(3)(2013), 631–641.
  • [51] Sun, H., On affine translation surfaces of constant mean curvature, Kumamoto J. Math. 13(2000), 49–57.
  • [52] Unal, I., On submanifolds of N(k)–quasi Einstein manifolds with a type of semi–symmetric metric connection, Univers. J. Math. Appl. 3(4)(2020), 167–172.
  • [53] Verstraelen, L., Walrave, J., Yaprak, S., The minimal translation surfaces in Euclidean space, Soochow J. Math. 20(1)(1994), 77–82.
  • [54] Wang, Y., Minimal translation surfaces with respect to semi–symmetric connections in $\mathbb{R}^{3}$ and $\mathbb{R}_{1}^{3}$, Bull. Korean Math. Soc. 58(4)(2021), 959–972.
  • [55] Yang, D., Fu, Y.,On affine translation surfaces in affine space, J. Math. Anal. Appl. 440(2)(2016), 437–450.
  • [56] Yang, D., Zhang, J., Fu, Y., A note on minimal translation graphs in Euclidean space, Mathematics 7(10)(2019).
  • [57] Yano, K., On semi–symmetric metric connection, Rev. Roumaine Math. Pures Appl. 15(1970), 1579–1586.
  • [58] Yano, K., Kon, M., Structures on Manifolds, Series in Pure Math., World Scientific, 1984.
  • [59] Yoon, D.W., On the Gauss map of translation surfaces in Minkowski 3-space, Taiwanese J. Math. 6(2002), 389–398.
  • [60] Yoon, D.W., Lee, C.W., Karacan, M.K., Some translation surfaces in the 3-dimensional Heisenberg group, Bull. Korean Math. Soc. 50(4)(2013), 1329–1343.
  • [61] Yucesan, A., Yasar, E., Non–degenerate hypersurfaces of a semi–Riemannian manifold with a semi–symmetric non–metric connection, Math. Rep. (Bucur) 14(64) (2012), no. 2, 209–219.
There are 61 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Ayla Erdur Kara 0000-0001-8328-4720

Muhittin Evren Aydın 0000-0001-9337-8165

Mahmut Ergüt 0000-0002-9098-8280

Publication Date December 31, 2021
Published in Issue Year 2021 Volume: 13 Issue: 2

Cite

APA Erdur Kara, A., Aydın, M. E., & Ergüt, M. (2021). Singular Minimal Translation Surfaces in Euclidean Spaces Endowed with Semi-symmetric Connections. Turkish Journal of Mathematics and Computer Science, 13(2), 270-281. https://doi.org/10.47000/tjmcs.882976
AMA Erdur Kara A, Aydın ME, Ergüt M. Singular Minimal Translation Surfaces in Euclidean Spaces Endowed with Semi-symmetric Connections. TJMCS. December 2021;13(2):270-281. doi:10.47000/tjmcs.882976
Chicago Erdur Kara, Ayla, Muhittin Evren Aydın, and Mahmut Ergüt. “Singular Minimal Translation Surfaces in Euclidean Spaces Endowed With Semi-Symmetric Connections”. Turkish Journal of Mathematics and Computer Science 13, no. 2 (December 2021): 270-81. https://doi.org/10.47000/tjmcs.882976.
EndNote Erdur Kara A, Aydın ME, Ergüt M (December 1, 2021) Singular Minimal Translation Surfaces in Euclidean Spaces Endowed with Semi-symmetric Connections. Turkish Journal of Mathematics and Computer Science 13 2 270–281.
IEEE A. Erdur Kara, M. E. Aydın, and M. Ergüt, “Singular Minimal Translation Surfaces in Euclidean Spaces Endowed with Semi-symmetric Connections”, TJMCS, vol. 13, no. 2, pp. 270–281, 2021, doi: 10.47000/tjmcs.882976.
ISNAD Erdur Kara, Ayla et al. “Singular Minimal Translation Surfaces in Euclidean Spaces Endowed With Semi-Symmetric Connections”. Turkish Journal of Mathematics and Computer Science 13/2 (December 2021), 270-281. https://doi.org/10.47000/tjmcs.882976.
JAMA Erdur Kara A, Aydın ME, Ergüt M. Singular Minimal Translation Surfaces in Euclidean Spaces Endowed with Semi-symmetric Connections. TJMCS. 2021;13:270–281.
MLA Erdur Kara, Ayla et al. “Singular Minimal Translation Surfaces in Euclidean Spaces Endowed With Semi-Symmetric Connections”. Turkish Journal of Mathematics and Computer Science, vol. 13, no. 2, 2021, pp. 270-81, doi:10.47000/tjmcs.882976.
Vancouver Erdur Kara A, Aydın ME, Ergüt M. Singular Minimal Translation Surfaces in Euclidean Spaces Endowed with Semi-symmetric Connections. TJMCS. 2021;13(2):270-81.