Some classifications for Gauss map of Tubular hypersurfaces in $\mathbb{E}^{4}_{1}$ concerning linearized operators $\mathcal{L}_{k}$

Year 2025, Volume: 54 Issue: 4, 1329 - 1344, 29.08.2025

Ahmet Kazan , Mustafa Altın , Nurettin Cenk Turgay

https://doi.org/10.15672/hujms.1468581

Cited By: 2

https://izlik.org/JA89AH96CM

Abstract

In this study, we deal with the Gauss map of tubular hypersurfaces in 4-dimensional Lorentz-Minkowski space concerning the linearized operators $\mathcal{L}_{1}$ (Cheng-Yau) and $\mathcal{L}_{2}$. We obtain the $\mathcal{L}_{1}$ (Cheng-Yau) operator of the Gauss map of tubular hypersurfaces that are formed as the envelope of a family of pseudo hyperspheres{ or pseudo hyperbolic hyperspheres} whose centers lie on timelike or spacelike curves with non-null Frenet vectors in $\mathbb{E}^{4}_{1}$ and give some classifications for these hypersurfaces which have generalized $\mathcal{L}_{k}$ 1-type Gauss map, first kind $\mathcal{L}_{k}$-pointwise 1-type Gauss map, second kind $\mathcal{L}_{k}$-pointwise 1-type Gauss map and $\mathcal{L}_{k}$-harmonic Gauss map, $k\in\{1,2\}$.

Keywords

Tubular hypersurface , $\mathcal{L}_{1}$ (Cheng-Yau) operator , $\mathcal{L}_{2}$ operator , generalized $\mathcal{L}_{k}$ 1-type Gauss map , first and second kind $\mathcal{L}_{k}$-pointwise 1-type Gauss map , $\mathcal{L}_{k}$-harmonic Gauss map

References

• [1] L. Alias, A. Ferrández and P. Lucas, Surfaces in the 3-dimensional Lorentz-Minkowski space satisfying $\Delta x=Ax+B$, Pacific J. Math. 156 (2), 201-208, 1992.
• [2] L.J. Alias and N. Gürbüz, An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures, Geom. Dedicata 121 (1), 113-127, 2006.
• [3] M. Altın, A. Kazan and D.W. Yoon, Canal Hypersurfaces generated by Non-null Curves in Lorentz-Minkowski 4-Space, Bull. Korean Math. Soc. 60 (5), 1299-1320, 2023.
• [4] B.-Y. Chen, Total Mean Curvature and Submanifold of Finite Type, World Scientific Publisher, 1984.
• [5] B.-Y. Chen, A report on submanifolds of finite type, Soochow J. Math. 22 (2), 117- 337, 1996.
• [6] B.-Y. Chen, J.-M. Morvan and T. Nore, Energy, tension and finite type maps, Kodai Math. J. 9 (3), 406-418, 1986.
• [7] B.-Y. Chen and M. Petrovic, On spectral decomposition of immersions of finite type, Bull. Aust. Math. Soc. 44 (1), 117-129, 1991.
• [8] B.-Y. Chen and P. Piccinni, Submanifolds with Finite Type Gauss Map, Bull. Austral. Math. Soc. 35 (2), 161-186, 1987.
• [9] S.-Y. Cheng and S.-T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225, 195-204, 1977.
• [10] M.K. Choi and Y.H. Kim, Characterization of the helicoid as ruled surface with pointwise 1-type Gauss map, Bull. Korean Math. Soc. 38 (4), 753-761, 2001.
• [11] A. Kazan, M. Altın and N.C. Turgay, Rotational hypersurfaces in $\mathbb{E}^{4}_{1}$ with Generalized $L_{k}$ 1-Type Gauss Map, arXiv:2403.19671v1, 2024.
• [12] A. Kelleci, Rotational surfaces with Cheng-Yau operator in Galilean 3-spaces, Hacet. J. Math. Stat. 50 (2), 365-376, 2021.
• [13] D-S. Kim, J.R. Kim and Y.H. Kim, Cheng-Yau Operator and Gauss Map of Surfaces of Revolution, Bull. Malays. Math. Sci. Soc. 39 (4), 1319-1327, 2016.
• [14] Y.H. Kim and N.C. Turgay, On the surfaces in $\mathbb{E}^{3}$ with $L_{1}$ pointwise 1-type Gauss map, (submitted).
• [15] W. Kuhnel, Differential geometry: curves-surfaces-manifolds, American Mathematical Soc., Braunschweig, Wiesbaden, 1999.
• [16] P. Lucas and H.F. Ramírez-Ospina, Hypersurfaces in the Lorentz-Minkowski space satisfying $L_{k}\psi=A\psi+b$, Geom. Dedicata 153 (1), 151-175, 2011.
• [17] J. Qian, X. Fu and S.D. Jung, Dual associate null scrolls with generalized 1-type Gauss maps, Mathematics 8 (7), 1111, 2020.
• [18] J. Qian, X. Fu, X. Tian and Y.H. Kim, Surfaces of Revolution and Canal Surfaces with Generalized Cheng-Yau 1-Type Gauss Maps, Mathematics 8 (10), 1728, 2020.
• [19] J. Qian, M. Su, Y.H. Kim, Canal surfaces with generalized 1-type Gauss map, Rev. Union Mat. Argent. 62 (1), 199-211, 2021.
• [20] T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (4), 380-385, 1966.
• [21] J. Walrave, Curves and surfaces in Minkowski space, Dissertation, K. U. Leuven, Fac. of Science, Leuven, 1995.
• [22] B. Yang and X. Liu, Hypersurfaces satisfying $L_{r}x=Rx$ in sphere $S^{n+1}$ or hyperbolic space $H^{n+1}$, Proc. Indian Acad. Sci. (Math. Sci.) 119 (4), 487-499, 2009.
• [23] R. Yegin Sen and U. Dursun, On submanifolds with 2-type pseudo-hyperbolic Gauss map in pseudo-hyperbolic space, Mediterr. J. Math. 14 (1), 1-20, 2017.
• [24] D.W. Yoon, Rotation surfaces with finite type Gauss Map in $E^{4}$, Indian J. Pure. Appl. Math. 32 (12), 1803-1808, 2001.
• [25] D.W. Yoon, D.S. Kim, Y.H. Kim and J.W. Lee, Hypersurfaces with generalized 1-type Gauss maps, Mathematics 6 (8), 130, 2018.
• [26] D.W. Yoon, D.S. Kim, Y.H. Kim and J.W. Lee, Classifications of flat surfaces with generalized 1-type Gauss map in $\mathbb{L}^{3}$, Mediterr. J. Math. 15, 78, 2018.