Hypersurfaces in a Euclidean space with a Killing vector field
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 Rn+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 S2m-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.
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…
In this paper, we examine torse-forming vector fields to characterize extrinsic spheres (that is, totally umbilical hypersurfaces with nonzero constant mean curvatures) in Riemannian and…
This paper investigates compact Riemannian hypersurfaces immersed in (n+1)-dimensional Riemannian or Lorentzian manifolds that admit concircular vector fields, also known as closed conformal…