Decomposition and minimality of Lagrangian submanifolds in nearly K��hler manifolds

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