Decomposition and minimality of lagrangian submanifolds in nearly Kähler manifolds

Abstract

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