An Optimal Inequality Related to Characterizations of the Contact Whitney Spheres in Sasakian Space Forms

Abstract

Let \(N^{2n+1}(c)\) be an \((2n+1)\)-dimensional Sasakian space form with Sasakian structure \((\varphi ,\xi , \eta ,g)\) and constant \(\varphi \)-sectional curvature c. An n-dimensional submanifold \(M^n\) of \(N^{2n+1}(c)\) is called integral if the contact form \(\eta \) restricted to \(M^n\) vanishes. In this paper, we established a general inequality for n-dimensional integral submanifolds in Sasakian space forms involving the norm of the covariant differentiation of both the second fundamental form h and the mean curvature vector field H. Our result is optimal in that we can classify all integral submanifolds realizing the equality case of the inequality. As direct consequence, we give a characterization of the contact Whitney spheres in the Sasakian space forms.