This article classifies closed G2-structures such that the induced metric is conformally flat. It is shown that any closed G2-structure with conformally flat metric is locally equivalent to one of three explicit examples. In particular, it follows from the classification that any closed G2-structure inducing a metric t…
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48 results for “G2-structure”
Found a new compact G2-structure on a 7-manifold.
problem Identifying compact G2-structures on manifolds.
method Lattice in solvable Lie group, Laplacian coflow analysis.
result Obtained a 4-parameter subfamily of expanding solitons.
Article explores G2-structures with quadratic conditions, finding new ERP and complete solitons.
problem Investigating closed G2-structures satisfying quadratic conditions.
method Analyzing a second-order PDE system involving parameter λ, producing new examples of ERP and complete solitons.
result First examples of ERP G2-structures, including complete inhomogeneous ERP G2-structure.
Among closed G2-structures there are two very distinguished classes: Laplacian solitons and Extremally Ricci-pinched G2-structures. We study the existence problem and explore possible interplays between these concepts in the context of left-invariant G2-structures on solvable Lie groups. Also, some Ricci pinching prope…
We consider non-infinitesimal deformations of G2-structures on 7-dimensional manifolds and derive an exact expression for the torsion of the deformed G2-structure. We then specialize to a case when the deformation is defined by a vector v and we explicitly derive the expressions for the different torsion components of …
Only two examples of extremally Ricci pinched G2-structures can be found in the literature and they are both homogeneous. We study in this paper the existence and structure of such very special closed G2-structures on Lie groups. Strong structural conditions on the Lie algebra are proved to hold. As an application, we …
We study the N=1 supersymmetric solutions of D=11 supergravity obtained as a warped product of four-dimensional anti-de-Sitter space with a seven-dimensional Riemannian manifold M. Using the octonion bundle structure on M we reformulate the Killing spinor equations in terms of sections of the octonion bundle on M. The …
We study the condition in which G2-structures are introduced by a non closed four-form, although they are satisfying locally conformal conditions.All solutions are found in the case when the Lee form of G2-structures is non-zero and gintroduces seven-dimensional Lie algebras, The main results are given in preposition1 …
We overview the properties of non-infinitesimal deformations of G2-structures on seven-manifolds, and in particular, focus on deformations that lie in the seven-dimensional representation of G2 and are thus defined by a vector. We then consider deformations from G2-structures with the torsion class having one-dimension…
A complete classification of left-invariant closed G2-structures on Lie groups which are extremally Ricci pinched, up to equivalence and scaling, is obtained. There are five of them, they are defined on five different completely solvable Lie groups and the G2-structure is exact in all cases except one, given by the onl…
New perspective on G2-structures flow from DeTurck Laplacian.
problem Understanding G2-structures and their flows.
method Introducing a new flow (DeTurck Laplacian flow) for G2-structures.
result DeTurck Laplacian flow is a flow of G2-structures.
We give a method to obtain new 7-dimensional Lie algebras endowed with closed and coclosed G2-structures starting from 6-dimensional Lie algebras with symplectic half- at SU(3)-structures and half- at SU(3)- structures, respectively. Finally, we describe all the 7-dimensional Lie algebras with a closed G2-structure tha…
We classify all seven-dimensional spaces which admit a homogeneous cosymplectic G2-structure. The motivation for this classification is that each of these spaces is a possible principal orbit of a parallel Spin(7)-manifold of cohomogeneity one.
Geodesic concavity and hypersymplectic structures in -structures space.
problem Analyzing the geodesic concavity and hypersymplectic structures in the space of closed -structures.
method Utilising the geodesic constructed in the previous article, we show geodesic concavity and decrease in length of Laplacian flow.
result Hitchin's volume functional is geodesically concave and the Laplacian flow decreases the length.
In this paper most of the classes of G2-structures with Einstein induced metric of negative, null or positive scalar curvature are realized. This is carried out by means of warped G2-structures with fiber an Einstein SU(3) manifold. The torsion forms of any warped G2-structure are explicitly described in terms of the t…
We give an answer to a question posed recently by R.Bryant, namely we show that a compact 7-dimensional manifold equipped with a G2-structure with closed fundamental form is Einstein if and only if the Riemannian holonomy of the induced metric is contained in G2. This could be considered to be a G2 analogue of the Gold…
We give the first examples of closed Laplacian solitons which are shrinking, and in particular produce closed Laplacian flow solutions with a finite-time singularity. Extremally Ricci pinched G2-structures (introduced by Bryant) which are steady Laplacian solitons have also been found. All the examples are left-invaria…
We use a G2-structure on a 7-dimensional Riemannian manifold with a fixed metric to define an octonion bundle with a fiberwise non-associative product. We then define a metric-compatible octonion covariant derivative on this bundle that is compatible with the octonion product. The torsion of the G2-structure is then sh…
We modify the Laplacian coflow of co-closed G2-structures - where is the closed dual 4-form of a -structure . The modified flow is now parabolic in the direction of closed forms upto diffeomorphisms. We then prove short time existence and uniqueness of solutions to the modified f…
Derives an integral formula for G2-structures.
problem Calculating properties of G2-structures.
method Applies an integral formula for G-structures to G2.
result Derives an integral formula relating curvatures and quadratic invariants.
We reduce the embedding problem for hypo SU(2) and SU(3)-structures to the embedding problem for hypo G2-structures into parallel Spin(7)-manifolds. The latter will be described in terms of gauge deformations. This description involves the intrinsic torsion of the initial G2-structure and allows us to prove that the ev…
Study 3-Sasakian and G2 structures on manifolds.
problem Investigate algebra of forms on 3-Sasakian manifolds.
method Examine cohomogeneity-one actions on spheres and Berger's space.
result Identify nearly-parallel G2 forms.
Let M be a 7-manifold with a G2-structure defined by φ\inΩ^{3}_{+}(M). We prove that φ is conformal-Killing with respect to the associated metric g(φ) if and only if the G2-structure is nearly parallel. Let M be an 8-manifold with a Spin7-structure defined by ψ\inΩ^{4}_{+}(M). We prove that ψ is conformal-Killing with …
We derive formulas for the mean curvature of associative 3-folds, coassociative 4-folds, and Cayley 4-folds in the general case where the ambient space has intrinsic torsion. Consequently, we are able to characterize those G2-structures (resp., Spin(7)-structures) for which every associative 3-fold (resp. coassociative…
Researchers compute the ν-invariant for specific G2-structures on nilmanifolds.
problem Detecting connected components of G2-structure moduli spaces.
method Defined and computed the ν-invariant using Mathai-Quillen currents, harmonic spinors, and η-invariants.
result Determined the parity of harmonic spinor dimensions and deduced ν vanishing on invariant spinors.
Study on dimensions of G2-structures on nilmanifolds, proving non-abelian automorphism groups.
problem Dimensions and automorphisms of G2-structures on nilmanifolds.
method Computational analysis of moduli spaces of G2-structures on 7-dimensional nilmanifolds.
result Coclosed G2-structures have non-abelian automorphism groups, unlike closed ones.
Completes classification of G2-structures on specific nilpotent Lie groups.
problem Classifying seven-dimensional nilpotent Lie groups with purely coclosed G2-structures.
method Analyzing nilpotent Lie groups of various steps and dimensions.
result Classification of indecomposable 5- and 6-step nilpotent Lie groups.
We construct propose an anzatz for Spin(7) metrics as an R-bundle over closed G2 structures. These G2 structures are R3 bundles over 4-dimensional compact quaternion Kahler spaces. The inspiration for the anzatz metric comes from the Bryant-Salamon construction of G2 holonomy metrics and from the fact that the twistor …
This paper is a review of current developments in the study of moduli spaces of G2 manifolds. G2 manifolds are 7-dimensional manifolds with the exceptional holonomy group G2. Although they are odd-dimensional, in many ways they can be considered as an analogue of Calabi-Yau manifolds in 7 dimensions. They play an impor…
Researchers compute the ν-invariant for specific G2-structures on Aloff-Wallach spaces.
problem Computing the ν-invariant for G2-structures on Aloff-Wallach spaces.
method Using Goette's formulas for η-invariants of homogeneous spaces, derived an explicit expression for ν.
result For the nearly-parallel G2-structures φ± on Nk,l, the ν-invariant is ν(φ±) = ±41.
Study of ACM(3)S structures on manifolds with G2 structure.
problem Understanding ACM(3)S structures on manifolds with G2 structure.
method Review and computation of torsion of associated SU(3) structure, study of ACM3S space, and observation of links with submanifolds.
result Initiation of study on the space of ACM3Ss with interesting topological features and links to submanifolds.
Characterizes G2-structures on Lie algebras with non-trivial center.
problem Classifying Lie algebras with G2-structures.
method Analyzing Lie algebras with non-trivial center, using contactization and symplectic properties.
result Six unimodular Lie algebras with non-trivial center admit closed G2-structures.
New compact manifolds with closed G2 structures found.
problem Existence of torsion-free G2 structures on compact manifolds.
method Equivariant resolution procedure for G2 orbifolds with Z2 isotropy.
result Construction of new compact manifolds with closed G2 structures.
Study on deformations of Einstein and nearly G2 structures in 3-Sasaki manifolds.
problem Deformation theory of Einstein and nearly G2 structures in 3-Sasaki manifolds.
method Systematic study of deformation theory, focusing on infinitesimal deformations and their parametrization via eigenfunctions of the basic Laplacian.
result Infinitesimal Einstein deformations of coincide with infinitesimal deformations of .
Study nearly parallel G2-structures with torus symmetry using multi-moment maps.
problem Characterize and construct nearly parallel G2-structures with torus symmetry.
method Use multi-moment map techniques and analyze the geometry of the base spaces.
result Locally, the construction may produce examples with four-torus symmetry.
Proves local solvability for -structures with Poisson equations.
problem Local solvability of Poisson equations for -structures.
method Proves local solvability for -structures with Poisson equations.
result Local solvability of Poisson equations for closed -structures.
Review of recent G₂-structures on 7D manifolds.
problem Constructing examples of closed G₂-structures.
method Discussion of recent results and examples.
result Review of recent findings on G₂-structures.
Neural and numerical methods approximate G2-structures on Calabi-Yau manifolds.
problem Approximating G2-structures on Calabi-Yau manifolds.
method Three stages: Ricci-flat metric computation, numerical approximations, and neural architecture training.
result Validated neural architecture for learning G2-structures and their metrics.
Geometric flows study nearly parallel G2-structures on 3-Sasakian 7-manifolds.
problem Analyzing geometric flows of G2-structures on 3-Sasakian manifolds.
method Study of Laplacian flow and Laplacian coflow of G2-structures on 3-Sasakian manifolds.
result Distinct behavior of flows, notably regarding stability of nearly parallel G2-structures.
The connection between Hitchin's stable forms and vector cross products is observed. Using this correspondence, we construct new examples of non-Kahler Calabi-Yau 3-folds and manifolds with G2-structure of class W3. We also generalize and refine results of Calabi and Gray in the paracomplex setting.
We study the moduli space of SU(3) structure manifolds X that form the internal compact spaces in four-dimensional N=1/2 domain wall solutions of heterotic supergravity with flux. Together with the direction perpendicular to the four-dimensional domain wall, X forms a non-compact 7-manifold Y with torsionful G2 structu…
Study coclosed G2-structures on SU(2)²-invariant manifolds.
problem Existence and classification of coclosed G2-structures on specific manifolds.
method Analysis of half-flat SU(3)-structures and boundary conditions.
result Existence of coclosed G2-structures on R⁴ × S³, no such structures on S⁴ × S³.
Study on flows of G2-structures on contact Calabi-Yau 7-manifolds.
problem Analyzing the behavior of G2-structures under Laplacian and Hitchin flows.
method Investigation of Laplacian and Hitchin flows on contact Calabi-Yau 7-manifolds.
result Ancient solutions of the Laplacian flow with finite time Type I singularity and immortal solutions of the Laplacian coflow with infinite time Type IIb singularity.
Study on Lie groups with exact G2 structures and closed eigenforms.
problem Characterizing Lie groups with exact G2 structures and closed eigenforms.
method Analyzing Lie algebras and structures on Lie groups.
result Compact solvmanifolds cannot have invariant exact G2 structures.
Study of associative submanifolds in Berger space SO(5)/SO(3).
problem Characterizing and classifying associative submanifolds in Berger space.
method Geometric correspondence with pseudo-holomorphic curves, analysis of special Gauss maps.
result Existence of infinitely many topological types of compact associative 3-folds.
Study on G2 structures with torsion and existence of solutions.
problem Existence and properties of strong G2-structures with torsion.
method Investigation of the twisted G2 equation and analysis of invariant structures.
result Non-existence of non-trivial solutions on compact solvmanifolds.
We characterize compact eight-manifolds M which arise as internal spaces in N=1 flux compactifications of M-theory down to AdS3 using the theory of foliations, for the case when the internal part of the supersymmetry generator is everywhere non-chiral. We prove that specifying such a supersymmetric background is equiva…
In this expository article we discuss the relations between Sasakian geometry, reduced holonomy and supersymmetry. It is well known that the Riemannian manifolds other than the round spheres that admit real Killing spinors are precisely Sasaki-Einstein manifolds, 7-manifolds with a nearly parallel G2 structure, and nea…