Abstract
We use a nontrivial concircular vector field u on the unit sphere Sn+1 in studying geometry of its hypersurfaces. An orientable hypersurface M of the unit sphere Sn+1 naturally inherits a vector field v and a smooth function rho. We use the condition that the vector field v is an eigenvector of the de-Rham Laplace operator together with an inequality satisfied by the integral of the Ricci curvature in the direction of the vector field v to find a characterization of small spheres in the unit sphere Sn+1. We also use the condition that the function rho is a nontrivial solution of the Fischer-Marsden equation together with an inequality satisfied by the integral of the Ricci curvature in the direction of the vector field v to find another characterization of small spheres in the unit sphere Sn+1.