The paper describes complex structures on Oeljeklaus-Toma manifolds.
problem Investigating different complex structures on Oeljeklaus-Toma manifolds.
method Algebraic description of biholomorphic manifolds obtained by conjugating complex embeddings.
result Compact non-Kähler manifolds with multiple rigid complex structures.
In this paper, we show the existence of (co-oriented) contact structures on certain classes of G2-manifolds, and that these two structures are compatible in certain ways. Moreover, we prove that any seven-manifold with a spin structure (and so any manifold with G2-structure) admits an almost contact structure. We…
Study describes conformal product structures on compact Kähler manifolds.
problem Characterizing compact Kähler manifolds with conformal product structures.
method Geometric description of conformal product structures on compact Kähler manifolds.
result Geometric characterization of compact Kähler manifolds with conformal product structures.
Study on 3D manifolds with circulant structures and their properties.
problem Characterizing 3D almost Einstein manifolds with circulant structures.
method Analyzing the curvature tensor and properties of the Levi-Civita connection.
result Determined geometric characteristics and examples of such manifolds.
Study angle structures on pseudo 3-manifolds, proving existence for some cases.
problem Determining if hyperbolic 3-manifolds can have angle structures.
method Examined triangulated pseudo 3-manifolds with area-curvature angle structures, establishing sufficient and necessary conditions.
result Compact hyperbolic 3-manifolds with totally geodesic boundary can have angle structures.
New proofs for complex Hopf manifolds using geometric structures.
problem Proving properties of complex Hopf manifolds.
method Constructing integrable holomorphic G-structures and flat holomorphic Cartan geometries.
result Provided a new proof of flat holomorphic Cartan geometries on complex Hopf manifolds.
The study explores new metric structures on manifolds, linking them to Einstein metrics.
problem Characterizing and understanding weak K-contact manifolds and their properties.
method Analyzing weak K-contact manifolds and their properties, including the parallel Ricci tensor and generalized Ricci soliton structures.
result Sufficient conditions for weak K-contact manifolds with specific properties to be Einstein manifolds.
Study the structure of equidistant decompositions in manifolds.
problem Geometric and topological structure of equidistant decompositions.
method Investigation of Riemannian manifolds.
result Detailed understanding of equidistant decompositions.
Defines Jacobi-Koszul-Vinberg structures on Jacobi-left-symmetric algebroids.
problem No specific problem stated; focuses on new structure definition.
method Definition and properties of Jacobi-Koszul-Vinberg structures on Jacobi-left-symmetric algebroids.
result Defines a new structure on Jacobi-left-symmetric algebroids.
New proof shows 4-manifolds can't support complex structures.
problem Proving 4-manifolds can't support complex structures.
method New proof of complex structure non-existence for real-hyperbolic 4-manifolds.
result Extended graph 4-manifolds with positive Euler characteristic cannot support a complex structure.
Introduces holed cone structures to generalize cone structures on 3-manifolds.
problem Generalizing cone structures to 3-manifolds with irreducible holonomy representations.
method Introduces holed cone structures and considers their deformation space.
result The deformation space of holed cone structures is a covering space of the character variety.
We study topological properties of log-symplectic structures and produce examples of compact manifolds with such structures. Notably we show that several symplectic manifolds do not admit log-symplectic structures and several log-symplectic manifolds do not admit symplectic structures, for example #m CP^2 # n bar(CP^2)…
Spin-structures on real Bott manifolds with Kähler structure are characterized.
problem Existence of spin-structures on real Bott manifolds with Kähler structure.
method Ishida characterization and techniques from \cite{PS16} using characteristic classes.
result Necessary and sufficient condition for the existence of spin-structures on M. Exotic hypercomplex structures on a torus are proven to not exist.
problem Existence of exotic hypercomplex structures on a torus.
method Classification of complete flat affine structures on real tori using the Obata connection.
result Exotic hypercomplex structures on a torus do not exist.
The paper defines a new connection on sub-Riemannian manifolds and explores conditions for almost quasi-Sasakian manifolds to be Einstein.
problem Exploring new connections and conditions for specific types of manifolds.
method Defining a skew-symmetric connection and studying its properties on sub-Riemannian manifolds, and examining conditions for almost quasi-Sasakian manifolds to be Einstein.
result Sufficient conditions are found for an almost quasi-Sasakian manifold to be an Einstein manifold.
Condition found for spinc structures on a specific type of manifold.
problem Existence of spinc structures on real Bott manifolds.
method Provided a necessary and sufficient condition.
result Condition for spinc structures on real Bott manifolds established.
Study on compact manifolds for exact G2-Structures without additional constraints.
problem Whether compact 7-manifolds support exact G2-Structures. method Investigate exact G2-Structures on compact manifolds, considering relationships with other conditions. result Initiate a study on exact G2-Structures on compact manifolds without additional constraints. Study on harmonicity of complex structure on product of trans-Sasakian manifolds.
problem Investigating harmonicity of complex structure on product of trans-Sasakian manifolds.
method Analysis of Levi-Civita connection and conditions for harmonicity on product manifold.
result Conditions for harmonicity of complex structure on product manifold of trans-Sasakian manifolds.
Study on 3D manifolds with specific tensor structures and their properties.
problem Characterizing 3D Riemannian manifolds with tensor structures.
method Investigation of locally conformal Riemannian product manifolds and their associated structures.
result Conditions for additional structures to be parallel and properties of almost Einstein and Einstein manifolds.
No Hantzsche-Wendt manifolds over 3D admit spin^c structures.
problem Existence of spin^c structures on Hantzsche-Wendt manifolds.
method Combinatorial description of Stiefel-Whitney classes for closed flat manifolds.
result No Hantzsche-Wendt manifold of dimension greater than three admits a spin^c structure.
The article gives a necessary and sufficient condition for a Frobenius manifold to be a CDV-structure. We show that there exists a positive definite CDV-structure on any semi-simple Frobenius manifold. We also compare three natural connections on a CDV-structure and conclude that the underlying Hermitian manifold of a …
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. Class I CR manifolds have initial G-structure a certain 4-dimensional subgroup of GL_3(C). Class II CR manifolds have initial G-structure a certain 10-dimensional subgroup of GL_4(C). Class III-1 CR manifolds have initial G-structure a certain 10-dimensional subgroup of GL_5(C). Class III-2 CR manifolds have initial G-…
Positive injectivity radius for manifolds with Lie structure at infinity.
problem Injectivity radius positivity for manifolds with specific boundary conditions.
method Lie groupoids to prove injectivity radius positivity.
result Injectivity radius is positive for manifolds with Lie structure at infinity.
Complex structures found on product of Sasakian manifolds.
problem Exploring complex structures on the product of two Sasakian manifolds.
method Examined a family of complex structures on the product of two compact Sasakian manifolds, computed Dolbeault cohomology groups.
result None of the complex structures in the family admit compatible locally conformally Kähler metrics if both Sasakian manifolds are of dimension greater than 1.
Study pseudo-harmonic maps on Weyl manifolds.
problem Characterize conditions for a Hermitian-Weyl manifold's complex structure to be a pseudo-harmonic map.
method Investigate geometric conditions for pseudo-harmonic maps in the context of Hermitian-Weyl manifolds.
result Find conditions for the complex structure to be a pseudo-harmonic map under specific dimensions or conformal structures.
The paper introduces a new form on Lie algebroids over multisymplectic manifolds.
problem Higher generalizations of Poisson structures and momentum maps.
method Introducing a compatible E-n-form on Lie algebroids.
result The introduced form satisfies a compatibility condition with Lie algebroid and multisymplectic structures.
Holomorphic structures on Oeljeklaus-Toma manifolds are shown to be locally homogeneous.
problem Characterizing holomorphic structures on Oeljeklaus-Toma manifolds.
method Proving local homogeneity for various holomorphic geometric structures.
result Holomorphic geometric structures on Oeljeklaus-Toma manifolds are locally homogeneous.
In this paper, we develop holomorphic Jacobi structures. Holomorphic Jacobi manifolds are in one-to-one correspondence with certain homogeneous holomorphic Poisson manifolds. Furthermore, holomorphic Poisson manifolds can be looked at as special cases of holomorphic Jacobi manifolds. We show that holomorphic Jacobi str…
Study shows Einstein structures on 4-manifolds are rigid.
problem Rigidity of Einstein structures in four dimensions.
method Examined deformations of the round four-sphere and analyzed self-dual structure of Einstein manifolds.
result Any deviation from the standard metric of the round four-sphere breaks the Einstein condition.
Study tight contact structures on specific 3-manifolds.
problem Counting and constructing tight contact structures on plumbed 3-manifolds.
method Algorithm to construct stein diagrams for tight structures without Giroux torsion.
result Explicit algorithm to construct tight contact structures.
The paper explores p-Kähler structures on complex manifolds and their properties.
problem Addressing conjectures on specific types of complex manifolds and their structures.
method Analyzing (n−2)-Kähler nilmanifolds and holomorphically parallelizable nilmanifolds, deriving necessary conditions for smooth curves of p-Kähler structures, studying cohomology classes, and providing examples. result Derives necessary conditions for the existence of smooth curves of p-Kähler structures and provides examples of compact complex manifolds with p-Kähler structures. Symplectic and Poisson structures proved for information geometry's Frobenius manifold.
problem Connecting disconnected theories in information geometry.
method Proving symplectic and Poisson structures on the Frobenius manifold.
result Established a bridge between Vinberg, Souriau, and Koszul's theories.
The paper classifies complex Dirac structures on flag manifolds.
problem Classifying invariant complex Dirac structures on flag manifolds.
method Described using roots of the Lie algebra and classified under B-transformations. result All invariant complex Dirac structures with constant real index on a maximal flag manifold are described.
The study provides obstructions and examples for p-symplectic structures on complex manifolds.
problem Obstructing the existence of p-symplectic structures on compact complex manifolds. method Analyzing properties of (p,p)-transverse forms and 2p-forms. result Found obstructions and examples of p-symplectic structures on compact complex manifolds. Study of affine and projective structures on foliated complex manifolds.
problem Formalizing and analyzing affine and projective structures on foliations.
method Formalizing concepts, providing local normal forms, proving index formulae, classifying structures.
result Compact algebraic manifolds of even dimension do not admit foliated projective structures.
Hydrodynamic structures linked to F-manifolds.
problem Hydrodynamic equations and their Hamiltonian structures.
method Introducing generalised (bi-)Hamiltonian structures and associating them with (bi-)flat F-manifolds.
result Generalised (bi-)Hamiltonian structures of hydrodynamic type can be associated with (bi-)flat F-manifolds.
In this paper we give some examples of almost para-hyperhermitian structures on the tangent bundle of an almost product manifold, on the product manifold M×R, where M is a manifold endowed with a mixed 3-structure and on the circle bundle over a manifold with a mixed 3-structure.
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.
Study Godbillon-Vey class for regular Jacobi foliations.
problem Characterizing foliations in Jacobi manifolds.
method Explicitly defined and computed Godbillon-Vey class for regular foliations.
result Expressed Godbillon-Vey class in terms of Jacobi structures.
Every real 3-manifold can be turned into a real contact structure.
problem Understanding and constructing real contact structures on real 3-manifolds.
method Surgery along invariant knots and contact operations starting from the standard real S3. result Every real 3-manifold admits a real contact structure.
It is introduced a differentiable manifold with almost contact 3-structure which consists of an almost contact metric structure and two almost contact B-metric structures. The product of this manifold and a real line is an almost hypercomplex manifold with Hermitian-Norden metrics. It is proven that the introduced mani…
The paper studies geometric structures on tangent and sphere bundles over statistical manifolds.
problem Defining geometric structures on tangent and sphere bundles over statistical manifolds.
method Using a statistical structure (g,abla), the paper defines a Riemannian structure on the tangent bundle and derives expressions for various curvatures. result Basic formulas for the geometry of sphere bundles are established, and rigidity results are proved for these structures.
For an almost contact metric manifold N, we find conditions for which either the total space of an S1-bundle over N or the Riemannian cone over N admits a strong Kähler with torsion (SKT) structure. In this way we construct new 6-dimensional SKT manifolds. Moreover, we study the geometric structure induced on …
Study on G2-structures with negative Ricci curvature on closed and noncompact manifolds.
problem Existence and properties of closed G2-structures with negative Ricci curvature. method Analyzing existence and nonexistence of closed G2-structures with negative Ricci curvature on closed and noncompact manifolds. result No closed manifold admits a closed G2-structure with negative Ricci curvature. For noncompact manifolds, restrictions on lengths of geodesics are found. 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…
The existence of some complex geometrical structures on a compact manifold such as complex structures, Kaehler (pseudo-Kaehler) structures often impose certain restrictions on its underling topological or differentiable manifold. In this article we survey recent developments in the study of the existence, classificatio…
The paper explores automorphism groups of parabolic structures on aspherical manifolds.
problem Characterizing the automorphism groups of parabolic structures on aspherical manifolds.
method Analyzing properties of closed aspherical parabolic ${\sfG}$-manifolds and their automorphism groups.
result Certain parabolic ${\sfG}$-structures impose strong restrictions on the topology of compact aspherical manifolds.