New characterizations of the Whitney spheres and the contact Whitney spheres

TL;DR

This paper establishes optimal integral inequalities for certain submanifolds in complex and Sasakian space forms, leading to new characterizations of Whitney spheres and contact Whitney spheres, including their conformal flatness and curvature properties.

Contribution

It introduces new integral inequalities involving Ricci curvature and second fundamental form derivatives, providing global characterizations of Whitney and contact Whitney spheres.

Findings

Characterization of Whitney spheres via equality cases

Contact Whitney spheres are locally conformally flat

Sectional curvatures of contact Whitney spheres are non-constant

Abstract

In this paper, based on the classical K. Yano's formula, we first establish an optimal integral inequality for compact Lagrangian submanifolds in the complex space forms, which involves the Ricci curvature in the direction $J\vec{H}$ and the norm of the covariant differentiation of the second fundamental form $h$, where $J$ is the almost complex structure and $\vec{H}$ is the mean curvature vector field. Second and analogously, for compact Legendrian submanifolds in the Sasakian space forms with Sasakian structure $(\varphi,\xi,\eta,g)$, we also establish an optimal integral inequality involving the Ricci curvature in the direction $\varphi\vec{H}$ and the norm of the modified covariant differentiation of the second fundamental form. The integral inequality is optimal in the sense that all submanifolds attaining the equality are completely classified. As direct consequences, we obtain new and global characterizations for the Whitney spheres in complex space forms as well as the contact Whitney spheres in Sasakian space forms. Finally, we show that, just as the Whitney spheres in complex space forms, the contact Whitney spheres in Sasakian space forms are locally conformally flat manifolds with sectional curvatures non-constant.