Decomposition and minimality of Lagrangian submanifolds in nearly Kähler manifolds

Details Decomposition and minimality of Lagrangian submanifolds in nearly Kähler manifolds Decomposition and minimality of Lagrangian submanifolds in nearly Kähler manifolds

2009-04-23

arxiv.org/abs/0904.3683v1

Mathematics

Differential Geometry

19 pages

Scientific paper

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.

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