Geodesics on spaces of almost hermitian structures

Almost Hermitian Ricci flow

We introduce a new curvature flow which matches with the Ricci flow on metrics and preserves the almost Hermitian condition. This enables us to use Ricci flow to study almost Hermitian manifolds.

殆Hermitian流形之间的Hermitian多重调和映射

Hermitian pluriharmonic maps between almost Hermitian manifolds

该研究在殆Hermitian流形的框架下提出Hermitian多重调和映射概念，证明全纯/反全纯映射属于该类映射，建立了映到Kähler流形的这类映射的部分能量单调性公式，并在部分能量增长假设下得到全纯性结果。

Hermitian流形上混合体积形式空间中测地线方程的正则性

Regularity of a Geodesic equation in the space of mixed Volume Forms on Hermitian Manifolds

研究Hermitian流形上混合体积形式空间中测地线方程的正则性，证明在椭圆性条件下该退化方程存在C^{1,1}解，并构造显式下解。

On a class of immersions between almost para-Hermitian manifolds

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 analogue of the idea of slant submanifold in almost Hermitian geometry. There are classified pointwise slant surfaces of four dimensional almost para-Hermitian manifold.

近四元斜Hermitian流形的子流形

Submanifolds of almost quaternionic skew-Hermitian manifolds

本文研究了近四元斜Hermitian流形的子流形，通过构造显式例子验证了不同类型的子流形结构。

Curvature of almost Hermitian manifolds and applications

In this paper, by introducing a notion of local quasi holomorphic frame, we obtain a curvature formula for almost Hermitian manifolds which is similar to that of Hermitian manifolds. Moreover, as applications of the curvature formula, we extend a result of H.S. Wu and a result of F. Zheng to almost Hermitian manifolds.

$4$-dimensional almost-Kähler Lie algebras of constant Hermitian holomorphic sectional curvature are Kähler

We prove that any $4$-dimensional almost-Kähler Lie algebra of constant Hermitian holomorphic sectional curvature with respect to the canonical Hermitian connection is Kähler.

Some haracterizations of almost Hermitian manifolds

We give a condition for an almost constant-type manifold to be a constant-type manifold, and holomorphic and $R$-invariant submanifolds of almost Hermitian manifolds are studied. Generalizations of some results in [5] are given.

Almost Hermitian Structures and Quaternionic Geometries

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 obtain a number of relations that give hyperKäher or locally conformal hyperKähler structures, thus generalising a result of Hitchin. We also study the types of almost quaternion-Hermitian geometries that arise and tabulate the results.

Schur's Theorem for Almost Hermitian Manifolds

The Schur's theorem of antiholomorphic type is proved for arbitrary almost Hermitian manifolds, namely: If a connected almost Hermitian manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then this curvature is a global constant.

Conformal generic submersions with total space an almost Hermitian manifold

Akyol, M. A and Şahin, B. [Conformal semi-invariant submersions, Commun. Contemp. Math. 19, 1650011 (2017).] introduced the notion of conformal semi-invariant submersions from almost Hermitian manifolds. The present paper deal with the study of conformal generic submersions from almost Hermitian manifolds which extends semi-invariant submersions, generic Riemannian submersions and conformal semi-invariant submersions a natural way. We mention some examples of such maps and obtain characterizations and investigate some properties, including the integrability of distributions, the geometry of fo…

Differential operators on almost-Hermitian manifolds and harmonic forms

We consider several differential operators on compact almost-complex, almost-Hermitian and almost-Kähler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces of harmonic forms and cohomologies with the classical de Rham, Dolbeault, Bott-Chern and Aeppli cohomologies.

Almost Hermitian Geometry on Six Dimensional Nilmanifolds

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.

Rigidity and Vanishing Theorems for Almost Even-Clifford Hermitian Manifolds

We prove the rigidity and vanishing of several indices of "geometrically natural" twisted Dirac operators on almost even-Clifford Hermitian manifolds admitting circle actions by automorphisms.

The axiom of coholomorphic (2n+1)-spheres in the almost Hermitian geometry

It is proved, that if an almost Hermitian manifold satisfies the axiom of coholomorphic spheres, it is conformal flat.

Almost Hermitian Identities

We study the local commutation relation between the Lefschetz operator and the exterior differential on an almost complex manifold with a compatible metric. The identity that we obtain generalizes the backbone of the local Kähler identities to the setting of almost Hermitian manifolds, allowing for new global results for such manifolds.

Stability of the Almost Hermitian Curvature Flow

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 stable? An almost hermitian structure $(ω,J)$ is dynamically stable if it is a fixed point of the flow and there exists a neighborhood $\mathcal{N}$ of $(ω,J)$ such that for any almost hermitian structure $(ω(0),J(0)) \in \mathcal{N}$ the solution of the Almost Hermitian Curvature fl…

Scalar Curvatures of Invariant Almost Hermitian Structures on Generalized Flag Manifolds

In this paper we study invariant almost Hermitian geometry on generalized flag manifolds. We will focus on providing examples of Kähler like scalar curvature metric, that is, almost Hermitian structures $(g,J)$ satisfying $s=2s_{\rm C}$, where $s$ is Riemannian scalar curvature and $s_{\rm C}$ is the Chern scalar curvature.

Almost hypercomplex manifolds with Hermitian-Norden metrics and 4-dimensional indecomposable real Lie algebras depending on two parameters

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 characteristics of the respective almost hypercomplex manifolds with Hermitian-Norden metrics are obtained.

Almost Yamabe Solitons on a Total Space of Almost Hermitian Submersions

This article presents the study of almost Hermitian submersion from an almost Yamabe soliton onto an almost Hermitian manifold. A non-trivial example is also mentioned in order to guarantee the existence of such solitons on the total space of almost Hermitian submersions. We mainly focus on Kaehler submersions from Kaehler manifolds which are special case of almost Hermitian submersions. Under certain conditions, we find out that the fibres and the base manifold of such submersions are almost Yamabe soliton. We give the characterizations for an almost Yamabe soliton of a Kaehler submersion to …

On the axiom of planes and the axiom of spheres in the almost Hermitian geometry

Criterions for constancy of the holomorphic sectional curvature and the antiholomorphic sectional curvature are proved for almost Hermitian manifolds. It is shown, that an almost Hermitian manifold satisfying the axiom of antiholomorphic planes or the axiom of antiholomorphic spheres is a real or a complex space form.

A note on the geometry and topology of almost even-Clifford Hermitian manifolds

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

The fundamental form of almost-quaternionic Hermitian manifolds

We prove that if the fundamental 4-form of an almost-quaternionic Hermitian manifold (M, Q, g) of dimension at least eight satisfies the conformal-Killing equation, then (M, Q, g) is quaternionic-Kahler.

Geometric realizations of curvature

We study geometric realization questions of curvature in the affine, Riemannian, almost Hermitian, almost para Hermitian, almost hyper Hermitian, almost hyper para Hermitian, Hermitian, and para Hermitian settings. We also express questions in Ivanov-Petrova geometry, Osserman geometry, and curvature homogeneity in terms of geometric realizations.

关于一类近埃尔米特4流形

On a class of almost Hermitian 4-manifolds

本文利用Sekigawa积分恒等式，从较弱条件刻画了近埃尔米特4流形类$\mathcal{AH}_1$，并推进其分类，证明了4维李代数上的唯一性结果。

Almost Hermitian Manifolds with Vanishing Bochner Curvature Tensor

It is proved that if an almost Hermitian manifold of dimension greater than 4 has vanishing (classical) Bochner curvature tensor and is not Kaehlerian at a point, then it is flat in a neighbourhood of this point.

Natural Diagonal Riemannian Almost Product and Para-Hermitian Cotangent Bundles

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 for the natural diagonal (almost) para-Kählerian structures on the total spaces of the cotangent bundle.

A connection with parallel torsion on almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics

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 constant almost complex structures and the case when the torsion of D is D-parallel. Curvature properties of these manifolds are studied. An example of 4-dimensional manifolds in the considered basic class is constructed and characterized.

Almost even-Clifford hermitian manifolds with large automorphism group

We study manifolds endowed with an (almost) even Clifford (hermitian) structure and admitting a large automorphism group. We classify them when they are simply connected and the dimension of the automorphism group is maximal, and also prove a gap theorem for the dimension of the automorphism group.

Scalar curvature rigidity of almost Hermitian manifolds which are asymptotic to $\mathbb{C}H^{2n}$

We show that an almost Hermitian manifold $(M,g)$ of real dimension $4n$ which is strongly asymptotic to $\mathbb{C}H^{2n}$ and satisfies a certain scalar curvature bound must be isometric to the complex hyperbolic space. Assuming Kähler instead of almost Hermitian this gives the already known rigidity result by H. Boualem and M. Herlich proved in \emph{Ann. Scuola Norm. Sup Pisa (Ser. V)}, vol. 1(2).

Almost h-conformal slant submersions from almost quaternionic Hermitian manifolds

We introduce the notions of h-conformal slant submersions and almost h-conformal slant submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, slant submersions, h-slant submersions, almost h-slant submersion and conformal slant submersions. We investigate several properties of these including the integrability of distributions, the geometry of foliations and the conditions for such maps to be totally geodesic. Further we give some examples of such maps.

An integral formula for Riemannian $G$-structures with applications to almost Hermitian and almost contact structures

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.

Curvature of (special) almost Hermitian manifolds

We study the curvature of almost Hermitian manifolds and their special analogues via intrinsic torsion and representation theory. By deriving different forumlae for the skew-symmetric part of the star-Ricci curvature, we find that some of these contributions are dependent on the approach used, and for the almost Hermitian case we obtain tables that differ from those of Falcitelli, Farinola and Salamon. We show how the exterior algebra may used to explain some of these variations.

Embeddings of almost Hermitian manifolds in almost hyperHermitian those. Complex and hypercomplex numbers in differential geometry

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 structure has been constructed on the tangent bundle TM with help of the Riemannian connection of an almost Hermitian structure on a manifold M then, we consider an embedding of the almost Hermitian manifold M in the corresponding normal tubular neighborhood of the null section in the…

Four-dimensional almost Hermitian manifolds with vanishing Tricerri-Vanhecke Bochner curvature tensor

We study curvature properties of four-dimensional almost Hermitian manifolds with vanishing Bochner curvature tensor as defined by Tricerri and Vanhecke. We give local structure theorems for such Kaehler manifolds, and find out several examples related to the theorems.

Curvature identities on almost Hermitian manifolds and applications

In this paper, we systematically compute the Bianchi identities for the canonical connection on an almost Hermitian manifold. Moreover, we also compute the curvature tensor of the Levi-Civita connection on almost Hermitian manifolds in terms of curvature and torsion of the canonical connection. As applications of the curvature identities, we obtain some results about the integrability of quasi Kaehler manifolds and some properties of nearly Kaehler manifolds.

The Almost Hermitian-Einstein Flow

Let $X$ be a compact Kähler manifold, $E\to X$ a Hermitian vector bundle and $L\to X$ an ample line bundle. We construct a non-linear heat flow corresponding to the almost Hermitian-Einstein equation introduced by N.C. Leung, and prove that the solution exists for a short time. We also construct a potential function $D_k$ for this flow. In particular, $D_k$ decreases along the flow.

Almost h-conformal semi-invariant submersions from almost quaternionic Hermitian manifolds

As a generalization of Riemannian submersions, horizontally conformal submersions, semi-invariant submersions, h-semi-invariant submersions, almost h-semi-invariant submersions, conformal semi-invariant submersions, we introduce h-conformal semi-invariant submersions and almost h-conformal semi-invariant submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We study their properties: the geometry of foliations, the conditions for total manifolds to be locally product manifolds, the conditions for such maps to be totally geodesic, etc. Finally, we give some example…

The axiom of θ-holomorphic 2-planes in the almost Hermitian geometry

The axiom of θ-holomorphic 2-planes is introduced. It is proved, that if an almost Hermitian manifold satisfies this axiom for a fixed θ, 0< θ< π/2, then it is a real space form.

On the boundedness of the sectional curvature of almost Hermitian manifolds

We prove the following results: An almost Hermitian manifold of indefinite metric is of pointwise constant holomorphic sectional curvature if the holomorphic sectional curvature is bounded from above and from below. If the antiholomorphic sectional curvature is bounded either from above or from below, then the manifold is of pointwise constant antiholomorphic sectional curvature. Similar results are obtained for almost Hermitian manifolds of definite metric.

Almost Hermitian Structures with Parallel Torsion

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 systematic investigation into the possible geometries. Numerous naturally reductive spaces are constructed and classified as well as examples on nilmanifolds given.

Scalar curvatures of invariant almost Hermitian structures on flag manifolds with two and three isotropic summands

In this paper we study invariant almost Hermitian geometry on generalized flag manifolds which the isotropy representation decompose into two or three irreducible components. We will provide a classification of such flag manifolds admitting Kähler like scalar curvature metric, that is, almost Hermitian structures $(g,J)$ satisfying $s=2s_C$ where $s$ is Riemannian scalar curvature and $s_C$ is the Chern scalar curvature.

Projective Curvature Tensors of Second Type Almost Geodesic Mappings

We consider equitorsion second type almost geodesic mappings of a non-symmetric affine connection space in this article. Using different computational methods, we obtained some invariants of these mappings. Last generalized Thomas projective parameter and Weyl projective tensor as invariants of a second type almost geodesic mapping of a non-symmetric affine connection space are further generalized here.

Tangent bundles with complete lift of the base metric and almost hypercomplex Hermitian-Norden structure

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.

Product of Almost-Hermitian Manfolds

This is a continuous work about the nonexistence of some complete metrics on the product of two manifolds studied by Tam-Yu [Asian Journal of Mathematics, 14(2010)]. Motivated by the result of Tossati [Comm.Anal.Geom. 15(2007)]. We generalize the corresponding results of Tam-Yu [Asian Journal of Mathematics, 14(2010)] to the almost-Hermitian case.

Natural almost Hermitian structures on conformally foliated 4-dimensional Lie groups with minimal leaves

Let $(G,g)$ be a 4-dimensional Riemannian Lie group with a 2-dimensional left-invariant, conformal foliation $\F$ with minimal leaves. Let $J$ be an almost Hermitian structure on $G$ adapted to the foliation $\F$. We classify such structures $J$ which are almost Kähler $(\A\K)$, integrable $(\I)$ or Kähler $(\K)$. Hereby we construct several new multi-dimensional examples in each class.

Anti-invariant Riemannian maps from almost Hermitian manifolds

As a generalization of anti-invariant Riemannian submersions, we introduce anti-invariant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds. We give examples and investigate the geometry of foliations which are arisen from the definition of an anti-Riemannian map. Then we give a decomposition theorem for the source manifold of such maps. We also find necessary and sufficient conditions for anti-invariant Riemannian maps to be totally geodesic and show that every pluriharmonic Lagrangian Riemannian map, which is a special anti-invariant Riemannian map, from a Kähler manifo…

Holomorphic harmonic morphisms from cosymplectic almost Hermitian manifolds

We study 4-dimensional Riemannian manifolds equipped with a minimal and conformal foliation $\mathcal F$ of codimension 2. We prove that the two adapted almost Hermitian structures $J_1$ and $J_2$ are both cosymplectic if and only if $\mathcal F$ is Riemannian and its horizontal distribution $\mathcal H$ is integrable.

Almost Hermitian Manifolds with Vanishing Generalized Bochner Curvature Tensor

We obtain a classification theorem for non Kaehler nearly Kaehler manifolds with vanishing Bochner curvature tensor (introduced by Tricerri and Vanhecke).

Slant submersions from almost Hermitian manifolds

We introduce slant submersions from almost Hermitian manifolds onto Riemannian manifolds. We give examples, investigate the geometry of foliations which are arisen from the definition of a Riemannian submersion and check the harmonicity of such submersions. We also find necessary and sufficient conditions for a slant submersion to be totally geodesic. Moreover, we obtain a decomposition theorem for the total manifold of such submersions.

On pseudoholomorphic map between almost Hermitian manifolds

In this paper, we use the canonical connection instead of Levi-Civita connection to study the smooth maps between almost Hermitian manifolds, especially, the pseudoholomorphic ones. By using the Bochner formulas, we obtian the $C^2$-estimate of canonical second fundamental form, Liouville type theorems of pseudoholomorphic maps, pseudoholomorphicity of pluriharmonic maps, and Simons integral inequality of pseudoholomorphic isometric immersion.

Manifolds with almost contact 3-structure and metrics of Hermitian-Norden type

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 manifold of cosymplectic type is at. Some examples of the studied manifolds are given.

A general Schwarz Lemma for almost-Hermitian manifolds

We prove a version of Yau's Schwarz Lemma for general almost-complex manifolds equipped with Hermitian metrics. This requires an extension to this setting of the Laplacian comparison theorem. As an application we show that the product of two almost-complex manifolds does not admit any complete Hermitian metric with bisectional curvature bounded between two negative constants that satisfies some additional mild assumptions.

Some four-dimensional almost hypercomplex pseudo-Hermitian manifolds

In this paper, a lot of examples of four-dimensional manifolds with an almost hypercomplex pseudo-Hermitian structure are constructed in several explicit ways. The received 4-manifolds are characterized by their linear invariants in the known aspects.