2013
DOI: 10.1007/s10455-013-9371-3
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Half-flat structures on $$ S^3\times S^3$$

Abstract: We describe left-invariant half-flat SU(3)-structures on S 3 × S 3 using the representation theory of SO (4) and matrix algebra. This leads to a systematic study of the associated cohomogeneity one Ricci-flat metrics with holonomy G 2 obtained on 7-manifolds with equidistant S 3 × S 3 hypersurfaces. The generic case is analysed numerically.

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Cited by 23 publications
(28 citation statements)
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“…(In fact, it is enough to impose the half-flat condition on the SU(3)-structure at some initial time and the evolution (2.2) will then preserve this condition.) For more on half-flat SU(3)-structures, in a case relevant to us, the reader can see [19]. The resulting G 2 -structure induces the metric g = dt 2 + g t , where g t is the metric on {t} × M induced by the SU(3)-structure (ω(t), 2 (t)).…”
Section: Evolution Equationsmentioning
confidence: 99%
See 2 more Smart Citations
“…(In fact, it is enough to impose the half-flat condition on the SU(3)-structure at some initial time and the evolution (2.2) will then preserve this condition.) For more on half-flat SU(3)-structures, in a case relevant to us, the reader can see [19]. The resulting G 2 -structure induces the metric g = dt 2 + g t , where g t is the metric on {t} × M induced by the SU(3)-structure (ω(t), 2 (t)).…”
Section: Evolution Equationsmentioning
confidence: 99%
“…The compatible metric determined by this SU(3) structure on {t} × M is [19] 13) and the resulting metric on R t × M, compatible with the G 2 -structure ϕ = dt ∧ω+ 1 , is given by g = dt 2 + g t . Recall also that this metric has holonomy in G 2 if and only if the SU(3)-structure above solves the Hitchin flow equations (2.2).…”
Section: Su(2) 2 -Invariant G 2 -Manifolds Of Cohomogeneity-1mentioning
confidence: 99%
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“…given by a one-parameter family of a half-flat structure on S 3 × S 3 [23]. A half-flat structure is one of SU (3)-structures, which is characterized by a pair (κ, γ), where κ is a symplectic form on S 3 × S 3 and γ is a real 3-form whose stabiliser is isomorphic to SL(3; C) and satisfies a condition…”
Section: A Cohomogeneity One Ansatzmentioning
confidence: 99%
“…SU(3)-structures whose torsion class is W − 1 ⊕ W − 2 are known as coupled SU(3)-structures [38] in the mathematical literature and are characterized by the fact that they are half-flat SU(3)-structures, i.e., both ψ + := ℜ(Ψ) and ω ∧ ω are closed forms, having dω proportional to ψ + . Coupled structures were recently considered in [16,30,36]. They are of interest for instance because their underlying almost Hermitian structure is quasi-Kähler and because they generalize the class of nearly Kähler SU(3)-structures, namely the half-flat structures having dω proportional to ψ + and dψ − proportional to ω ∧ ω, where ψ − := ℑ(Ψ).…”
Section: Introductionmentioning
confidence: 99%