@article{Guediri2024, 
author = {Mohammed Guediri and Sharief Deshmukh},
title = {Hypersurfaces in a Euclidean space with a Killing vector field},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {1},
pages = {1899-1910},
keywords = {Euclidean space, hypersurface, Killing vector field},
url = {https://www.sciopen.com/article/10.3934/math.2024093},
doi = {10.3934/math.2024093},
abstract = {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              R              n      +      1      ,    n  >  1 admits a unit Killing vector field    ξ that leaves the shape operator    S invariant and has sectional curvatures of plane sections containing    ξ positive which satisfies    S  (  ξ  )  =  α  ξ,    α mean curvature if, and only if,    n  =  2  m  −  1,    α is constant and the hypersurface is isometric to the sphere        S          2      m      −      1        (      α    2    ). Similarly, we found another characterization of the unit sphere        S    2    (      α    2    ) using the smooth function    σ  =  g  (  S  (  ξ  )  ,  ξ  ) on the hypersurface.}
}