An odd-dimensional sphere admits a killing vector field, induced by the transform of the unit normal by the complex structure of the ambiant Euclidean space. In this paper, we studied orientable hypersurfaces in a Euclidean space that admits a unit Killing vector field and finds two characterizations of odd-dimensional spheres. In the first result, we showed that a complete and simply connected hypersurface of Euclidean space $ \mathbb{R}^{n+1} $, $ n > 1 $ admits a unit Killing vector field $ \xi $ that leaves the shape operator $ S $ invariant and has sectional curvatures of plane sections containing $ \xi $ positive which satisfies $ S(\xi) = \alpha \xi $, $ \alpha $ mean curvature if, and only if, $ n = 2m-1 $, $ \alpha $ is constant and the hypersurface is isometric to the sphere $ S^{2m-1}(\alpha^2) $. Similarly, we found another characterization of the unit sphere $ S^2(\alpha^2) $ using the smooth function $ \sigma = g(S(\xi), \xi) $ on the hypersurface.
Citation: Mohammed Guediri, Sharief Deshmukh. Hypersurfaces in a Euclidean space with a Killing vector field[J]. AIMS Mathematics, 2024, 9(1): 1899-1910. doi: 10.3934/math.2024093
An odd-dimensional sphere admits a killing vector field, induced by the transform of the unit normal by the complex structure of the ambiant Euclidean space. In this paper, we studied orientable hypersurfaces in a Euclidean space that admits a unit Killing vector field and finds two characterizations of odd-dimensional spheres. In the first result, we showed that a complete and simply connected hypersurface of Euclidean space $ \mathbb{R}^{n+1} $, $ n > 1 $ admits a unit Killing vector field $ \xi $ that leaves the shape operator $ S $ invariant and has sectional curvatures of plane sections containing $ \xi $ positive which satisfies $ S(\xi) = \alpha \xi $, $ \alpha $ mean curvature if, and only if, $ n = 2m-1 $, $ \alpha $ is constant and the hypersurface is isometric to the sphere $ S^{2m-1}(\alpha^2) $. Similarly, we found another characterization of the unit sphere $ S^2(\alpha^2) $ using the smooth function $ \sigma = g(S(\xi), \xi) $ on the hypersurface.
[1] | H. Alodan, S. Deshmukh, A characterization of spheres in a Euclidean space, New Zealand Journal of Mathematics, 36 (2007), 93–99. |
[2] | V. N. Berestovskii, Y. G. Nikonorov, Killing vector fields of constant length on Riemannian manifolds, Siberian Math. J., 49 (2008), 395–407. https://doi.org/10.1007/s11202-008-0039-3 doi: 10.1007/s11202-008-0039-3 |
[3] | M. Berger, Trois remarques sur les vairétés Riemanniennes à courbure positive, C. R. Acad. Sci. Paris Ser. A-B, 263 (1966), 76–78. |
[4] | A. L. Besse, Einstein manifolds, Heidelberg: Springer Berlin, 1987. https://doi.org/10.1007/978-3-540-74311-8 |
[5] | M. P. do Carmo, Riemannian geometry, Boston: Brikhäuser, 1992. |
[6] | S. Deshmukh, Compact hypersurfaces in a Euclidean space, Q. J. Math., 49 (1998), 35–41. https://doi.org/10.1093/qmathj/49.1.35 doi: 10.1093/qmathj/49.1.35 |
[7] | S. Deshmukh, A note on Euclidean spheres, Balk. J. Geom. Appl., 11 (2006), 44–49. |
[8] | S. Deshmukh, Real hypersurfaces in a Euclidean complex space form, Q. J. Math., 58 (2007), 313–317. https://doi.org/10.1093/qmath/ham015 doi: 10.1093/qmath/ham015 |
[9] | S. Deshmukh, Characterizations of Einstein manifolds and odd-dimensional spheres, J. Geom. Phys., 61 (2011), 2058–2063. https://doi.org/10.1016/j.geomphys.2011.06.009 doi: 10.1016/j.geomphys.2011.06.009 |
[10] | S. Deshmukh, O. Belova, On Killing vector fields on Riemannian manifolds, Mathematics, 9 (2021), 259. https://doi.org/10.3390/math9030259 doi: 10.3390/math9030259 |
[11] | S. Deshmukh, M. Guediri, Characterization of Euclidean spheres, AIMS Mathematics, 6 (2021), 7733–7740. https://doi.org/10.3934/math.2021449 doi: 10.3934/math.2021449 |
[12] | W. C. Lynge, Sufficient conditions for periodicity of a killing vector field, P. Am. Math. Soc., 38 (1973), 614–616. |
[13] | X. Rong, Positive curvature, local and global symmetry, and fundamental groups, Am. J. Math., 121 (1999), 931–943. https://doi.org/10.1353/ajm.1999.0036 doi: 10.1353/ajm.1999.0036 |
[14] | S. Yorozu, Killing vector fields on complete Riemannian manifolds, P. Am. Math. Soc., 84 (1982), 115–120. |