Study introduces semi-integrable almost hyperhermitian structures.
problem Characterizing and constructing almost hyperhermitian structures.
method Divided structures into four classes and constructed on Lie algebras.
result Found semi-integrable structures on all compact Lie algebras of dimension 4n.
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.
Tubular neighborhoods play an important role in differential topology. We have applied these constructions to geometry of almost Hermitian manifolds. At first, we consider deformations of tensor structures on a normal tubular neighborhood of a submanifold in a Riemannian manifold.Further, an almost hyperHermitian struc…
We study the classification of special almost hermitian manifolds in Gray and Hervella's type classes. We prove that the exterior derivatives of the symplectic form and the complex volume form contain all the information about the intrinsic torsion of the $\SUn(n)$-structure. Furthermore, we apply the obtained results …
We study the problem of existence of geometric structures on compact complex surfaces that are related to split quaternions. These structures, called para-hypercomplex, para-hyperhermitian and para-hyperkähler are analogs of the hypercomplex, hyperhermitian and hyperkähler structures in the definite case. We show that …
In this note we discuss the problem of existence of para-hyperhermitian structures on compact complex surfaces. We construct examples of para-hypercomplex structures on Inoue surfaces of type S− which do not admit compatible metrics.
Study torsion and curvature in ACYT and AHKT 8-manifolds.
problem Characterize torsion and curvature in ACYT and AHKT 8-manifolds.
method Analyzes properties of Nijenhuis tensors, Ricci tensors, and torsion forms on ACYT and AHKT 8-manifolds.
result Closed torsion condition and Ricci flatness for ACYT 8-manifolds.
This is a survey on quaternion Hermitian Weyl (locally conformally quaternion Kähler) and hyperhermitian Weyl (locally conformally hyperkähler) manifolds. These geometries appear by requesting the compatibility of some quaternion Hermitian or hyperhermitian structure with a Weyl structure. The motivation for such a stu…
Study elliptic equations on hyperhermitian manifolds with flat hyperkähler metric.
problem Solving elliptic equations on compact hyperhermitian manifolds.
method Adapting Székelyhidi's approach to the hypercomplex setting.
result Prove a priori estimates for solutions to elliptic equations.
Sharp C0 bound for quaternionic Monge-Ampere on hyperhermitian manifolds.
problem Sharp C0 estimate for quaternionic Monge-Ampere equation on hyperhermitian manifolds. method Sharp uniform estimate for quaternionic PDEs using Guo and Phong's method.
result The estimate depends only on Lp norm of the right-hand side for any p>2 and holds for any hyperhermitian initial metric. Investigates special metrics in hypercomplex geometry.
problem Characterizing and understanding special hyperhermitian metrics.
method Characterization of hypercomplex structures with Obata holonomy, investigation of quaternionic Gauduchon and balanced metrics, incompatibility results, and introduction of Einstein-type conditions.
result Joyce's manifolds always admit special metrics.
Study null conformal Killing vector fields on complex surfaces.
problem Characterize pseudo-Hermitian surfaces with null vector fields.
method Analyze topological types and use vector fields to define para-hyperhermitian structures.
result Classify compact four-manifolds with orthogonal null Killing vector fields.
Study fully nonlinear elliptic equations on compact hyperhermitian manifolds.
problem Solving fully nonlinear elliptic equations on compact hyperhermitian manifolds.
method Adapting Székelyhidi's approach to the hypercomplex setting, proving a priori estimates.
result Proves solvability of quaternionic Hessian and Monge-Ampère equations on compact flat hyperkähler manifolds.
We provide a local classification of self-dual Einstein Riemannian four manifolds admitting a positively oriented Hermitian structure and characterize those which carry a hyperhermitian, non-hyperkählerian structure compatible with the negative orientation. We finally show that self-dual Einstein 4-manifolds obtained a…
Extends parabolic study to flat hyperkähler manifolds.
problem Study problems in hyperhermitian geometry.
method Extends elliptic approach to parabolic setting.
result Solves problems in hyperhermitian geometry.
Solves long-time solutions for a specific equation on hyperkähler manifolds.
problem Finding solutions to a specific equation on hyperkähler manifolds.
method Introduced a parabolic quaternionic Monge-Ampère equation and proved its long-time solvability.
result Smooth convergence to a solution of the quaternionic Monge-Ampère equation.
Harmonic almost complex structures on specific Lie groups and solvmanifolds identified.
problem Characterizing harmonic almost complex structures on almost abelian Lie groups and solvmanifolds.
method Adapted Gray-Hervella classification to almost abelian Lie groups, characterized harmonic structures.
result Examples of harmonic almost complex structures in different Gray-Hervella classes on compact almost abelian solvmanifolds.
Study shows almost complex structures with certain tensor properties are prevalent.
problem Characterizing almost complex structures with specific tensor properties.
method Analyzes the space of almost complex structures on compact manifolds.
result The space of almost complex structures with rank at least k Nijenhuis tensor is either empty or dense in each component.
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. New structure with B-metric extends classical almost contact structures.
problem Extending classical almost contact structures with new geometric properties.
method Introduced and studied weak almost contact structures with B-metric.
result Several geometric properties and special classes are obtained.
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…
In this article, we study an almost contact metric structure on a G2-manifold constructed by Arikan, Cho and Salur in via the classification of almost contact metric structures given by Chinea and Gonzalez. In particular, we characterize when this almost contact metric structure is cosymplectic and narrow down the p…
Conditions for Penrose-Ward transformation on specific manifolds.
problem Conditions for Penrose-Ward transformation on almost G2-manifolds with almost twistorial structures. method Necessary and sufficient conditions derived through Penrose-Ward transformation.
result Conditions for Penrose-Ward transformation on almost G2-manifolds with almost twistorial structures. Study on biharmonic almost complex structures on compact manifolds.
problem Existence and regularity of biharmonic almost complex structures.
method Analyzes biharmonic almost complex structures on compact almost Hermitian manifolds, focusing on dimension four.
result Existence of energy-minimizing biharmonic almost complex structures for various topologies and homotopy classes.
We systematically analyze Riemannian manifolds M that admit rigid supersymmetry, focusing on four-dimensional N=1 theories with a U(1)_R symmetry. We find that M admits a single supercharge, if and only if it is a Hermitian manifold. The supercharge transforms as a scalar on M. We then consider the restrictions imposed…
Two constructions link path geometries to almost Grassmann structures.
problem Linking path geometries to almost Grassmann structures.
method Introducing two Fefferman-type constructions.
result Characterizing conditions for almost Grassmann structures arising from these constructions.
Study on Kodaira dimension of SU(m)-structures on almost complex manifolds.
problem Understanding the Kodaira dimension of almost complex manifolds with SU(m)-structures.
method Introduced almost complex structure of splitting type and associated SU(m)-structure. Provided constructions for non-invariant almost complex structures with specific Kodaira dimensions.
result Found non-invariant almost complex structures with Kodaira dimensions 0 and -∞.
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 explores families of almost complex structures and transverse (p,p)-forms.
problem Understanding and producing families of almost p-Kähler structures.
method Producing families of almost p-Kähler structures (Jt,Ωt) on $\C^3$, $\C^4$, and T6. result The almost complex structures Jt cannot be locally compatible with any symplectic form for teq0. In this paper, the notion of an almost contact Kählerian structure is introduced. The interior geometry of almost contact Kählerian spaces is investigated. On the zero-curvature distribution of an almost contact metric structure, as on the total space of a vector bundle, an almost contact Kählerian structure is obtaine…
Study harmonicity of normal almost contact structures on Riemannian manifolds.
problem Understanding harmonicity of normal almost contact structures.
method Analyzing harmonicity through associated sections of a twistor bundle and rewriting equations in terms of curvature tensor.
result Conditions relating harmonicity of almost contact metric and almost complex structures.
The paper studies lifts of complex structures on a manifold.
problem Understanding higher-order lifts of extended almost complex structures.
method Proved theorems on Nijenhuis tensor and introduced a new tensor field.
result Basic results on almost analytic complex vectors are investigated.
The study of harmonicity for almost contact metric structures was initiated by Vergara-Díaz and Wood and continued by González-Dávila and the present author. By using the intrinsic torsion and some restriction on the type of almost contact metric structure, González-Dávila and the present author have characterised harm…
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…
Study computes invariants on six-dimensional solvmanifolds, providing symplectic structure obstructions.
problem Computing invariants on six-dimensional solvmanifolds.
method Computed almost-complex and almost-Hermitian invariants on families of solvmanifolds.
result Provides obstructions to symplectic structures on compact almost-complex manifolds.
In this work, almost product and almost golden structures are studied. Conditions for those structures being Integrable and parallel are investigated. Also harmonicity of a map between almost pruduct or almost golden manifolds with pure or hyperbolic metric is discussed under certain conditions.
We study almost contact metric structures induced by 2-fold vector cross products on manifolds with G2 structures. We get some results on possible classes of almost contact metric structures. Finally we give examples.
Almost complex structures found on many homotopy complex projective spaces.
problem Finding almost complex structures on homotopy complex projective spaces.
method New proof using Chern classes and homotopy properties.
result Classification of almost complex structures on homotopy CPn for 3≤n≤6. 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.
Study of complex and Hermitian structures on specific Lie groups.
problem Classifying Lie groups with specific geometric structures.
method Analysis of left-invariant structures on almost abelian Lie groups.
result Classification of six-dimensional generalized Kähler almost abelian Lie groups.
An almost contact metric structure is parametrized by a section of an associated homogeneous fibre bundle, and conditions for this to be a harmonic section, and a harmonic map, are studied. These involve the characteristic vector field, and the almost complex structure in the contact subbundle. Several examples are giv…
Gray & Hervella gave a classification of almost Hermitian structures (g,I) into 16 classes. We systematically study the interaction between these classes when one has an almost hyper-Hermitian structure (g,I,J,K). In general dimension we find at most 167 different almost hyper-Hermitian structures. In particular, we ob…
The study classifies natural almost Hermitian structures on specific Lie groups.
problem Classifying natural almost Hermitian structures on conformally foliated Lie groups.
method Examining 4-dimensional Riemannian Lie groups with a 2-dimensional conformal foliation and minimal leaves, constructing new examples of multi-dimensional structures.
result Constructing several new multi-dimensional examples of almost Kähler, integrable, and Kähler structures.
Improved optimal regularity for harmonic almost complex structures.
problem Establishing optimal regularity for harmonic almost complex structures.
method Quantitative stratification method and rectifiability of singular strata.
result Optimal regularity theory for energy minimizing harmonic almost complex structures.
Study transverse Dolbeault cohomology for almost complex structures.
problem Understanding cohomology of transverse structures on manifolds.
method Define transverse Dolbeault cohomology, extend transverse complex structure, introduce involutive limit distribution.
result Cohomology spaces of (p,0) for almost complex structures coincide with transverse Dolbeault cohomology.
Study on Kodaira dimension of specific solvmanifolds without complex structures.
problem Analyzing Kodaira dimension for almost complex 4D solvmanifolds without integrable structures.
method Classification of solvmanifolds and computation of Kodaira dimension for specific structures.
result Showed that Kodaira dimension is not a deformation invariant for some solvmanifolds.
Study on almost complex structures with maximal Nijenhuis tensor rank and cohomological properties.
problem Maximally non-integrable almost complex structures and their cohomological properties.
method h-principle and topological invariants characterization.
result Existence of almost complex structures with maximal Nijenhuis tensor rank on parallelizable and certain manifolds.
For a Riemannian G-structure, we compute the divergence of the vector field induced by the intrinsic torsion. Applying the Stokes theorem, we obtain the integral formula on a closed oriented Riemannian manifold, which we interpret in certain cases. We focus on almost harmitian and almost contact metric structures.