@article{Alohali2024, 
author = {Hanan Alohali and Sharief Deshmukh},
title = {Some generic hypersurfaces in a Euclidean space},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {6},
pages = {15008-15023},
keywords = {hypersurfaces, Killing vector fields, concircular vector fields, n-sphere, Ricci curvature},
url = {https://www.sciopen.com/article/10.3934/math.2024727},
doi = {10.3934/math.2024727},
abstract = {In this paper, we find three nontrivial characterizations of Euclidean spheres. In the first result, we show that the existence of a nonzero nontrivial concircular vector field        ω   on a compact and connected hypersurface    N of the Euclidean space        R          m      +      1       with a mean curvature    α constant along the integral curves of        ω   and a shape operator    T satisfying    T  (      ω    )    =    α    ω   implies that    α is a constant and    N is isometric to a sphere, and the converse also holds. In the second result, we show that the presence of a unit Killing vector field        v   on a compact and connected hypersurface    N of a Euclidean space        R          m      +      1       gives a nonzero function    σ  =  g      (          T              v            ,              v              )   with shape operator    T, and the integral of the function    m  α  σ  R  i  c      (                  v            ,              v              )   has a certain lower bound, and is isometric to an odd-dimensional sphere, and the converse holds too. Finally, we show that for a compact and connected hypersurface    N with support    ρ and basic vector field        u  , the integral of the Ricci curvature    R  i  c      (                  u            ,              u              )   has a specific lower bound and is necessarily isometric to a sphere, and the converse also holds.}
}