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

TL;DR

This paper proves that Lagrangian submanifolds in certain nearly K"ahler and twistor spaces are minimal and can be decomposed into products of simpler Lagrangian submanifolds, providing a classification in specific dimensions.

Contribution

It establishes minimality of Lagrangian submanifolds in six-dimensional nearly K"ahler and related spaces, and introduces a splitting theorem for their structure.

Findings

Lagrangian submanifolds in six-dimensional nearly K"ahler manifolds are minimal.

Any Lagrangian submanifold splits into a product of two Lagrangian submanifolds in different parts of the manifold.

Classification of Lagrangian submanifolds in dimensions six, eight, and ten.

Abstract

We show that Lagrangian submanifolds in six-dimensional nearly K\"ahler (non K\"ahler) manifolds and in twistor spaces $Z\sp{4n+2}$ over quaternionic K\"ahler manifolds $Q\sp{4n}$ 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.