Closed G2-structures with conformally flat metric

Ball, G.L.

doi:10.48550/arxiv.2002.01634

Cited by 3 publications

(39 citation statements)

References 13 publications

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“…The structure equations for closed G 2 -structures can be developed further [2]. There exist functions…”

Section: The Adjoint Representation One Can Easily Check Thatmentioning

confidence: 99%

“…where J(H, T ) ijkl , r(T ) ijkl , and L(H, T ) ijm are explicit functions linear in the components H and quadratic in the components of T given by formulas (3.23) and (3.24) of [2].…”

Section: The Adjoint Representation One Can Easily Check Thatmentioning

confidence: 99%

“…The tensors H, C, and S form a complete set of second-order invariants for a closed G 2 -structure. In particular, it is possible to express the full Riemann curvature tensor of g ϕ in terms of H, C, S, and T v [2]. The Ricci tensor of g ϕ depends only on T and H, and it is thus possible to express it in terms of τ and its exterior derivative dτ alone.…”

Section: The Adjoint Representation One Can Easily Check Thatmentioning

confidence: 99%

“…Bryant [8] gives one example of a homogeneous ERP structure, Lauret [22] another, and Lauret-Nicolini [23,24] have classified left-invariant ERP structures on Lie groups, providing three other distinct examples. The only known inhomogeneous examples of λ-quadratic closed G 2 -structures in the literature are two examples each for λ = −1/8 and λ = 2/5 given by the author [2] in a previous article.…”

Section: Introductionmentioning

confidence: 99%

“…The torsion 2-form τ of a closed G 2 -structure ϕ takes values in a 14-dimensional subbundle of Λ 2 T * M that is modeled on the Lie algebra g 2 . Under the adjoint action of G 2 , the generic element of g 2 is stabilised by a maximal torus, but there are two exceptional G 2 -orbits in g 2 consisting of elements stabilised by subgroups of G 2 isomorphic to U (2). The elements β of one of these G 2 -orbits, thought of as 2-forms β, satisfy β 3 = 0 and we call these elements positive, while the elements of the other G 2 -orbit satisfy |β 3 | 2 = 2 3 |β| 6 and we call these elements negative.…”

Section: Introductionmentioning

confidence: 99%

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Exact Filters and Joins of Closed Sublocales

Ball

Moshier

Pultr

2020

Appl Categor Struct

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This article studies closed G 2 -structures satisfying the quadratic condition, a secondorder PDE system introduced by Bryant involving a parameter λ. For certain special values of λ the quadratic condition is equivalent to the Einstein condition for the metric induced by the closed G 2 -structure (for λ = 1/2), the extremally Ricci-pinched (ERP) condition (for λ = 1/6), and the condition that the closed G 2 -structure be an eigenform for the Laplace operator (for λ = 0). Prior to the work in this article, solutions to the quadratic system were known only for λ = 1/6, −1/8, and 2/5, and for these values only a handful of solutions were known.In this article, we produce infinitely many new examples of ERP G 2 -structures, including the first example of a complete inhomogeneous ERP G 2 -structure and a new example of a compact ERP G 2 -structure. We also give a classification of homogeneous ERP G 2 -structures. We provide the first examples of quadratic closed G 2 -structures for λ = −1, 1/3, and 3/4, as well as infinitely many new examples for λ = −1/8 and 2/5. Our constructions involve the notion of special torsion for closed G 2 -structures, a new concept that is likely to have wider applicability.In the final section of the article, we provide two large families of inhomogeneous complete steady gradient solitons for the Laplacian flow, the first known such examples.

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“…The structure equations for closed G 2 -structures can be developed further [2]. There exist functions…”

Section: The Adjoint Representation One Can Easily Check Thatmentioning

confidence: 99%

“…where J(H, T ) ijkl , r(T ) ijkl , and L(H, T ) ijm are explicit functions linear in the components H and quadratic in the components of T given by formulas (3.23) and (3.24) of [2].…”

Section: The Adjoint Representation One Can Easily Check Thatmentioning

confidence: 99%

Section: The Adjoint Representation One Can Easily Check Thatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Exact Filters and Joins of Closed Sublocales

Ball

Moshier

Pultr

2020

Appl Categor Struct

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Quadratic closed G2$\mathrm{G}_2$‐structures

Ball

2023

Journal of London Math Soc

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This article studies closed G 2 -structures satisfying the quadratic condition, a second-order PDE system introduced by Bryant involving a parameter 𝜆. For certain special values of 𝜆 the quadratic condition is equivalent to the Einstein condition for the metric induced by the closed G 2 -structure (for 𝜆 = 1∕2), the extremally Ricci-pinched (ERP) condition (for 𝜆 = 1∕6), and the condition that the closed G 2 -structure be an eigenform for the Laplace operator (for 𝜆 = 0). Prior to the work in this article, solutions to the quadratic system were known only for 𝜆 = 1∕6, −1∕8, and 2∕5, and for these values only a handful of solutions were known. In this article, we produce infinitely many new examples of ERP G 2 -structures, including the first example of a complete inhomogeneous ERP G 2 -structure and a new example of a compact ERP G 2 -structure. We also give a classification of homogeneous ERP G 2 -structures.We provide the first examples of quadratic closed G 2structures for 𝜆 = −1, 1∕3 and 3∕4, as well as infinitely many new examples for 𝜆 = −1∕8 and 2∕5. Our constructions involve the notion of special torsion for closed G 2 -structures, a new concept that is likely to have wider applicability. In the final section of the article, we provide two large families of inhomogeneous complete steady gradient solitons for the Laplacian flow, the first known such examples.

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Closed $G_2$-eigenforms and exact $G_2$-structures

Freibert

Salamon

2021

Rev. Mat. Iberoam.

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A study is made of left-invariant G 2 -structures with an exact 3-form on a Lie group G whose Lie algebra g admits a codimension-one nilpotent ideal h. It is shown that such a Lie group G cannot admit a left-invariant closed G 2 -eigenform for the Laplacian and that any compact solvmanifold nG arising from G does not admit an (invariant) exact G 2 -structure. We also classify the seven-dimensional Lie algebras g with codimension-one ideal equal to the complex Heisenberg Lie algebra which admit exact G 2 -structures with or without special torsion. To achieve these goals, we first determine the six-dimensional nilpotent Lie algebras h admitting an exact SL.3; C/-structure or a half-flat SU.3/-structure .!; / with exact , respectively.

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Closed G2-structures with conformally flat metric

Cited by 3 publications

References 13 publications

Exact Filters and Joins of Closed Sublocales

Exact Filters and Joins of Closed Sublocales

Quadratic closed G2$\mathrm{G}_2$‐structures

Closed $G_2$-eigenforms and exact $G_2$-structures

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