Given an immersed hypersurface $ M^{n} $ in the Euclidean space $ E^{n+1} $, the tangential component $\boldsymbol{\omega }$ of the position vector field of the hypersurface is called the basic vector field, and the smooth function of the normal component of the position vector field gives a function $ \sigma $ on the hypersurface called the support function of the hypersurface. In the first result, we show that on a complete and simply connected hypersurface $ M^{n} $ in $ E^{n+1} $ of positive Ricci curvature with shape operator $ T $ invariant under $\boldsymbol{\omega }$ and the support function $ \sigma $ satisfies the static perfect fluid equation if and only if the hypersurface is isometric to a sphere. In the second result, we show that a compact hypersurface $ M^{n} $ in $ E^{n+1} $ with the gradient of support function $ \sigma $, an eigenvector of the shape operator $ T $ with eigenvalue function the mean curvature $ H $, and the integral of the squared length of the gradient $ \nabla \sigma $ has a certain lower bound, giving a characterization of a sphere. In the third result, we show that a compact and simply connected hypersurface $ M^{n} $ of positive Ricci curvature in $ E^{n+1} $ has an incompressible basic vector field $\boldsymbol{\omega }$, if and only if $ M^{n} $ is isometric to a sphere.
Citation: Yanlin Li, Nasser Bin Turki, Sharief Deshmukh, Olga Belova. Euclidean hypersurfaces isometric to spheres[J]. AIMS Mathematics, 2024, 9(10): 28306-28319. doi: 10.3934/math.20241373
Given an immersed hypersurface $ M^{n} $ in the Euclidean space $ E^{n+1} $, the tangential component $\boldsymbol{\omega }$ of the position vector field of the hypersurface is called the basic vector field, and the smooth function of the normal component of the position vector field gives a function $ \sigma $ on the hypersurface called the support function of the hypersurface. In the first result, we show that on a complete and simply connected hypersurface $ M^{n} $ in $ E^{n+1} $ of positive Ricci curvature with shape operator $ T $ invariant under $\boldsymbol{\omega }$ and the support function $ \sigma $ satisfies the static perfect fluid equation if and only if the hypersurface is isometric to a sphere. In the second result, we show that a compact hypersurface $ M^{n} $ in $ E^{n+1} $ with the gradient of support function $ \sigma $, an eigenvector of the shape operator $ T $ with eigenvalue function the mean curvature $ H $, and the integral of the squared length of the gradient $ \nabla \sigma $ has a certain lower bound, giving a characterization of a sphere. In the third result, we show that a compact and simply connected hypersurface $ M^{n} $ of positive Ricci curvature in $ E^{n+1} $ has an incompressible basic vector field $\boldsymbol{\omega }$, if and only if $ M^{n} $ is isometric to a sphere.
| [1] | M. Dajczer, D. Gromoll, Rigidity of complete Euclidean hypersurfaces, J. Differ. Geom., 31 (1990), 401–416. |
| [2] |
S. Deshmukh, Isometric immersion of a compact Riemannian manifold into a Euclidean space, Bull. Aust. Math. Soc., 46 (1992), 177–178. https://doi.org/10.1017/S0004972700011801 doi: 10.1017/S0004972700011801
|
| [3] | C. Tompkins, Isometric embedding of flat manifolds in Euclidean spaces, Duke Math. J., 5 (1939), 58–61. |
| [4] |
N. B. Turki, A note on incompressible vector fields, Symmetry, 15 (2023), 1479. https://doi.org/10.3390/sym15081479 doi: 10.3390/sym15081479
|
| [5] |
G. Wei, Complete hypersurfaces in a Euclidean space $R^{n+1}$ with constant mth mean curvature, Differ. Geom. Appl., 26 (2008), 298–306. https://doi.org/10.1080/10916460600805996 doi: 10.1080/10916460600805996
|
| [6] |
H. Jacobowicz, Isometric embedding of a compact Riemannian manifold into Euclidean space, P. Am. Math. Soc., 40 (1973), 245–246. https://doi.org/10.1090/S0002-9939-1973-0375173-3 doi: 10.1090/S0002-9939-1973-0375173-3
|
| [7] |
B. Y. Chen, M. I. Munteanu, Biharmonic ideal hypersurfaces in Euclidean spaces, Diff. Geom. Appl., 31 (2013), 1–16. https://doi.org/10.1016/j.difgeo.2012.10.008 doi: 10.1016/j.difgeo.2012.10.008
|
| [8] |
B. Y. Chen, Euclidean submanifolds with incompressible canonical vector field, Sib. Math. J., 43 (2017), 321–334. https://doi.org/10.48550/arXiv.1801.07196 doi: 10.48550/arXiv.1801.07196
|
| [9] |
B. Y. Chen, Some open problems and conjectures on submanifolds of finite type: Recent development, Tamkang J. Math., 45 (2014), 87–108. https://doi.org/10.48550/arXiv.1401.3793 doi: 10.48550/arXiv.1401.3793
|
| [10] | B. Y. Chen, Geometry of submanifolds, New York: Marcel Dekker, Inc, 1973. |
| [11] |
M. Aminian, S. M. B. Kashani, $L_{k}$-Biharmonic hypersurfaces in the Euclidean space, Taiwan. J. Math., 19 (2015), 861–874. https://doi.org/10.11650/tjm.19.2015.4830 doi: 10.11650/tjm.19.2015.4830
|
| [12] |
N. Hicks, Closed vector fields, Pac. J. Math., 15 (1965), 141–151. https://doi.org/10.2140/pjm.1965.15.141 doi: 10.2140/pjm.1965.15.141
|
| [13] | F. Defever, Hypersurfaces of E$^{4}$ satisfying $ \Delta\overrightarrow{H}$ $ = \lambda \overrightarrow{H}$, Mich. Math. J., 44 (1997), 355–363. |
| [14] | F. Defever, G. Kaimakamis, V. Papantoniou, Biharmonic hypersurfaces of the $4$-dimensional semi-Euclidean space E$_{s}^{4}$, J. Math. Anal. Appl., 315 (2006), 276–286. |
| [15] |
T. Cecil, Classifications of Dupin hypersurfaces in Lie sphere geometry, Acta Math. Sci., 44 (2024), 1–36. https://doi.org/10.1007/s10473-024-0101-7 doi: 10.1007/s10473-024-0101-7
|
| [16] | T. Cecil, P. Ryan, Geometry of hypersurfaces, New York, NY: Springer monographs in mathematics, 2015. |
| [17] |
T. Cecil, G. Jensen, Dupin hypersurfaces with three principal curvatures, Invent. Math., 132 (1998), 121–178. https://doi.org/10.1007/s002220050220 doi: 10.1007/s002220050220
|
| [18] | T. Cecil, G. Jensen, Dupin hypersurfaces with four principal curvatures, Geometriae Dedicata, 79 (2000), 1–49. |
| [19] |
T. Cecil, Using Lie sphere geometry to study Dupin Hypersurfaces in $R^n$, Axioms, 13 (2024), 399. https://doi.org/10.3390/axioms13060399 doi: 10.3390/axioms13060399
|
| [20] |
Y. L. Li, H. S. Abdel-Aziz, H. M. Serry, F. M. El-Adawy, M. K. Saad, Geometric visualization of evolved ruled surfaces via alternative frame in Lorentz-Minkowski 3-space, AIMS Math., 9 (2024), 25619–25635. https://doi.org/10.3934/math.20241251 doi: 10.3934/math.20241251
|
| [21] |
Y. Li, E. Güler, M. Toda, Family of right conoid hypersurfaces with light-like axis in Minkowski four-space, AIMS Math., 9 (2024), 18732–18745. https://doi.org/10.3934/math.2024911 doi: 10.3934/math.2024911
|
| [22] |
Y. Li, E. Güler, Right conoids demonstrating a time-like axis within Minkowski four-dimensional space, Mathematics, 12 (2024), 2421. https://doi.org/10.3390/math12152421 doi: 10.3390/math12152421
|
| [23] |
B. Y. Chen, E. Güler, Y. Yaylı, H. H. Hacısalihoğlu, Differential geometry of 1-type submanifolds and submanifolds with 1-type Gauss map, Int. Electron. J. Geom., 16 (2023), 4–47. https://doi.org/10.36890/iejg.1216024 doi: 10.36890/iejg.1216024
|
| [24] |
B. Y. Chen. Chen's biharmonic conjecture and submanifolds with parallel normalized mean curvature vector, Mathematics, 7 (2019), 710. https://doi.org/10.3390/math7080710 doi: 10.3390/math7080710
|
| [25] |
Y. Li, M. Aquib, M. Khan, I. Al-Dayel, K. Masood, Analyzing the Ricci tensor for slant submanifolds in locally metallic product space forms with a semi-symmetric metric connection, Axioms, 13 (2024), 454. https://doi.org/10.3390/axioms13070454 doi: 10.3390/axioms13070454
|
| [26] |
Y. Li, M. Aquib, M. Khan, I. Al-Dayel, M. Youssef, Geometric inequalities of slant submanifolds in locally metallic product space forms, Axioms, 13 (2024), 486. https://doi.org/10.3390/axioms13070486 doi: 10.3390/axioms13070486
|
| [27] |
Y. Li, A. Gezer, E. Karakas, Exploring conformal soliton structures in tangent bundles with Ricci-quarter symmetric metric connections, Mathematics, 12 (2024), 2101. https://doi.org/10.3390/math12132101 doi: 10.3390/math12132101
|
| [28] |
J. D. Moore, T. Schulte, Minimal disks and compact hypersurfaces in Euclindea space, P. Am. Math. Soc., 49 (1985), 321–328. https://doi.org/10.1016/S0002-9459(24)09937-6 doi: 10.1016/S0002-9459(24)09937-6
|
| [29] | M. Obata, The conjectures about conformal transformations, J. Differ. Geom., 6 (1971), 247–258. |
| [30] | J. Qing, W. Yuan, A note on static spaces and related problems, J. Geom. Phys., 74 (2013), 18–27. |
| [31] | K. Yano, Integral formulas in Riemannian geometry, Marcel Dekker, 1970. |
Yanlin Li, Nasser Bin Turki, Sharief Deshmukh, Olga Belova. Euclidean hypersurfaces isometric to spheres[J]. AIMS Mathematics, 2024, 9(10): 28306-28319. doi: 10.3934/math.20241373
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