Study of trigonal curves in abelian differentials with specific divisor properties.
problem Characterizing locally closed subspaces of abelian differentials.
method Using linear systems on Segre-Hirzebruch surfaces to describe orbifold structure and orbifold fundamental groups.
result Identified the orbifold fundamental group of a specific subspace as a quotient of the Artin group of type E8. Study introduces new indices for special fibered surfaces of genus 4.
problem Characterizing relatively minimal fibered surfaces of genus 4.
method Introduces Horikawa index and local signature for surfaces with unique trigonal structure.
result New indices provide insights into the structure of surfaces.
Let K be a link of Conway's normal form C(m), m≥0, or C(m,n) with $mn\textgreater{}0$, and let D be a trigonal diagram of K. We show that it is possible to transform D into an alternating trigonal diagram, so that all intermediate diagrams remain trigonal, and the number of crossings never increases.
We show that some of centeral fibers of degenerations of hyperelliptic curves are realized as those trigonal curves. In particular, any hyperelliptic curve can be the central fiber of a degeneration of trigonal curves.
In this paper, we study trigonal minimal surfaces in flat tori. First, we show a topological obstruction similar to that of hyperelliptic minimal surfaces. Actually, the genus of trigonal minimal surface in 3-dimensional flat torus must be 1 (mod 3). Next, we construct an explicit example in the higher codimensional ca…
A cumbersome hypothesis for Viro patchworking of real algebraic curves is the convexity of the given subdivision. It is an open question in general to know whether the convexity is necessary. In the case of trigonal curves we interpret Viro method in terms of dessins d'enfants. Gluing the dessins d'enfants in a coheren…
Consider the moduli space Mg of Riemann surfaces of genus g≥2 and its Deligne-Munford compactification Mgˉ. We are interested in the branch locus Bg for g>2, i.e., the subset of Mg consisting of surfaces with automorphisms. It is well-known that…
We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group Γ and use it to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of the same element. As an application, we also obtain exponentially large families of {\…
We study the degree of polynomial representations of knots. We give the lexicographic degree of all two-bridge knots with 11 or fewer crossings. First, we estimate the total degree of a lexicographic parametrisation of such a knot. This allows us to transform this problem into a study of real algebraic trigonal plane c…
Defines compatibility between Riemannian and Jacobi structures.
problem Understanding the relationship between Riemannian and Jacobi structures.
method Introducing a notion of compatibility and proving results for specific examples.
result Establishes compatibility for Poisson, contact, and locally conformally symplectic structures.
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.
Defines compatibility between Riemannian and Jacobi structures.
problem Understanding compatibility between Riemannian and Jacobi structures.
method Introducing a notion of compatibility and proving it for specific structures.
result Fundamental examples of Jacobi structures lead to known structures like Riemann-Poisson, Kenmotsu, and locally conformally Kähler.
Extending Jacobi and Riemannian compatibility to Lie algebroids.
problem Generalizing compatibility between Jacobi and Riemannian structures.
method Generalizing previous work on fundamental examples to Lie algebroids.
result Compatibility results for Lie algebroids.
Study projective and direct limits of Banach structures with connections to G-structures.
problem Understanding connections between Banach structures and G-structures. method Endow projective and direct limits with Fréchet or convenient structures and study connections.
result Illustrated examples demonstrate the study of projective and direct limits.
Defines structure constants for specific geometric structures on Lie groups.
problem No specific problem stated; focuses on defining structure constants.
method Not explicitly detailed in the abstract.
result Defines structure constants for almost complex, almost symplectic, and Riemannian structures on a local Lie group.
Study on G2∗ structures and almost para-contact structures in 7D.
problem Understanding the relation between G2∗ structures and almost para-contact structures. method Calculating projections using properties of G2∗ structures. result Determined the class of almost para-contact structures induced by G2∗ structures. 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. 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.
Bihamiltonian structures lead to tau structures for integrable hierarchies.
problem Classifying deformations of bihamiltonian structures.
method Starting from flat exact semisimple bihamiltonian structures, we derive Frobenius manifolds and tau structures.
result Deformations of the principal hierarchy with tau structures are classified.
The paper explores geometric structures on Hom-Lie groups and algebras.
problem Exploring Kähler-Norden structures on Hom-Lie groups and algebras.
method Analyzing the relationship between holomorphic Norden structures and Kähler-Norden structures on Hom-Lie groups.
result Left-invariant holomorphic Hom-Lie groups with abelian complex structures are flat.
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.
Study on submanifolds in metallic structures with new results and structures.
problem Investigating submanifolds in metallic structures.
method Analyzing hypersurfaces and products spaces, defining new structures, and expressing fundamental theorems.
result New fundamental theorems for submanifolds in metallic structures.
New metric structures generalize Sasakian and cosymplectic structures, proving rigidity and finding conditions.
problem Generalizing Sasakian and cosymplectic structures to new metric structures.
method Introducing weak structures and proving rigidity of Sasakian structures.
result Any weak Sasakian structure is homothetically equivalent to a Sasakian structure.
Study GL(2)-structures on manifolds leading to complex structures.
problem Understanding GL(2)-structures and their relation to complex structures. method Explored GL(2)-structures on differential manifolds, proving their relation to almost-complex structures and providing a canonical connection. result Established a twistor-like construction for GL(2)-geometry. 3D projective structures can be metrized with conformal structures.
problem Weyl metrizability of 3D projective structures.
method Interpreting Weyl metrizability as CR submanifolds in 7D.
result Beltrami's theorem extends to conformal structures in 3D.
Hypersymplectic structures with torsion on Lie algebroids are investigated. We show that each hypersymplectic structure with torsion on a Lie algebroid determines three Nijenhuis morphisms. From a contravariant point of view, these structures are twisted Poisson structures. We prove the existence of a one-to-one corres…
Spin-harmonic structures on low-dimensional manifolds.
problem Defining geometric structures on low-dimensional manifolds.
method Introducing spin-harmonic structures defined by harmonic unitary spinors.
result Spin-harmonic structures are equivalent to balanced Spin(7) structures in dimension 8.
Study equivalence between Hessian and Born structures on tangent bundles.
problem Equivalence between Hessian and Born structures on tangent bundles.
method Analyzing conditions for Hessian structures and integrability of induced almost Born structures.
result Conditions for equivalence between Hessian and Born structures are established.
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.
Introduces compatibility between Dirac structures and Nijenhuis tensors.
problem No specific problem stated; focuses on extending Poisson-Nijenhuis structures.
method Introduces compatibility between Dirac structures and (1,1)-tensor fields.
result Properties of Dirac-Nijenhuis structures studied, including connections and integrations.
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.
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.
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.
Introduces VB-structures for geometric objects on manifolds.
problem Properties of higher tangent lifts of geometric structures.
method Introduces weighted structures for various geometric objects on a manifold with a homogeneity structure.
result Proves interesting properties of various weighted structures.
New Clifford-Weyl structures defined on conformal manifolds.
problem Understanding the geometry of even Clifford structures on conformal manifolds.
method Introduced Clifford-Weyl structures and showed conditions for their closure.
result Weyl structures are closed except in low-dimensional cases.
The paper studies nilpotent structures in oriented neutral vector bundles and neutral hyperKähler structures.
problem Nilpotent structures in oriented neutral vector bundles and their relation to neutral hyperKähler structures.
method Defined H-nilpotent structures for Lie subgroups of SO(2n,2n) related to neutral hyperKähler structures. result Existence of complex and paracomplex structures forming neutral hyperKähler structures if and only if there exists an H-nilpotent structure. 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. New complex orthogonal structures found on a 3D manifold.
problem Finding new geometric structures on a 3D manifold.
method Constructing uniformizable complex orthogonal and projective structures.
result A manifold has multiple uniformizable complex orthogonal structures.
Parabolic almost conformally symplectic structures were introduced in the first part of this series of articles as a class of geometric structures which have an underlying almost conformally symplectic structure. If this underlying structure is conformally symplectic, then one obtains a PCS-structure. In the current ar…
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.
Unique birational structure proven on Inoue surfaces.
problem Proving uniqueness of birational structures on Inoue surfaces.
method Generalizing a result by Bruno Klingler, proving uniqueness of structures.
result The natural (Aff2(C),C2)-structure on an Inoue surface is the unique (Bir(P2),P2(C))-structure. Defines gauge transformations for Jacobi structures and their effects on contact groupoids.
problem Understanding transformations of Jacobi structures and their implications.
method Definition and discussion of gauge transformations for Jacobi structures and their impact on contact groupoids.
result Gauge transformations affect the contact structure of contact groupoids.
Expanding on previous work, this note generalizes geometric structures results.
problem Generalizing geometric structures results.
method Generalization to a class of geometric structures including integrable almost-complex structures.
result Main results generalized to a broader class of geometric 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…
Two Kähler structures are PCR equivalent in the Siegel domain.
problem Equivalence of two Kähler structures in the Siegel domain.
method Construction of complex hyperbolic and Kähler structures from Sasakian structure.
result PCR Kähler equivalent structures in Siegel domain.
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, …