  In this paper, we study Lagrangian submanifolds satisfying ${\rm \nabla^*}
T=0$ introduced by Zhang \cite{Zh} in the complex space forms $N(4c)(c=0\ or \
1)$, where $T ={\rm \nabla^*}\tilde{h}$ and $\tilde{h}$ is the Lagrangian
trace-free second fundamental form. We obtain some Simons' type integral
inequalities and rigidity theorems for such Lagrangian submanifolds. Moreover
we study Lagrangian submanifolds in $\mathbb{C}^n$ satisfying
$\nabla^*\nabla^*T=0$ and introduce a flow method related to them.
