Half-flat structures on decomposable lie groups

Freibert, Marco; Schulte-Hengesbach, Fabian

doi:10.1007/s00031-011-9168-z

Cited by 13 publications

(38 citation statements)

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“…It is worth emphasizing here that for each of the s i , the inner product, with respect to which As a consequence of the previous Theorem, for the 6-dimensional homogeneous Einstein manifolds of nonpositive sectional curvature they showed what follows. It is worth recalling here an obstruction to the existence of half-flat structures on 6dimensional Lie algebras shown by Freibert and Schulte-Hengesbach in [17]. This result is a refinement of the one obtained by Conti in [11].…”

Section: Einstein Half-flat Structures On 6-solvmanifoldssupporting

confidence: 67%

“…These classifications will be useful in the proof of the following Proof. We will prove the theorem as follows: in the list of Einstein solvable metric Lie algebras we first exclude the ones that do not admit a half-flat structure using the results of [17] and [18], then we will show the result by direct computations in the remaining cases.…”

Section: Einstein Half-flat Structures On 6-solvmanifoldsmentioning

confidence: 99%

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Half-flat structures inducing Einstein metrics on homogeneous spaces

Raffero

2015

Ann Glob Anal Geom

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In this paper, we consider half-flat $SU(3)$-structures and the subclasses of coupled and double structures. In the general case we show that the intrinsic torsion form $w_1^-$ is constant in each of the two subclasses. We then consider the problem of finding half-flat structures inducing Einstein metrics on homogeneous spaces. We give an example of a left invariant half-flat (non coupled and non double) structure inducing an Einstein metric on $S^3\times S^3$ and we show there does not exist any left invariant coupled structure inducing an ${\rm Ad}(S^1)$-invariant Einstein metric on it. Finally, we show that there are no coupled structures inducing the Einstein metric on Einstein solvmanifolds and on homogeneous Einstein manifolds of nonpositive sectional curvature.Comment: 17 pages, to appear in Annals of Global Analysis and Geometr

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Section: Einstein Half-flat Structures On 6-solvmanifoldssupporting

confidence: 67%

Section: Einstein Half-flat Structures On 6-solvmanifoldsmentioning

confidence: 99%

Half-flat structures inducing Einstein metrics on homogeneous spaces

Raffero

2015

Ann Glob Anal Geom

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“…, e 6 } is the basis of g * in which the Lie algebra is expressed. Notice that we will follow the notation given in [14,15,28] and [31] to name the Lie algebras; for instance, the notation e(2) ⊕ e(1, 1) = (0, −e 13 , e 12 , 0, −e 46 , −e 45 ) means that e(2) ⊕ e(1, 1) is the (decomposable) Lie algebra determined by a basis {e i } 6 i=1 such that de 1 = 0, de 2 = −e 13 , de 3 = e 12 , de 4 = 0, de 5 = −e 46 , de 6 = −e 45 . The next two concrete examples show how we will proceed in general in the proofs of Propositions 2.6 and 2.7 below in order to exclude candidates.…”

Section: The Classificationmentioning

confidence: 99%

Six-Dimensional Solvmanifolds with Holomorphically Trivial Canonical Bundle

Fino

Otal

Ugarte

2015

Int Math Res Notices

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We determine the 6-dimensional solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong Kähler with torsion (SKT), generalized Gauduchon, balanced and strongly Gauduchon metrics is studied. As an application we construct a holomorphic family (M, Ja) of compact complex manifolds such that (M, Ja) satisfies the ∂∂lemma and admits a balanced metric for any a = 0, but the central limit neither satisfies the ∂∂-lemma nor admits balanced metrics.

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“…, e 6 } satisfies de 1 = e 15 , de 2 = −e 25 , de 3 = −e 35 , de 4 = e 45 , de 5 = 0, de 6 = 0, where d is the Chevalley-Eilenberg differential of g, and e ij··· is a shorthand for the wedge product e i ∧ e j ∧ · · · . It is known (see [18,20]) that on g there exists a symplectic half-flat SU(3)-structure defined by the differential forms ω = −e 13 + e 24 + e 56 , ψ + = −e 126 − e 145 − e 235 − e 346 .…”

Section: Laplacian Flow On Riemannian Product Manifoldsmentioning

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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Half-flat structures on decomposable lie groups

Cited by 13 publications

References 14 publications

Half-flat structures inducing Einstein metrics on homogeneous spaces

Half-flat structures inducing Einstein metrics on homogeneous spaces

Six-Dimensional Solvmanifolds with Holomorphically Trivial Canonical Bundle

Closed warped G_2-structures evolving under the Laplacian flow

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