New Characterizations of the Whitney Spheres and the Contact Whitney Spheres

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