Decomposition and minimality of lagrangian submanifolds in nearly Kähler manifolds

Abstract

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

Isotropic Lagrangian submanifolds in the homogeneous nearly Kähler S3 × S3

“…Noting that the cubic form g(h(·, ·), J·) is totally symmetric (cf. Proposition 3.2 of [24]), we can choose an orthonormal basis {e 1 , e 2 .e 3 } of T p M as in Lemma 1 of [19], such that…”

“…Solving this equation for µ j , we get µ j = −µ, or µ j = 1 2 µ. From Theorem A of [24] we know that M is minimal, thus we have 0 = g(h(e 1 , e 1 ) + h(e 2 , e 2 ) + h(e 3 , e 3 ),…”

“…The curvature tensorR of the homogeneous NK S 3 × S 3 is given by (cf. [1]): [24] (see also [15]), M is orientable and minimal, and that G(X, Y ) is a normal vector for any X, Y ∈ T M . We denote by ∇ and ∇ ⊥ respectively the induced connection and normal connection on M .…”

Almost Contact Lagrangian Submanifolds of Nearly Kaehler 6-Sphere

“…L. Vrancken has pointed out (private communication) that the minus sign in the first term of equation 9is missing in [8] and also on p. 403 in [3]. The correct form appears in the Lemma 3.2 of the paper [13] of Schafer-Smoczyk.…”

Properties of the nearly kähler S3×S3

“…One should remark that only very recently, the first complete non homogeneous nearly Kähler structures were discovered on S 6 and S 3 × S 3 in [6]. As far as their submanifolds are concerned, strict 6-dimensional nearly Kähler manifolds have the surprising property that their Lagrangian submanifolds are always minimal (see [10,13]).…”