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

Abstract

Abstract We use a nontrivial concircular vector field u on the unit sphere S n + 1 $\mathbf{S}^{n+1}$ in studying geometry of its hypersurfaces. An orientable hypersurface M of the unit sphere S n + 1 $\mathbf{S}^{n+1}$ 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 S n + 1 $\mathbf{S}^{n+1}$ . 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 S n + 1 $\mathbf{S}^{n+1}$ .

Keywords

• Sphere
• de-Rham Laplace operator
• Fischer–Marsden differential equation
• Small sphere