Deformations of nearly Kähler structures

Moroianu, Andrei; Nagy, Paul-Andi; Semmelmann, Uwe

doi:10.2140/pjm.2008.235.57

Cited by 27 publications

(58 citation statements)

References 9 publications

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“…Similarly, differentiating the equality η ∧ ω + * η = 0 and using Lemma 2.4(i) yields Λdη = −Jd * η, cf. [17,Lemma 4.3].…”

Section: Differential and Co-differential On Nearly Kähler Manifoldsmentioning

confidence: 99%

“…In this paper we study the deformation theory of nearly Kähler manifolds. In [17] Moroianu, Nagy and Semmelmann studied infinitesimal deformations of nearly Kähler structures, identifying them with an eigenspace of the Laplacian acting on co-closed primitive (1, 1)-forms. The question of whether nearly Kähler 6-manifolds have smooth, unobstructed deformations was however left open.…”

Section: Introductionmentioning

confidence: 99%

“…The proof of the theorem (and the paper itself) is divided into two distinct parts. In the first step, we will obtain a deeper understanding of the deformation theory of nearly Kähler structures in general beyond the infinitesimal level considered in [17]. The main tool is a certain Dirac-type operator on nearly Kähler manifolds, which appears as a certain combination of differential, co-differential and type decomposition acting on differential forms.…”

Section: Introductionmentioning

confidence: 99%

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Deformation theory of nearly Kähler manifolds

Foscolo

2017

J. London Math. Soc.

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Nearly Kähler manifolds are the Riemannian 6-manifolds admitting real Killing spinors. Equivalently, the Riemannian cone over a nearly Kähler manifold has holonomy contained in G2. In this paper we study the deformation theory of nearly Kähler manifolds, showing that it is obstructed in general. More precisely, we show that the infinitesimal deformations of the homogeneous nearly Kähler structure on the flag manifold are all obstructed to second order.

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“…Similarly, differentiating the equality η ∧ ω + * η = 0 and using Lemma 2.4(i) yields Λdη = −Jd * η, cf. [17,Lemma 4.3].…”

Section: Differential and Co-differential On Nearly Kähler Manifoldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Deformation theory of nearly Kähler manifolds

Foscolo

2017

J. London Math. Soc.

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“…We shall follow this usage, often omitting "strictly" and "6-dimensional". In more recent literature (such as Mororianu, Nagy and Semmelmann [24]), these objects are called Gray manifolds.…”

Section: Nearly Kähler 6-manifoldsmentioning

confidence: 99%

Hodge theory on nearly Kähler manifolds

Verbitsky¹

2011

Geom. Topol.

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Let .M; I; !; / be a nearly Kähler 6-manifold, that is, an SU.3/-manifold with .3; 0/-form and Hermitian form ! which satisfiesfor a nonzero real constant . We develop an analogue of the Kähler relations on M , proving several useful identities for various intrinsic Laplacians on M . When M is compact, these identities give powerful results about cohomology of M . We show that harmonic forms on M admit a Hodge decomposition, and prove that

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“…In an article by [18], the deformation space of nearly Kähler structures on six-dimensional nearly Kähler manifolds has been related to the space of coclosed eigenforms of the Hodge-Laplacian. As we will show below, a similar statement holds for the deformation of Lagrangian submanifolds in strict nearly (pseudo-)Kähler six-manifolds.…”

Section: Deformations Of Lagrangian Submanifolds In Nearly Kähler Manmentioning

confidence: 99%

Decomposition and minimality of lagrangian submanifolds in nearly Kähler manifolds

Schäfer

Smoczyk

2009

Ann Glob Anal Geom

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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.

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Deformations of nearly Kähler structures

Cited by 27 publications

References 9 publications

Deformation theory of nearly Kähler manifolds

Deformation theory of nearly Kähler manifolds

Hodge theory on nearly Kähler manifolds

Decomposition and minimality of lagrangian submanifolds in nearly Kähler manifolds

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