Conformal Riemannian P-manifolds with connections whose curvature tensors are Riemannian P-tensors

The largest class of Riemannian almost product manifolds, which is closed with respect to the group of the conformal transformations of the Riemannian metric, is the class of the conformal Riemannian P-manifolds. This class is an analogue of the class of the conformal Kähler manifolds in almost Hermitian geometry. The main aim of this work is to obtain properties of manifolds of this class with connections, whose curvature tensors have similar properties as the Kähler tensors in Hermitian geometry.     

Quaternionic Kähler manifolds with Hermitian and Norden metrics

Almost hypercomplex manifolds with Hermitian and Norden metrics and more specially the corresponding quaternionic Kähler manifolds are considered. Some necessary and sufficient conditions for the investigated manifolds to be isotropic hyper-Kählerian and flat are found. It is proved that the quaternionic Kähler manifolds with the considered metric structure are Einstein for dimension at least 8. The class of the non-hyper-Kähler quaternionic Kähler manifolds of the considered type is determined.     

Einstein almost cokähler manifolds

We study an odd‐dimensional analogue of the Goldberg conjecture for compact Einstein almost Kähler manifolds. We give an explicit non‐compact example of an Einstein almost cokähler manifold that is not cokähler. We prove that compact Einstein almost cokähler manifolds with nonnegative *‐scalar curvature are cokähler (indeed, transversely Calabi–Yau); more generally, we give a lower and upper bound for the *‐scalar curvature in the case that the structure is not cokähler. We prove similar bounds for almost Kähler Einstein manifolds that are not Kähler.     

Kähler manifolds of quasi-constant holomorphic sectional curvatures

The Kähler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kähler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizing such manifolds is found. The biconformal group of transformations whose elements transform Kähler metrics into Kähler ones is introduced and biconformal tensor invariants are obtained. This makes it possible to classify the manifolds under consideration locally. The class of locally biconformal flat Kähler metrics is shown to be exactly the class of Kähler metrics whose potential function is only a function of the distance from the origin in ? ⁿ . Finally we show that any rotational even dimensional hypersurface carries locally a natural Kähler structure which is of quasi-constant holomorphic sectional curvatures.     

On the structure of nearly pseudo-Kähler manifolds

Firstly we give a condition to split off the Kähler factor from a nearly pseudo-Kähler manifold and apply this to get a structure result in dimension 8. Secondly we extend the construction of nearly Kähler manifolds from twistor spaces to negatively curved quaternionic Kähler manifolds and para-quaternionic Kähler manifolds. The class of nearly pseudo-Kähler manifolds obtained from this construction is characterized by a holonomic condition. The combination of these results enables us to give a classification result in (real) dimension 10. Moreover, we show that a strict nearly pseudo-Kähler six-manifold is Einstein.     

Some recent progress in non-Kähler geometry

Science China Mathematics - In this paper, we discuss some recent progress in the study of non-Kähler manifolds, in particular the Hermitian geometry of flat canonical connections and...     

Nearly Kähler Submanifolds of a Space Form

In this article, we study isometric immersions of nearly Kähler manifolds into a space form (especially Euclidean space) and show that every nearly Kähler submanifold of a space form has an umbilic foliation whose leafs are 6-dimensional nearly Kähler manifolds. Moreover, using this foliation we show that there is no non-homogeneous 6-dimensional nearly Kähler submanifold of a space form. We prove some results towards a classification of nearly Kähler hypersurfaces in standard space forms.     

Pseudo-holomorphic curves in nearly Kähler manifolds

We study pseudo-holomorphic curves in general nearly Kähler manifolds. For that purpose, we first introduce the fundamental equations of submanifold geometry in terms of the characteristic connection of the nearly Kähler structure. Then we classify pseudo-holomorphic curves with parallel second fundamental form in Chern-flat nearly Kähler manifolds. Moreover, we give a new Simons type identity. As an application of this identity, we show that the closed pseudo-holomorphic curves in Chern-flat nearly Kähler manifolds with a second fundamental form of controlled growth are totally geodesic.     

Decomposition and minimality of lagrangian submanifolds in nearly Kähler manifolds

We show that Lagrangian submanifolds in six-dimensional nearly Kähler (non-Kähler) manifolds and in twistor spaces Z ⁴ⁿ⁺² over quaternionic Kähler manifolds Q ⁴ⁿ are minimal. Moreover, we prove that any Lagrangian submanifold L in a nearly Kähler manifold M splits into a product of two Lagrangian submanifolds for which one factor is Lagrangian in the strict nearly Kähler part of M and the other factor is Lagrangian in the Kähler part of M. Using this splitting theorem, we then describe Lagrangian submanifolds in nearly Kähler manifolds of dimensions six, eight, and ten.     

On locally conformal almost Kähler manifolds

In the first section of this note, we discuss locally conformal symplectic manifolds, which are differentiable manifoldsV ²ⁿ endowed with a nondegenerate 2-form Ω such thatdΩ=θ ∧ Ω for some closed form θ. Examples and several geometric properties are obtained, especially for the case whendΩ ≠ 0 at every point. In the second section, we discuss the case when Ω above is the fundamental form of an (almost) Hermitian manifold, i.e. the case of the locally conformal (almost) Kähler manifolds. Characterizations of such manifolds are given. Particularly, the locally conformal Kähler manifolds are almost Hermitian manifolds for which some canonically associated connection (called the Weyl connection) is almost complex. Examples of locally conformal (almost) Kähler manifolds which are not globally conformal (almost) Kähler are given. One such example is provided by the well-known Hopf manifolds.     

A note on <Emphasis Type="Italic">tt</Emphasis><Superscript>*</Superscript>-bundles over compact nearly Kähler manifolds

First, we generalize a rigidity result for harmonic maps of Gordon (Gordon (1972) Proc AM Math Soc 33: 433–437) to generalized pluriharmonic maps. We give the construction of generalized pluriharmonic maps from metric tt ^*-bundles over nearly Kähler manifolds. An application of the last two results is that any metric tt ^*-bundle over a compact nearly Kähler manifold is trivial (Theorem A). This result we apply to special Kähler manifolds to show that any compact special Kähler manifold is trivial. This is Lu’s theorem (Lu (1999) Math Ann 313: 711–713) for the case of compact special Kähler manifolds. Further we introduce harmonic bundles over nearly Kähler manifolds and study the implications of Theorem A for tt ^*-bundles coming from harmonic bundles over nearly Kähler manifolds.     

An Ohsawa-Takegoshi theorem on compact Khler manifolds

We prove two extension theorems of Ohsawa-Takegoshi type on compact Khler manifolds.In our proof,there are many complications arising from the regularization process of quasi-psh functions on compact Khler manifolds,and unfortunately we only obtain particular cases of the expected result.We remark that the two special cases we proved are natural,since they occur in many situations.We hope that the new techniques we develop here will allow us to obtain the general extension result of Ohsawa-Takegoshi type on compact Khler manifolds in a near future.     

Locally conformally flat Kähler and para-Kähler manifolds

We complete the classification of locally conformally flat Kähler and para-Kähler manifolds, describing all possible non-flat curvature models for Kähler and para-Kähler surfaces.

     

Kähler magnetic fields on Kähler manifolds of negative curvature

On a Kähler manifold we have natural uniform magnetic fields which are constant multiples of the Kähler form. Trajectories, which are motions of electric charged particles, under these magnetic fields can be considered as generalizations of geodesics. We give an overview on a study of Kähler magnetic fields and show some similarities between trajectories and geodesics on Kähler manifolds of negative curvature.     

Hessian Comparison and Spectrum Lower Bound of Almost Hermitian Manifolds

The authors obtain a complex Hessian comparison for almost Hermitian manifolds, which generalizes the Laplacian comparison for almost Hermitian manifolds by Tossati, and a sharp spectrum lower bound for compact quasi Kähler manifolds and a sharp complex Hessian comparison on nearly Kähler manifolds that generalize previous results of Aubin, Li Wang and Tam-Yu.     

Unstable products of smooth curves

We give examples of smooth manifolds with negative first Chern class which are slope unstable with respect to certain polarisations, and so have Kähler classes that do not admit any constant scalar curvature Kähler metrics. These manifolds also have unstable Hilbert and Chow points. We compare this to the work of Song-Weinkove on the J-flow.     

The Einstein condition on nearly Kähler six-manifolds

We review basic facts on the structure of nearly Kähler manifolds, focussing in particular on the six-dimensional case. A self-contained proof that nearly Kähler six-manifolds are Einstein is given by combining different known results. We finally rephrase the definition of nearly Kähler six-manifold in terms of a pair of partial differential equations.     

Virtually abelian Kähler and projective groups

We characterise the virtually abelian groups which are fundamental groups of compact Kähler manifolds and of smooth projective varieties. We show that a virtually abelian group is Kähler if and only if it is projective. In particular, this allows to describe the Kähler condition for such groups in terms of integral symplectic representations.     

A note on the almost-one-half holomorphic pinching

Motivated by a previous work by Zheng and the second-named author, we study pinching constants of compact Kähler manifolds with positive holomorphic sectional curvature. In particular, we prove a gap theorem on Kähler manifolds with almost-one-half pinched holomophic sectional curvature. The proof is motivated by the work of Petersen and Tao on Riemannian manifolds with almost-quarter-pinched sectional curvature.     

Curvature identities on almost Hermitian manifolds and applications

We systematically derive 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 K¨ahler manifolds and nearly K¨ahler manifolds.