Characterizing non-totally geodesic spheres in a unit sphere

Abstract

A concircular vector field $ \mathbf{u} $ on the unit sphere $ \mathbf{S}^{n+1} $ induces a vector field $ \mathbf{w} $ on an orientable hypersurface $ M $ of the unit sphere $ \mathbf{S}^{n+1} $, simply called the induced vector field on the hypersurface $ M $. Moreover, there are two smooth functions, $ f $ and $ \sigma $, defined on the hypersurface $ M $, where $ f $ is the restriction of the potential function $ \overline{f} $ of the concircural vector field $ \mathbf{u} $ on the unit sphere $ \mathbf{S}^{n+1} $ to $ M $ and $ \sigma $ is defined as $ g\left(\mathbf{u}, N\right) $, where $ N $ is the unit normal to the hypersurface. In this paper, we show that if function $ f $ on the compact hypersurface satisfies the Fischer–Marsden equation and the integral of the squared length of the vector field $ \mathbf{w} $ has a certain lower bound, then a characterization of a small sphere in the unit sphere $ \mathbf{S}^{n+1} $ is produced. Additionally, we find another characterization of a small sphere using a lower bound on the integral of the Ricci curvature of the compact hypersurface $ M $ in the direction of the vector field $ \mathbf{w} $ with a non-zero function $ \sigma $.

Keywords

• small sphere
• concircular vector field
• the fischer–marsden equation
• the ricci curvature