@article{Li2024, 
author = {Yanlin Li and Nasser Bin Turki and Sharief Deshmukh and Olga Belova},
title = {Euclidean hypersurfaces isometric to spheres},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {10},
pages = {28306-28319},
keywords = {shape operator, n-sphere, Euclidean space, static perfect fluid equation, incompressible vector fields},
url = {https://www.sciopen.com/article/10.3934/math.20241373},
doi = {10.3934/math.20241373},
abstract = {Given an immersed hypersurface  Mn in the Euclidean space  En+1, the tangential component  ω 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  σ on the hypersurface called the support function of the hypersurface. In the first result, we show that on a complete and simply connected hypersurface  Mn in  En+1 of positive Ricci curvature with shape operator  T invariant under  ω and the support function  σ 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  Mn in  En+1 with the gradient of support function  σ, an eigenvector of the shape operator  T with eigenvalue function the mean curvature  H, and the integral of the squared length of the gradient  ∇σ has a certain lower bound, giving a characterization of a sphere. In the third result, we show that a compact and simply connected hypersurface  Mn of positive Ricci curvature in  En+1 has an incompressible basic vector field  ω, if and only if  Mn is isometric to a sphere.}
}