Study para-Kähler Lie algebroids and generalize pseudo-Hessian structures.
problem Generalize para-Kähler Lie algebras to Lie algebroids.
method Study exact para-Kähler Lie algebroids and pseudo-Hessian manifolds.
result Orbits of action on dual space are pseudo-Hessian manifolds.
Study of contravariant pseudo-Hessian manifolds and their Poisson structures.
problem Understanding properties of contravariant pseudo-Hessian manifolds.
method Investigation of flat connections and symmetric bivector fields satisfying a contravariant Codazzi equation.
result Association of a Poisson tensor to contravariant pseudo-Hessian manifolds.
The paper defines left-symmetric bialgebroids and their Manin triples.
problem No specific problem stated; focuses on definitions and constructions.
method Introduced left-symmetric bialgebroids and Manin triples, constructed from pseudo-Hessian manifolds.
result Established a relation between Maurer-Cartan type equations and Dirac structures.
Develops a bialgebra theory for post-Lie algebras using geometric interpretations and bilinear forms.
problem Characterizing and understanding post-Lie algebras and their associated structures.
method Utilizes Manin triples and generalized Hessian Lie groups to define and characterize post-Lie algebras with nondegenerate symmetric invariant bilinear forms.
result Establishes a bialgebra theory for post-Lie algebras via the Manin triple approach, including new algebraic structures like pp-post-Lie algebras.
The paper introduces new structures for left-symmetric algebroids.
problem Developing new mathematical structures for left-symmetric algebroids.
method Introducing Koszul-Vinberg-Nijenhuis structures and related concepts.
result Koszul-Vinberg-Nijenhuis structures provide a hierarchy of structures.
Study submanifolds in Koszul-Vinberg geometry, a blend of Poisson and pseudo-Riemannian structures.
problem Understanding submanifolds in Koszul-Vinberg geometry.
method Analyzing submanifolds within the framework of Koszul-Vinberg manifolds, considering developments in Poisson submanifolds.
result Developed methods to analyze submanifolds in this geometric setting.
Unified method for studying geometric structures of pseudo-Riemannian manifolds.
problem Study of geometric structures on pseudo-Riemannian manifolds.
method Introduce homogeneous pairs and use conformal transformations to analyze sectional curvature and geodesics.
result Unified approach to studying geometric structures on various moduli spaces.
Defines a new Poisson structure for generalized Sasakian spaces.
problem No specific problem stated; focuses on new structure definition.
method Defines a canonical Poisson structure on generalized contact metric spaces.
result Shows distinction between generalized Sasakian and coKähler structures.
Study on types of generalized hypercomplex structures on tori and Kodaira-Thurston surface.
problem Characterizing types of generalized hypercomplex structures.
method Analysis of S2-family of generalized complex structures and study of twistor spaces. result Existence of generalized hypercomplex structures on 4n-dimensional tori with non-maximal types. Conditions for hypersurfaces of generalized Kähler manifolds to have specific structures are established.
problem Conditions for hypersurfaces of generalized Kähler manifolds to have specific structures.
method Established conditions for induced generalized metric F structure to be a generalized CRFK structure.
result Characterization of binormality and examples of binormal structures.
We introduce generalized almost contact structures which admit the B-field transformations on odd dimensional manifolds. We provide definition of generalized Sasakain structures from the view point of the generalized almost contact structures. We obtain a generalized Sasakian structure on a non-compact manifold which…
Introduces semi-abelian generalized complex structures.
problem Deformation theory of abelian complex structures.
method Definition and examples of semi-abelian generalized complex structures.
result Illustration of new concept with examples.
Characterizes structures on generalized tangent bundles and CRF-structures.
problem Understanding structures on generalized tangent bundles and CRF-structures.
method Equivalent characterizations and spinor formalism for CRF-structures.
result Characterization of generalized complex manifolds as products and infinitesimal deformations of CRF-structures.
We study integrability of generalized almost contact structures, and find conditions under which the main associated maximal isotropic vector bundles form Lie bialgebroids. These conditions differentiate the concept of generalized contact structures from a counterpart of generalized complex structures on odd-dimensiona…
Regular Jacobi structures on line bundles lead to generalized contact bundles.
problem Understanding the relationship between Jacobi structures and generalized contact bundles.
method Investigating weakly regular Jacobi structures and their transverse complex structures.
result Conditions for a pair of a regular Jacobi structure and a transverse complex structure to form a generalized contact structure.
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.
Characterizes integrability of generalized structures on Courant algebroids.
problem Integrability of generalized structures on Courant algebroids.
method Characterization via torsion-free generalized connections and Dirac generating operators.
result Criterion for integrability of generalized almost Hermitian structures and hyper-Hermitian structures.
A new method generates graphs with hierarchical structures.
problem Generating graphs with natural hierarchical structures.
method Recursively generates community structures at multiple resolutions, parallel generation of all sub-structures.
result Improves generative performance on multiple graph datasets.
New geometric structures on 3-manifolds discovered and proven for all closed orientable ones.
problem Existence of specific geometric structures on 3-manifolds.
method Generalized geometry and surgery techniques.
result Any closed orientable 3-manifold admits a B3-generalized complex structure. No direct generalized complex structure can be induced from S6's nearly Kähler structure.
problem Existence of generalized complex structures on S6. method Defined integrability in terms of the Dorfman bracket and studied S6's nearly Kähler structure. result No generalized complex structure can be induced from S6's nearly Kähler structure. Defines generalized Sasakian structures in contact geometry.
problem Understanding k-contact manifolds and CR manifolds.
method Introduced generalized Sasakian structures and proved their equivalence to Sasakian structures.
result k-contact manifolds are generalized Sasakian if and only if they are classically Sasakian.
Proposes a deep structured generative model for complex scenes.
problem Difficulty in generating images with complicated structures.
method Boosts GANs with structure information encoded by a stochastic and-or graph (sAOG).
result Successfully captures and generates images with complex structures.
Study geometric structures and their interactions under different metrics.
problem Understanding interactions between geometric structures under various metrics.
method Analyzing generalized polynomial structures and their behavior under different metrics on the generalized tangent bundle.
result Showed the commutation or anti-commutation of generalized polynomial structures forming triple structures.
We define a generalized almost para-Hermitian structure to be a commuting pair (F,J) of a generalized almost para-complex structure and a generalized almost complex structure with an adequate non-degeneracy condition. If the two structures are integrable the pair is called a generalized para-Kähle…
We characterize the Dirac structures that are parallel with respect to Gualtieri's canonical connection of a generalized Riemannian metric. On the other hand, we discuss Dirac structures that are images of generalized tangent structures. These structures turn out to be Dirac structures that, if seen as Lie algebroids, …
We construct a generalization of Courant algebroids which are classified by the third cohomology group H3(A,V), where A is a Lie Algebroid, and V is an A-module. We see that both Courant algebroids and E1(M) structures are examples of them. Finally we introduce generalized CR structures on a manif…
In this paper we define the notion of a generalized coKähler structure and prove that the product M1×M2 of generalized contact metric manifolds (Mi,Φi,E±,i,Gi), i=1,2, where M1×M2 is endowed with the product generalized complex structure induced from Φ1 and Φ2, is gener…
Study metallic structures on generalized tangent bundles.
problem Properties of generalized metallic structures.
method Generalized geometry and suitable connections.
result Conditions for integrability of generalized metallic structures.
Eta-Einstein and (κ,μ)-structures studied in dimension 3.
problem Characterizing and understanding Eta-Einstein and (κ,μ)-structures in 3D. method Analyzing closed manifolds and constructing examples.
result Almost regular Eta-Einstein structures not D-homothetic to Einstein structures exist.
Let X be a compact Kahler manifold with a non-trivial holomorphic Poisson structure. Then there exist deformations of non-trivial generalized Kahler structures with one pure spinor on X. We prove that every Poisson submanifold of X is a generalized Kahler submanifold with respect to the deformed generalized Kahler stru…
Study integrability of specific geometric structures on odd Courant algebroids.
problem Characterize integrability of B_n-generalized structures on odd exact Courant algebroids.
method Characterize integrability in terms of existence of adapted generalized connections.
result Describe affine spaces of adapted generalized connections for integrable structures.
The paper explores new structures in generalized geometry and their relationships.
problem Exploring new structures in generalized geometry.
method Discussing the relation between VB-Courant algebroids and E-Courant algebroids, introducing generalized complex structures, and studying their properties.
result Generalized complex structures on E-Courant algebroids unify different types of structures.
A generalized F-structure is a complex, isotropic subbundle E of TcM⊕Tc∗M ($T_cM=TM\otimes_{\mathds{R}}\mathds{C}$ and the metric is defined by pairing) such that E∩Eˉ⊥=0. If E is also closed by the Courant bracket, E is a generalized CRF-structure. We show that a generalized F-structur…
Classifies complex Dirac structures with invariants and local structure.
problem Classifying complex Dirac structures.
method Introducing invariants (order, type), proving existence and splitting theorems.
result Pointwise classification and local structure of complex Dirac structures.
The study characterizes real flag manifolds with invariant generalized almost complex structures.
problem Characterizing real flag manifolds with invariant generalized almost complex structures.
method Characterization through invariant B-transformations and classification of structures. result No GM2-maximal real flag manifolds admit integrable invariant generalized almost complex structures. The paper finds a criterion for generating commuting pairs of structures.
problem Defining commuting pairs of generalized structures on product spaces.
method Proves a theorem for generating commuting pairs of generalized almost complex structures.
result Simple criterion for generating commuting pairs of generalized structures.
New method constructs stable generalized complex structures.
problem Creating stable generalized complex structures.
method Developed Gompf-Thurston symplectic techniques adapted to Lie algebroids, used to construct stable structures from log-symplectic structures.
result Introduced boundary Lefschetz fibrations to obtain stable structures from genus one Lefschetz fibrations.
Extends corner structure study to general case, constructs normal Trans-Sasakian structures.
problem Extending corner structure study to general case without conditions.
method Extends corner structure to general case, constructs Trans-Sasakian structures from non-normal corner structures.
result Constructs normal Trans-Sasakian structures from non-normal corner structures.
An (I,J,K)-generalized Finsler structure on a 3-manifold is a generalization of a Finslerian structure, introduced in order to separate and clarify the local and global aspects in Finsler geometry making use of the Cartan's method of exterior differential systems. In this paper, we show that there is a close relation b…
In this paper we obtain a stability theorem of generalized Kahler structures with one pure spinor under small deformations of generalized complex structures. (This is analogous to the stability theorem of Kahler manifolds by Kodaira-Spencer.) We apply the stability theorem to a class of compact Kahler manifolds which a…
New examples of structures found using twistor space.
problem Finding non-trivial generalized paracomplex structures.
method Twistor space construction scheme.
result Non-trivial examples of generalized paracomplex structures constructed.
It is known that holomorphic Poisson structures are closely related to theories of generalized Kähler geometry and bi-Hermitian structures. In this article, we introduce quantization of holomorphic Poisson structures which are closely related to generalized Kähler structures /bi-Hermitian structures. By resulting nonco…
New method to deform generalized Kähler structures.
problem Deforming generalized Kähler structures.
method Hamiltonian deformations of holomorphic Poisson structures.
result Any generalized Kähler structure can be deformed while keeping one holomorphic Poisson structure fixed.
The notion of generalized almost paracontact structure on the generalized tangent bundle TM⊕T∗M is introduced and its properties are investigated. The case when the manifold M carries an almost paracontact metric structure is also discussed. Conditions for its transformed under a β- or a B-field transfor…
Stable generalized complex structures on certain surfaces are constant.
problem Existence of stable generalized complex structures on ruled surfaces.
method Analysis of sphere bundles over surfaces of genus ≥2.
result Stable generalized complex structures on these surfaces are of constant type.
New structures allow for self-crossing singularities, leading to new families of stable generalized complex manifolds.
problem Stable generalized complex structures in higher dimensions with self-crossing singularities.
method Extending stable generalized complex structures to include anticanonical sections with normal self-crossings.
result Construction of large families of stable generalized complex manifolds in four dimensions.
New geometric structure on surfaces generalizing complex structures.
problem Defining and analyzing new geometric structures on surfaces.
method Using the punctual Hilbert scheme of the plane to define higher complex structures.
result Moduli space of higher complex structures is a generalization of Teichmüller space and conjecturally isomorphic to Hitchin's component.
On a smooth manifold M, generalized complex (generalized paracomplex) structures provide a notion of interpolation between complex (paracomplex) and symplectic structures on M. Given a complex manifold (M,j), we define six families of distinguished generalized complex or paracomplex structures on M. Each one of them in…