Locally conformal parallel $G_2$ and Spin(7) manifolds

Ivanov, Stefan; Parton, Maurizio; Piccinni, Paolo

doi:10.4310/mrl.2006.v13.n2.a1

Cited by 34 publications

(43 citation statements)

References 25 publications

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“…A characterization of compact locally conformal parallel G 2 -manifolds as fiber bundles over S 1 with compact nearly Kähler fiber was obtained in [6] (see also [7]). It was also shown there that for compact seven-dimensional locally conformal parallel G 2 -manifolds (M, ϕ) with co-closed Lee form θ , the Lee flow preserves the Gauduchon G 2 -structure, i.e., L θ # ϕ = 0, where θ # is the dual of θ with respect to g ϕ .…”

Section: Remark 34 Since By Remark 23 We Can Always Suppose That Thmentioning

confidence: 98%

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Locally conformal calibrated $$G_2$$ G 2 -manifolds

Fernández

Fino

Raffero

2015

Annali di Matematica

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We study conditions for which the mapping torus of a 6-manifold endowed with an SU(3)-structure is a locally conformal calibrated G 2 -manifold, that is, a 7-manifold endowed with a G 2 -structure ϕ such that dϕ = −θ ∧ ϕ for a closed nonvanishing 1-form θ . Moreover, we show that if (M, ϕ) is a compact locally conformal calibrated G 2 -manifold with L θ # ϕ = 0, where θ # is the dual of θ with respect to the Riemannian metric g ϕ induced by ϕ, then M is a fiber bundle over S 1 with a coupled SU(3)-manifold as fiber.

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Section: Remark 34 Since By Remark 23 We Can Always Suppose That Thmentioning

confidence: 98%

“…In [6], a characterization of compact locally conformal parallel G 2 -manifolds as fiber bundles over S 1 with compact nearly Kähler fiber was obtained (see also [7]). …”

Section: Introductionmentioning

confidence: 98%

Locally conformal calibrated $$G_2$$ G 2 -manifolds

Fernández

Fino

Raffero

2015

Annali di Matematica

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“…Since M is compact, S is complete, and by Myers theorem, S is actually compact; see [Verbitskiȋ 2004, Remark 10.7]. Now, the argument which proves Theorem 12.1 of [Verbitskiȋ 2004] can be used to show that dim H 1 (M, ‫)ޑ‬ = 1, and M = C(S)/‫.ޚ‬ This gives the following structure theorem, which is proven independently in [Ivanov et al 2006]. …”

Section: Nearly Kähler Manifolds: An Introductionmentioning

confidence: 97%

“…It is easy to check that the cone C(M) of a nearly Kähler manifold is equipped with a parallel G 2 -structure, and, conversely, every conical singularity of a parallel G 2 -manifold is obtained as C(M), for some nearly Kähler manifold M [Hitchin 2001;Ivanov et al 2006]. For completeness' sake, we give a sketch of a proof of this result in Proposition 4.5.…”

Section: Nearly Kähler Manifolds: An Introductionmentioning

confidence: 99%

“…The correspondence between conical singularities of G 2 -manifolds and nearly Kähler geometry can be used further to study the locally conformally parallel G 2 -manifolds (see also [Ivanov et al 2006]). A locally conformally parallel G 2 -manifold is a 7-manifold M with a coveringM equipped with a parallel G 2 -structure, with the deck transform acting onM by homotheties.…”

Section: Nearly Kähler Manifolds: An Introductionmentioning

confidence: 99%

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An intrinsic volume functional on almost complex 6-manifolds and nearly Kähler geometry

Verbitsky¹

2008

Pacific J. Math.

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Let (M, I) be an almost complex 6-manifold. The obstruction to integrability of the almost complex structure (the so-called Nijenhuis tensor) N : Λ 0,1 (M ) −→ Λ 2,0 (M ) maps a 3-dimensional bundle to a 3-dimensional one. We say that Nijenhuis tensor is non-degenerate if it is an isomorphism. An almost complex manifold (M, I) is called nearly Kähler if it admits a Hermitian form ω such that ∇(ω) is totally antisymmetric, ∇ being the Levi-Civita connection. We show that a nearly Kähler metric on a given almost complex 6-manifold with non-degenerate Nijenhuis tensor is unique (up to a constant). We interpret the nearly Kähler property in terms of G 2 -geometry and in terms of connections with totally antisymmetric torsion, obtaining a number of equivalent definitions.We construct a natural diffeomorphism-invariant functional I −→ M Vol I on the space of almost complex structures on M , similar to the Hitchin functional, and compute its extrema in the following important case. Consider an almost complex structure I with non-degenerate Nijenhuis tensor, admitting a Hermitian connection with totally antisymmetric torsion. We show that the Hitchin-like functional I −→ M Vol I has an extremum in I if and only if (M, I) is nearly Kähler.

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Purely coclosed G₂‐structures on nilmanifolds

Bazzoni

Garvı́n

Muñoz

2023

Mathematische Nachrichten

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We classify seven‐dimensional nilpotent Lie groups, decomposable or of nilpotency step at most 4, endowed with left‐invariant purely coclosed G2‐structures. This is done by going through the list of all seven‐dimensional nilpotent Lie algebras given by Gong, providing an example of a left‐invariant 3‐form φ which is a pure coclosed G2‐structure (i.e., it satisfies d∗φ=0$d*\varphi =0$, φ∧dφ=0$\varphi \wedge d\varphi =0$) for those nilpotent Lie algebras that admit them; and by showing the impossibility of having a purely coclosed G2‐structure for the rest of them.

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Locally conformal parallel $G_2$ and Spin(7) manifolds

Cited by 34 publications

References 25 publications

Locally conformal calibrated $$G_2$$ G 2 -manifolds

Locally conformal calibrated $$G_2$$ G 2 -manifolds

An intrinsic volume functional on almost complex 6-manifolds and nearly Kähler geometry

Purely coclosed G₂‐structures on nilmanifolds

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Locally conformal parallel $G_2$ and Spin(7) manifolds

Cited by 34 publications

References 25 publications

Locally conformal calibrated $$G_2$$ G 2 -manifolds

Locally conformal calibrated $$G_2$$ G 2 -manifolds

An intrinsic volume functional on almost complex 6-manifolds and nearly Kähler geometry

Purely coclosed G2‐structures on nilmanifolds

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Purely coclosed G₂‐structures on nilmanifolds