Decomposition and minimality of Lagrangian submanifolds in nearly K\"ahler manifolds

We show that Lagrangian submanifolds in six-dimensional nearly K"ahler (non K"ahler) manifolds and in twistor spaces $Z\text{sp}4n+2$ over quaternionic K"ahler manifolds $Q\text{sp}4n$ are minimal. Moreover, we will prove that any Lagrangian submanifold $L$ in a nearly K"ahler manifold $M$ splits into a product of two Lagrangian submanifolds for which one factor is Lagrangian in the strict nearly K"ahler part of $M$ and the second factor is Lagrangian in the K"ahler part of $M$. Using this splitting theorem we then describe Lagrangian submanifolds in nearly K"ahler manifolds of dimensions six, eight and ten.