@article{Al-Dayel2023, 
author = {Ibrahim Al-Dayel and Sharief Deshmukh and Olga Belova},
title = {Characterizing non-totally geodesic spheres in a unit sphere},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {9},
pages = {21359-21370},
keywords = {small sphere, concircular vector field, the Fischer–Marsden equation, the Ricci curvature},
url = {https://www.sciopen.com/article/10.3934/math.20231088},
doi = {10.3934/math.20231088},
abstract = {A concircular vector field        u   on the unit sphere              S              n      +      1       induces a vector field        w   on an orientable hypersurface    M of the unit sphere              S              n      +      1      , simply called the induced vector field on the hypersurface    M. Moreover, there are two smooth functions,    f and    σ, defined on the hypersurface    M, where    f is the restriction of the potential function        f    ¯   of the concircural vector field        u   on the unit sphere              S              n      +      1       to    M and    σ is defined as    g      (                  u            ,      N        )  , where    N is the unit normal to the hypersurface. In this paper, we show that if function    f on the compact hypersurface satisfies the Fischer–Marsden equation and the integral of the squared length of the vector field        w   has a certain lower bound, then a characterization of a small sphere in the unit sphere              S              n      +      1       is produced. Additionally, we find another characterization of a small sphere using a lower bound on the integral of the Ricci curvature of the compact hypersurface    M in the direction of the vector field        w   with a non-zero function    σ.}
}