Nilmanifolds with a calibrated G2-structure

Conti, Diego; Fernández, Marisa

doi:10.1016/j.difgeo.2011.04.030

Cited by 49 publications

(81 citation statements)

References 16 publications

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“…Proof. Seven-dimensional nilpotent Lie algebras are classified by Gong [18]; we refer to that classification as reproduced in [9]. Case by case calculations show that the Lie algebras in Gong's list that satisfy Der(g) ⊂ sl(g) are precisely those of Table 1.…”

Section: Further Nonexistence Resultsmentioning

confidence: 99%

Einstein nilpotent Lie groups

Conti

Rossi

2019

Journal of Pure and Applied Algebra

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We study the Ricci tensor of left-invariant pseudoriemannian metrics on Lie groups. For an appropriate class of Lie groups that contains nilpotent Lie groups, we introduce a variety with a natural GL(n, R) action, whose orbits parametrize Lie groups with a left-invariant metric; we show that the Ricci operator can be identified with the moment map relative to a natural symplectic structure. From this description we deduce that the Ricci operator is the derivative of the scalar curvature s under gauge transformations of the metric, and show that Lie algebra derivations with nonzero trace obstruct the existence of Einstein metrics with s = 0.Using the notion of nice Lie algebra, we give the first example of a leftinvariant Einstein metric with s = 0 on a nilpotent Lie group. We show that nilpotent Lie groups of dimension ≤ 6 do not admit such a metric, and a similar result holds in dimension 7 with the extra assumption that the Lie algebra is nice.

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Section: Further Nonexistence Resultsmentioning

confidence: 99%

Einstein nilpotent Lie groups

Conti

Rossi

2019

Journal of Pure and Applied Algebra

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“…The 2-form τ is known as intrinsic torsion form of ϕ. Examples of closed G 2 -structures were obtained for instance in [7,9,15].…”

Section: Preliminaries On Su(3)-and G 2 -Structuresmentioning

confidence: 99%

Closed warped G_2-structures evolving under the Laplacian flow

Fino¹,

Raffero²

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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We study the behaviour of the Laplacian flow evolving closed G2-structures on warped products of the form M 6 ×S 1 , where the base M 6 is a compact 6-manifold endowed with an SU(3)-structure. In the general case, we reinterpret the flow as a set of evolution equations on M 6 for the differential forms defining the SU(3)-structure and the warping function. When the latter is constant, we find sufficient conditions for the existence of solutions of the corresponding coupled flow. This provides a method to construct immortal solutions of the Laplacian flow on the product manifolds M 6 × S 1 . The application of our results to explicit cases allows us to obtain new examples of expanding Laplacian solitons.2010 Mathematics Subject Classification. 53C44, 53C10, 53C30. Key words and phrases. Laplacian flow, G2-structure, SU(3)-structure, warped product, soliton. The authors were supported by GNSAGA of INdAM. The first author was also supported by PRIN 2015 "Real and complex manifolds: geometry, topology and harmonic analysis" of MIUR. 1 2 ANNA FINO AND ALBERTO RAFFEROconditions. Then, there are various methods to define a G 2 -structure on the product M 6 ×S 1 by means of the SU(3)-structure on M 6 , and in each case this 7-manifold turns out to be a Riemannian product or a warped product (see e.g. [6,8,25]).Recall that a G 2 -structure on a 7-manifold M 7 is defined by a stable 3-form ϕ giving rise to a Riemannian metric g ϕ and to a volume form dV ϕ . The intrinsic torsion of a G 2structure ϕ is completely determined by dϕ and d * ϕ ϕ, * ϕ being the Hodge operator defined by g ϕ and dV ϕ , and it vanishes identically if and only if both ϕ and * ϕ ϕ are closed [3,17]. When this happens, the Riemannian holonomy group Hol(g ϕ ) is a subgroup of G 2 , and g ϕ is Ricci-flat. G 2 -structures whose defining 3-form is both closed and co-closed are called torsion-free, and they play a central role in the construction of metrics with holonomy G 2 . The first complete examples of such metrics were obtained by Bryant and Salamon in [4], while compact examples of Riemannian manifolds with holonomy G 2 were constructed first by Joyce [23], and then by Kovalev [26], and by Corti, Haskins, Nordström, Pacini [12]. A potential method to obtain new results in this direction is represented by geometric flows evolving G 2 -structures.Let M 7 be a 7-manifold endowed with a closed G 2 -structure ϕ, i.e., satisfying dϕ = 0. The Laplacian flow starting from ϕ is the initial value problemwhere ∆ ϕ(t) denotes the Hodge Laplacian of the Riemannian metric g ϕ(t) induced by ϕ(t). This geometric flow was introduced by Bryant in [3] as a tool to find torsion-free G 2structures on compact manifolds. Short-time existence and uniqueness of the solution when M 7 is compact were proved by Bryant and Xu in the unpublished paper [5]. Recently, Lotay and Wei investigated the properties of the Laplacian flow in the series of papers [32,33,34].In [25], a flow evolving the 4-form * ϕ ϕ in the direction of minus its Hodge Laplacian was introduced, and its beha...

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“…Example 2.5. In [4], the authors obtained the classification of seven-dimensional nilpotent Lie algebras admitting closed G 2 -structures. An inspection of all possible cases shows that the Lie algebras whose second Betti number is lower than seven are those appearing in Table 1 Let n be one of the Lie algebras in Table 1, and consider a closed non-parallel G 2 -structure ϕ on it.…”

Section: The Automorphism Groupmentioning

confidence: 99%

On the automorphism group of a closed G2-structure

Podestà

Raffero

2018

The Quarterly Journal of Mathematics

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We study the automorphism group of a compact 7-manifold M endowed with a closed non-parallel G2-structure, showing that its identity component is abelian with dimension bounded by min{6, b2(M )}. This implies the non-existence of compact homogeneous manifolds endowed with an invariant closed non-parallel G2-structure. We also discuss some relevant examples.2010 Mathematics Subject Classification. 53C10.

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Nilmanifolds with a calibrated G2-structure

Cited by 49 publications

References 16 publications

Einstein nilpotent Lie groups

Einstein nilpotent Lie groups

Closed warped G_2-structures evolving under the Laplacian flow

On the automorphism group of a closed G2-structure

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