arXiv research

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 Almost Hermitian Curvature flow was introduced by Streets and Tian in order to study almost hermitian structures, with a particular interest in symplectic structures. This flow is given by a diffusion-reaction equation. Hence it is natural to ask the following: which almost hermitian structures are dynamically stab…

Harmonic almost complex structures on specific Lie groups and solvmanifolds identified.

The study classifies natural almost Hermitian structures on specific Lie groups.

Almost para-Hermitian manifold it is manifold equipped with almost para-complex structure and compatible pseudo-metric of neutral signature. It is considered a class of immersions of almost para-Hermitian manifolds into almost para-Hermitian manifolds. Such immersions are called slant submanifolds. The concept is an an…

Study computes invariants on six-dimensional solvmanifolds, providing symplectic structure obstructions.

Study of Hermitian and Gauduchon connections on Lie groups with almost Hermitian structures.

The paper explores Kähler-like metrics on generalized flag manifolds.

The paper classifies natural almost Hermitian structures on Lie groups with minimal conformal leaves.

Study on slant submanifolds with new conditions and transitivity.

We present a characterization, in terms of torsion-free generalized connections, for the integrability of various generalized structures (generalized almost complex structures, generalized almost hypercomplex structures, generalized almost Hermitian structures and generalized almost hyper-Hermitian structures) defined …

The fundamental 2-form of an invariant almost Hermitian structure on a 6-dimensional Lie group is described in terms of an action by SO(4)xU(1) on complex projective 3-space. This leads to a combinatorial description of the classes of almost Hermitian structures on the Iwasawa and other nilmanifolds.

The paper proves Liouville theorems for holomorphic maps on pseudo-Hermitian manifolds.

The paper studies twisted almost Hermitian structures on the 6-sphere.

Construct a Hermitian metric on non-Hermitian Yang--Mills moduli spaces near the Hermitian locus.

New algebraic structures for Hermitian geometry cohomologies.

In this article, we consider the almost Hermitian structure on $TM$ induced by a pair of a metric and an affine connection on $M$. We find the conditions under which $TM$ admits almost Kähler structures, Kähler structures and Einstein metrics, respectively. Moreover, we give two examples of Kähler-Einstein structures o…

We compute the structure groups of almost even-Clifford Hermitian manifolds and determine when such groups lead to Spin structures.

The paper classifies flag manifolds with specific isotropy components and finds conditions for Kähler-like scalar curvature.

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 of complex and Hermitian structures on specific Lie groups.

This paper classifies LCSKT almost abelian Lie algebras in 6 dimensions.

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…

The paper classifies invariant structures on complex almost Abelian groups.

Study balanced Hermitian structures on almost abelian Lie algebras, classifying six-dimensional cases.

We study harmonic almost contact structures in the context of contact metric manifolds, and an analysis is carried out when such a manifold fibres over an almost Hermitian manifold, as exemplified by the Boothby-Wang fibration. Two types of almost contact metric warped products are also studied, relating their harmonic…

The study of quasi-Kähler Chern-flat almost Hermitian manifolds is strictly related to the study of anti-bi-invariant almost complex Lie algebras. In the present paper we show that quasi-Kähler Chern-flat almost Hermitian structures on compact manifolds are in correspondence to complex parallelisable Hermitian structur…

The study proves conditions for Hermitian metrics on compact almost complex manifolds.

Compactify complex hyperbolic almost Hermitian manifolds.

An almost quaternion-Hermitian structure on a Riemannian manifold $(M^{4n},g)$ is a reduction of the structure group of $M$ to $\mathrm{Sp}(n)\mathrm{Sp}(1)\subset \mathrm{SO}(4n)$. In this paper we show that a compact simply connected homogeneous almost quaternion-Hermitian manifold of non-vanishing Euler characterist…

We obtain the natural diagonal almost product and locally product structures on the total space of the cotangent bundle of a Riemannian manifold. We find the Riemannian almost product (locally product) and the (almost) para-Hermitian cotangent bundles of natural diagonal lift type. We prove the characterization theorem…

We study almost Hermitian structures admitting a Hermitian connexion with totally skew-symmetric torsion or equivalently, those almost Hermitian structures with totally skew-symmetric Nijenhuis tensor. We investigate up to what extent the Nijenhuis tensor fails to be parallel with respect to the characteristic connexio…

Almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics are considered. A linear connection $D$ is introduced such that the structure of these manifolds is parallel with respect to D. Of special interest is the class of the locally conformally equivalent manifolds of the manifolds with covariantly const…

We consider and resolve the gap problem for almost quaternion-Hermitian structures, i.e. we determine the maximal and submaximal symmetry dimensions, both for Lie algebras and Lie groups, in the class of almost quaternion-Hermitian manifolds. We classify all structures with such symmetry dimensions. Geometric propertie…

Study lightlike hypersurfaces in a specific type of manifold.

We obtain conditions on the Lee form under which a holomorphic map between almost Hermitian manifolds is a harmonic map or morphism. Then we discuss under what conditions (i) the image of a holomorphic map from a cosymplectic manifold is also cosymplectic, (ii) a holomophic map with Hermitian image defines a Hermitian …

Study shows that dimension of Dolbeault harmonic forms is not always equal to B- on certain 4-manifolds.

The tangent bundle of an almost Norden manifold and the complete lift of the Norden metric is considered as a 4n-manifold. It is equipped with an almost hypercomplex Hermitian-Norden structure. It is characterized geometrically. The case when the base manifold is an h-sphere is considered.

The canonical connection on a Riemannian almost product manifold is an analogue to the Hermitian connection on an almost Hermitian manifold. In this paper we consider the canonical connection on a class of Riemannian almost product manifolds with non-integrable almost product structure. We construct and characterize an…

Study of complex structures on specific solvmanifolds, proving existence and non-existence results.

The subject of investigations are the almost hypercomplex manifolds with Hermitian and anti-Hermitian (Norden) metrics. A linear connection D is introduced such that the structure of these manifolds is parallel with respect to D and its torsion is totally skew-symmetric. The class of the nearly Kaehler manifolds with r…

In this paper we consider a manifold $(M,\nabla )$ with a symmetric linear connection $\nabla$ which induces on the cotangent bundle $T^*M$ of $M$ a semi-Riemannian metric $\overline g$ with a neutral signature. The metric $\overline g$ is called natural Riemann extension and it is a generalization (made by M. Sekizaw…

On a 4-dimensional compact symplectic manifold, we consider a smooth family of compatible almost-complex structures such that at time zero the induced metric is Hermite-Einstein almost-Kähler metric with zero or negative Hermitian scalar curvature. We prove, under certain hypothesis, the existence of a smooth family of…

The object of investigations are almost hypercomplex structures with Hermitian-Norden metrics on 4-dimensional Lie groups considered as smooth manifolds. There are studied both the basic classes of a classification of 4-dimensional indecomposable real Lie algebras depending on two parameters. Some geometric characteris…

The paper explores curvature constraints on Kodaira dimension for specific almost Hermitian manifolds.

The paper characterizes curvature of quaternionic skew-Hermitian manifolds and constructs related geometric structures.

The characteristic connection of an almost hermitian structure is a hermitian connection with totally skew-symmetric torsion. The case of parallel torsion in dimension six is of particular interest. In this work, we give a full classification of the algebraic types of the torsion form, and, based on this, undertake a s…

Study integrability of specific geometric structures on odd Courant algebroids.