  In \cite{Zh} \cite{LY} Zhang, Luo and Yin initiated the study of Lagrangian
submanifolds satisfying ${\rm \nabla^*} T=0$ or ${\rm \nabla^*\nabla^*}T=0$ in
$\mathbb{C}^n$ or $\mathbb{CP}^n$, where $T ={\rm \nabla^*}\tilde{h}$ and
$\tilde{h}$ is the Lagrangian trace-free second fundamental form. They proved
several rigidity theorems for Lagrangian surfaces satisfying ${\rm \nabla^*}
T=0$ or ${\rm \nabla^*\nabla^*}T=0$ in $\mathbb{C}^2$ under proper small energy
assumption and gave new characterization of the Whitney spheres in
$\mathbb{C}^2$. In this paper we extend these results to Lagrangian
submanifolds in $\mathbb{C}^n$ of dimension $n\geq3$ and to Lagrangian
submanifolds in $\mathbb{CP}^n$.
