The article classifies G2-structures with conformally flat metrics.
problem Identifying G2-structures with specific geometric properties.
method Classifying closed G2-structures with conformally flat metrics.
result Any closed G2-structure with conformally flat metric is locally equivalent to one of three explicit examples.
G2-manifolds with a cohomogeneity-one action of a compact Lie group G are studied. For G simple, all solutions with holonomy G2 and weak holonomy G2 are classified. The holonomy G2 solutions are necessarily Ricci-flat and there is a one-parameter family with SU(3)-symmetry. The weak holonomy G2 solutions are Einstein o…
Study on G2-structures on solvmanifolds, focusing on Laplacian solitons and Ricci pinching.
problem Existence and interplay of Laplacian solitons and Ricci pinched G2-structures on solvmanifolds.
method Exploration of left-invariant G2-structures on solvable Lie groups, analysis of Ricci pinching properties.
result Obtained Ricci pinching properties and extremal values for G2-structures on solvmanifolds.
The paper explores Einstein warped G2 and Spin(7) manifolds.
problem Understanding Einstein warped G2 and Spin(7) manifolds.
method Warped G2 and Spin(7) structures with Einstein fiber are studied.
result Explicit descriptions of torsion forms for these structures.
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…
New G2-structures found on Lie groups with strong structural conditions.
problem Existence and structure of extremally Ricci pinched G2-structures on Lie groups.
method Strong structural conditions on Lie algebra, deformation and rigidity studies.
result Three new examples of extremally Ricci pinched G2-structures, all steady Laplacian solitons.
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.
Study G2-structures with non-closed four-forms under local conformal conditions.
problem Characterizing G2-structures with non-closed four-forms under local conformal conditions.
method Investigated G2-structures with non-closed four-forms and non-zero Lee form, focusing on Lie algebra properties.
result Found solutions for G2-structures in preposition1 and theorem1.
Study G2-structures in N=1 AdS4 solutions of M-theory.
problem Characterize G2-structures in N=1 AdS4 solutions of M-theory.
method Reformulate Killing spinor equations using octonion bundle structure; study G2-structures and their torsion.
result Define single complexified G2-structure or two real G2-structures on M.
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…
New 7D Lie algebras with G2-structures created from 6D Lie algebras.
problem Creating new Lie algebras with specific geometric structures.
method Starting from 6D Lie algebras with symplectic structures, constructing 7D Lie algebras with G2-structures.
result Described all 7D Lie algebras with closed G2-structures from 6D solvable Lie algebras.
Classification of G2-structures on Lie groups with Ricci pinched conditions.
problem Classifying G2-structures on Lie groups under specific geometric conditions.
method Complete classification of left-invariant closed G2-structures on Lie groups, extremally Ricci pinched, up to equivalence and scaling.
result Five distinct G2-structures on five different completely solvable Lie groups, with one unimodular case being exact.
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.
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…
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 …
Let M be a G2-manifold. We consider an almost CR-structure on the sphere bundle of unit tangent vectors on M, called the CR twistor space. This CR-structure is integrable if and only if M is a holonomy G2 manifold. We interpret G2-instanton bundles as CR-holomorphic bundles on its twistor space.
We introduce the concept of G2(2)-structure on an orientable 3-manifold M using the setting of generalized geometry of type Bn, study their local deformation by making use of a Moser-type argument, and give a description of the cone of G2(2)-structures.
Defines octonion bundles and connections related to G2-structures.
problem Exploring octonion bundles and connections in G2-manifolds.
method Using G2-structures and octonion bundles, defining metric-compatible covariant derivatives and torsions.
result Critical points of the energy functional correspond to divergence-free torsion, an octonionic analog of the Coulomb gauge.
A flying saucer's flight is mathematically modeled with geometric structures.
problem Modeling the complex flight maneuvers of a flying saucer.
method Imposing nonlinear restrictions on the saucer's velocity to define geometric structures.
result Three types of flat parabolic geometries are defined for the saucer's configuration space.
Geodesic concavity and hypersymplectic structures in G2-structures space.
problem Analyzing the geodesic concavity and hypersymplectic structures in the space of closed G2-structures. method Utilising the geodesic constructed in the previous article, we show geodesic concavity and decrease in length of G2 Laplacian flow. result Hitchin's volume functional is geodesically concave and the G2 Laplacian flow decreases the length. 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 …
New examples of Laplacian solitons found on solvable Lie groups.
problem Finding closed Laplacian solitons with finite-time singularities.
method Left-invariant G2-structures on solvable Lie groups.
result First examples of closed Laplacian solitons with finite-time singularities.
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.
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…
New formulas for mean curvature of submanifolds in geometries with torsion.
problem Characterizing submanifolds in geometries with intrinsic torsion.
method Deriving formulas for mean curvature of associative, coassociative, and Cayley submanifolds.
result New obstructions to the local existence of coassociative 4-folds in G2-structures with torsion.
We study the moduli space of torsion-free G2-structures on a fixed compact manifold, and define its associated universal intermediate Jacobian J. We define the Yukawa coupling and relate it to a natural pseudo-Kahler structure on J. We consider natural Chern-Simons type functionals, whose critical points give associati…
This article studies the geometry of moduli spaces of G2-manifolds, associative cycles, coassociative cycles and deformed Donaldson-Thomas bundles. We introduce natural symmetric cubic tensors and differential forms on these moduli spaces. They correspond to Yukawa couplings and correlation functions in M-theory. We ex…
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 find G2-manifolds with specific asymptotic properties.
problem Existence and structure of G2-manifolds with ALC asymptotics.
method Robust Fredholm theory for ALC spaces, proving existence and rigidity results.
result Existence of a G2-analogue of the Atiyah-Hitchin metric and good moduli theory for ALC G2-holonomy metrics.
We modify the Laplacian coflow of co-closed G2-structures - dtdψ=Δψ where ψ is the closed dual 4-form of a G2-structure φ. 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…
Constructs deformed G2-instantons on specific G2-manifolds.
problem Finding non-trivial deformed G2-instantons on G2-manifolds.
method Explicit construction of deformed G2-instantons on specified manifolds.
result First non-trivial examples of deformed G2-instantons on G2-manifolds.
We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) or G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For virtual knots and graphs these invariants provide new graphical information that allo…
We show how certain diffeomorphism-invariant functionals on differential forms in dimensions 6,7 and 8 generate in a natural way special geometrical structures in these dimensions: metrics of holonomy G2 and Spin(7), metrics with weak holonomy SU(3) and G2, and a new and unexplored example in dimension 8. The general f…
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 …
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.
Study finite singularities in G2 structure flows using Shi-type estimates.
problem Finite time singularities in G2 structure flows.
method Extend Shi-type estimates to G2 structure flows and prove κ-non-collapsing theorem.
result Prove finite time singularities of G2 structure flows.
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.
The heterotic G2 system is solved on Calabi-Yau 3-orbifolds.
problem Solving the heterotic G2 system on Calabi-Yau 3-orbifolds.
method Adjusting circle bundles and powers of α' to obtain cocalibrated G2 structures.
result Non-trivial solutions to the heterotic G2 system are found.
Classifies 7D manifolds with specific G2-structures and automorphisms.
problem Classifying 7D manifolds with closed G2-structures and transitive automorphisms.
method Complete classification through detailed analysis of manifold properties and group actions.
result The center of the automorphism group is one-dimensional and the manifold is a product of a flat factor and a homogeneous six-manifold with a specific SU(3)-structure.
New nearly parallel G2-structures found on S³×R⁴.
problem Existence of nearly parallel G2-structures with large symmetry.
method One-parameter family of nearly parallel G2-structures on S³×R⁴ with SU(2)³ symmetry.
result One-parameter family connects two locally homogeneous nearly parallel G2-structures.
New geometric forms and manifolds discovered using stable forms and vector cross products.
problem Understanding geometric structures and their applications.
method Using Hitchin's stable forms and vector cross products to construct new geometric examples.
result New non-Kahler Calabi-Yau 3-folds and manifolds with G2-structure of class W3.
Study deformations of G2-instantons on nearly G2 manifolds.
problem Deformations of G2-instantons on nearly G2 manifolds.
method Formulated in terms of spinors and Dirac operators, proved isomorphism of infinitesimal deformations to kernel of an elliptic operator.
result Proved abelian instantons are rigid and described the deformation space of the canonical connection on specific nearly G2 manifolds.
Study finds closed G2-structures on non-solvable Lie groups.
problem Existence of closed G2-structures on non-solvable Lie groups.
method Investigation of left-invariant closed G2-structures on specific Lie groups.
result First examples of closed G2-structures on non-solvable Lie groups.
New generalized G2-structures derived from T-duality.
problem Deriving new G2-structures from existing ones.
method T-duality construction applied to classical G2-structures.
result Obtained integrable generalized G2-structures with non-zero three-form.
Study on G2-manifolds shows automorphism group is abelian.
problem Understanding automorphism groups of G2-manifolds.
method Analyzing compact 7-manifolds with closed non-parallel G2-structures.
result The identity component of automorphism group is abelian with dimension bounded.