Characterizing small spheres in a unit sphere by Fischer–Marsden equation

Abstract

We use a nontrivial concircular vector field u on the unit sphere mathbf{S}^{n+1} in studying geometry of its hypersurfaces. An orientable hypersurface M of the unit sphere mathbf{S}^{n+1} naturally inherits a vector field v and a smooth function ρ. 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 mathbf{S}^{n+1}. We also use the condition that the function ρ 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 mathbf{S}^{n+1}.