On Pseudo-Hermitian Magnetic Curves in Sasakian Manifolds

Harmonic maps and para-Sasakian geometry

The purpose of this paper is to study the harmonicity of maps to or from para-Sasakian manifolds. We derive the condition for the tension field of paraholomorphic map between almost para-Hermitian manifold and para-Sasakian manifold. The necessary and sufficient condition for a paraholomorphic map between para-Sasakian manifolds to be parapluriharmonic are shown and a non-trivial example is presented for its illustrations.

Convergence of Closed Pseudo-Hermitian Manifolds

Based on uniform CR Sobolev inequality and Moser iteration, this paper investigates the convergence of closed pseudo-Hermitian manifolds. In terms of the subelliptic inequality, the set of closed normalized pseudo-Einstein manifolds with some uniform geometric conditions is compact. Moreover, the set of closed normalized Sasakian $η$-Einstein $(2n+1)$-manifolds with Carnot-Carathéodory distance bounded from above, volume bounded from below and $L^{n + \frac{1}{2}}$ norm of pseudo-Hermitian curvature bounded is $C^\infty$ compact. As an application, we will deduce some pointed convergence of co…

On the sharp dimension estimate of CR holomorphic functions in Sasakian Manifolds

This is the very first paper to focus on the CR analogue of Yau's uniformization conjecture in a complete noncompact pseudohermitian $(2n+1)$-manifold of vanishing torsion (i.e. Sasakian manifold) which is an odd dimensional counterpart of Kähler geometry. In this paper, we mainly deal with the problem of the sharp dimension estimate of CR holomorphic functions in a complete noncompact pseudohermitian manifold of vanishing torsion with nonnegative pseudohermitian bisectional curvature.

Existence of nonconstant CR-holomorphic functions of polynomial growth in Sasakian Manifolds

In this paper, we show that there exists a nonconstant CR holomorphic function of polynomial growth in a complete noncompact Sasakian manifold of nonnegative pseudohermitian bisectional curvature with the CR maximal volume growth property. This is the very first step toward the CR analogue of Yau uniformization conjecture which states that any complete noncompact Sasakian manifold of positive pseudohermitian bisectional curvature is CR biholomorphic to the standard Heisenberg group.

紧 Sasakian 流形的一致化

Uniformizations of compact Sasakian manifolds

本文建立紧 Sasakian 流形可形变为 Hermitian 对称空间反典范丛圆丛局部模型的判据，将 Simpson 的 Hodge 结构变分与 Higgs 丛一致化定理推广到 Sasakian 几何。

Invariant and anti-invariant submanifolds of special quasi-sasakian manifolds

The present paper deals with the study of Chaki-pseudo parallel and Deszcz-pseudo parallel invariant submanifolds of SQ-Sasakian manifolds with respect to Levi-Civita connection and semisymmetric metric connection and obtain that these two classes are equivalent with a certain condition. Also the invariant and anti-invariant submanifolds of SQ-Sasakian manifolds with respect to Levi-Civita connection as well as semisymmetric metric connection whose metrics are Ricci solitons are studied.

Generalized Sasakian space forms and Riemannian manifolds of quasi constant sectional curvature

In this paper, we show that a generalized Sasakian space form of dimension greater than three is either of constant sectional curvature; or a canal hypersurface in Euclidean or Minkowski spaces; or locally a certain type of twisted product of a real line and a flat almost Hermitian manifold; or locally a wapred product of a real line and a generalized complex space form; or an $α$-Sasakian space form; or it is of five dimension and admits an $α$-Sasakian Einstein structure. In particular, a local classification for generalized Sasakian space forms of dimension greater than five is obtained. A …

Jacobi fields of the Tanaka-Webster connection on Sasakian manifolds

We build a variational theory of geodesics of the Tanaka-Webster connection on a strictly pseudoconvex CR manifold.

$\mathcal{Z}$-symmetries of $(ε)$-para-Sasakian 3-manifolds

The object of this paper is study $(ε)$-para-Sasakian 3-manifolds satisfying certain conditions on the $\mathcal{Z}$ tensor. We characterize, $\mathcal{Z}$-symmetric; $\mathcal{Z}$-semisymmetric; $\mathcal{Z}$-pseudosymmetric; and projectively $\mathcal{Z}$-semisymmetric conditions on an $(ε)$-para-Sasakian 3-manifold.

Horizontal Gradient Estimate of Positive Pseudo-Harmonic Functions on Complete Noncompact Pseudo-Hermitian Manifolds

In this paper, we will give a horizontal gradient estimate of positive solutions of $Δ_b u = - λu$ on complete noncompact pseudo-Hermitian manifolds. As a consequence, we recapture the Liouville theorem of positive pseudo-harmonic functions on Sasakian manifolds with nonnegative pseudo-Hermitian Ricci curvature.

Invariant pseudo-Sasakian and $K$-contact structures on seven-dimensional nilpotent Lie groups

We study the question of the existence of left-invariant Sasaki contact structures on the seven-dimensional nilpotent Lie groups. It is shown that the only Lie group allowing Sasaki structure with a positive definite metric tensor is the Heisenberg group. We find a complete list of the 22 classes of seven-dimensional nilpotent Lie groups which admit pseudo-Sasaki structure. We also present a list of 25 classes of seven-dimensional nilpotent Lie groups admitting a $K$-contact structure, but not the pseudo-Sasaki structure. All the contact structures considered are central extensions of six-dime…

On the three-circle theorem and its applications in Sasakian manifolds

This paper mainly focuses on the CR analogue of the three-circle theorem in a complete noncompact pseudohermitian manifold of vanishing torsion being odd dimensional counterpart of Kähler geometry. In this paper, we show that the CR three-circle theorem holds if its pseudohermitian sectional curvature is nonnegative. As an application, we confirm the first CR Yau's uniformization conjecture and obtain the CR analogue of the sharp dimension estimate for CR holomorphic functions of polynomial growth and its rigidity when the pseudohermitian sectional curvature is nonnegative. This is also the fi…

CR结构、k-接触结构与广义Sasakian结构

CR structures, k-contact structures, and generalized Sasakian structures

本文在广义接触几何背景下定义了广义Sasakian结构，并证明k-接触流形为广义Sasakian当且仅当其为经典Sasakian，同时证明严格伪凸CR流形总是广义Sasakian的。

On $(\varepsilon)$-para Sasakian 3-manifolds

In this paper we study the 3-dimensional $(\varepsilon) $-para Sasakian manifolds. We obtain an necessary and sufficient condition for an $(\varepsilon ) $-para Sasakian 3 -manifold to be an indefinite space form. We show that a Ricci-semi-symmetric $(\varepsilon) $-para Sasakian 3 -manifold is an indefinite space form. We investigate the necessary and sufficient condition for an $(\varepsilon) $-para Sasakian 3 -manifold to be locally $φ$-symmetric. It is proved that in an $ (\varepsilon) $-para Sasakian 3-manifold with $η$ -parallel Ricci tensor the scalar curvature is constant. It is also s…

*-Conformal η-Ricci Soliton on Sasakian manifold

In this paper we study *-Conformal η-Ricci soliton on Sasakian manifolds. Here, we discuss some curvature properties on Sasakian manifold admitting *-Conformal η-Ricci soliton. We obtain some significant results on *-Conformal η-Ricci soliton in Sasakian manifolds satisfying R(ξ,X).S = 0, S(ξ,X).R = 0, {\overline}P(ξ,X).S = 0, where {\overline}P is Pseudo-projective curvature tensor.The conditions for *-Conformal η-Ricci soliton on Φ-conharmonically flat and Φ-projectively flat Sasakian manifolds have been obtained in this article. Lastly we give an example of 5-dimensional Sasakian manifolds …

On the Obata Theorem in a weighted Sasakian manifold

In this paper, we generalize the CR Obata theorem to a compact strictly pseudoconvex CR manifold with a weighted volume measure. More precisely, we first derive the weighted CR Reilly's formula associated with the Witten sub-Laplacian and obtain the corresponding first eigenvalue estimate. With its applications, we obtain the CR Obata theorem in a compact weighted Sasakian manifold with or without boundary.

On the Obata theorem for the weighted Kohn Laplacian in a closed weighted Sasakian manifold

In this paper, we generalize the CR Obata theorem for the Kohn Laplacian to a closed strictly pseudoconvex CR manifold with a weighted volume measure. More precisely, we first derive the weighted CR Reilly's formula associated with the weighted Kohn Laplacian and obtain the corresponding first eigenvalue estimate. With its application, we obtain the CR Obata theorem in a closed weighted Sasakian manifold.

On the CR Poincaré-Lelong equation, Yamabe steady solitons and structures of complete noncompact Sasakian manifolds

In this paper, we solve the so-called CR Poincaré-Lelong equation by solving the CR Poisson equation on a complete noncompact CR $(2n+1)$-manifold with nonegative pseudohermitian bisectional curvature tensors and vanishing torsion which is an odd dimensional counterpart of Kähler geometry. With applications of this solution plus the CR Liouvelle property, we study the structures of complete noncompact Sasakian manifolds and CR Yamabe steady solitons.

CR Sub-Laplacian Comparison and Liouville-type Theorem in a Complete Noncompact Sasakian Manifold

In this paper, we first obtain the sub-Laplacian comparison theorem in a complete noncompact pseudohermitian manifold of vanishing torsion (i.e. Sasakian manifold). Secondly, we derive the sub-gradient estimate for positive pseudoharmonic functions in a complete noncompact pseudohermitian manifold which satisfies the CR sub-Laplacian comparison property. It is served as the CR analog of Yau's gradient estimate. As a consequence, we have the natural CR analogue of Liouville-type theorems in a complete noncompact Sasakian manifold of nonnegative pseudohermitian Ricci curvature tensors.

Sasakian structures. A foliated approach

Recent renewed interest in Sasakian manifolds is due mainly to the fact that they can provide examples of generalized Einstein manifolds, manifolds which are of great interest in mathematical models of various aspects of physical phenomena. Sasakian manifolds are odd dimensional counterparts of Kählerian manifolds to which they are closely related. The book of Ch. Boyer and K. Galicki, Sasakian Geometry is both the best introduction to the subject and at the same time it gathers state of the art information and results on these manifolds. However, although the authors are well aware that a Sas…

Sasakian 7-流形上的瞬子

Instantons on Sasakian 7-manifolds

本文研究 7 维 Sasakian 流形上的接触瞬子方程，用 Fredholm 理论建立模空间局部模型并给出维数公式，证明自对偶接触瞬子模空间为 Kähler，并在接触 Calabi-Yau 情形下统一了接触瞬子、可积 G_2-瞬子与 HYM 联络。

Sasakian structures on CR-manifolds

A contact manifold $M$ can be defined as a quotient of a symplectic manifold $X$ by a proper, free action of $\R^{>0}$, with the symplectic form homogeneous of degree 2. If $X$ is, in addition, Kaehler, and its metric is also homogeneous of degree 2, $M$ is called Sasakian. A Sasakian manifold is realized naturally as a level set of a Kaehler potential on a complex manifold, hence it is equipped with a pseudoconvex CR-structure. We show that any Sasakian manifold $M$ is CR-diffeomorphic to an $S^1$-bundle of unit vectors in a positive line bundle on a projective Kähler orbifold. This induces a…

Holomorphicity and Walczak formula on Sasakian manifolds

Walczak formula is a very nice tool for understanding the geometry of a Riemannian manifold equipped with two orthogonal complementary distributions. Svensson [7] has shown that this formula simplifies to a Bochner type formula when we are dealing with Kähler manifolds and holomorphic (integrable) distributions. Here, with adapted notions as invariant distribution and (contact) holomorphicity, we derive the special form of the Walczack formula on a Sasaki manifold. Then we apply a standard Bochner argument in the study of (contact) holomorphic distributions. Some other applications for (pseudo…

Weakly symmetry of a class of g-natural metrics on tangent bundles

Considering the class G of g-natural metrics on the tangent bundle of a Riemannian manifold (M, g), it is shown that the flatnees for g is a necessary and sufficient condition of weakly symmetry (recurrent or pseudo-symmetry) of G. In particular, the cases of weakly symmetric Sasakian lift metric studied by Bejan and Crasmareanu and recurrent or pseudo-symmetric Sasakian lift metric studied by Binh and Tamassy are obtained.

Higgs bundles and flat connections over compact Sasakian manifolds

Given a compact Kähler manifold $X$, there is an equivalence of categories between the completely reducible flat vector bundles on $X$ and the polystable Higgs bundles $(E,\, θ)$ on $X$ with $c_1(E)= 0= c_2(E)$ \cite{SimC}, \cite{Cor}, \cite{UY}, \cite{DonI}. We extend this equivalence of categories to the context of compact Sasakian manifolds. We prove that on a compact Sasakian manifold, there is an equivalence between the category of semi-simple flat bundles on it and the category of polystable basic Higgs bundles on it with trivial first and second basic Chern classes. We also prove that a…

Symmetries of Contact Metric Manifolds

We study the Lie algebra of infinitesimal isometries on compact Sasakian and K--contact manifolds. On a Sasakian manifold which is not a space form or 3--Sasakian, every Killing vector field is an infinitesimal automorphism of the Sasakian structure. For a manifold with K--contact structure, we prove that there exists a Killing vector field of constant length which is not an infinitesimal automorphism of the structure if and only if the manifold is obtained from the Konishi bundle of a compact pseudo--Riemannian quaternion--Kaehler manifold after changing the sign of the metric on a maximal ne…

Pseudo-Einstein structure, eigenvalue estimate for the CR Paneitz operator and its applications to uniformization theorem

In this note, we mainly focus on the existence of pseudo-Einstein contact forms, an upper bound eigenvalue estimate for the CR Paneitz operator and its applications to the uniformization theorem for Sasakian space form in an embeddable closed strictly pseudoconvex CR 3-manifold. Firstly, the existence of pseudo-Einstein contact form is confirmed if the CR 3-manifold is Sasakian. Secondly, we derive an eigenvalue upper bound estimate for the CR Paneitz operator and obtain the CR uniformization theorem for a class of CR 3-manifolds. At the end, under the positivity assumption of the pseudohermit…

伪Hermitian流形上$Δ_b u + au^{p+1} = 0$的梯度估计

Gradient estimates for $Δ_b u + au^{p+1} = 0$ on pseudo-Hermitian manifolds

本文推导了完备非紧伪Hermitian流形上方程$Δ_b u + au^{p+1} = 0$正解的梯度估计，并在Sasakian型流形上获得了Liouville型定理。

Warped Product Pointwise Semi-slant Submanifolds of Almost Contact Manifolds

Recently, B.-Y. Chen and O. J. Garay studied pointwise slant submanifolds of almost Hermitian manifolds. By using the notion of pointwise slant submanifolds, we investigate the geometry of pointwise semi-slant submanifolds and their warped products in Sasakian and cosymplectic manifolds. We prove that there exist no proper pointwise semi-slant warped product submanifold other than contact CR-warped products in Sasakian manifolds. We give non-trivial examples of such submanifolds in cosypmlectic manifolds and obtain several fundamental results, including a characterization for warped product po…

On generalized Gauduchon metrics

We study a class of Hermitian metrics on complex manifolds, recently introduced by J. Fu, Z. Wang and D. Wu, which are a generalization of Gauduchon metrics. This class includes the one of Hermitian metrics for which the associated fundamental 2-form is $\partial \bar\partial$-closed. Examples are given on nilmanifolds, on products of Sasakian manifolds, on $S^1$-bundles and via the twist construction introduced by A. Swann.

Associative Submanifolds of Squashed 3-Sasakian Manifolds

Every compact 3-Sasakian 7-manifold $M$ admits a canonical 2-parameter family of co-closed $\text{G}_2$-structures $φ_{a,b}$ for $a,b > 0$, as well as a foliation by $φ_{a,b}$-associative 3-folds whose leaf space $X$ is a positive quaternion-Kähler 4-orbifold. We prove that associative 3-folds in $(M,φ_{a,b})$ that are ruled by a certain type of geodesic are in correspondence with pseudo-holomorphic curves in the almost-complex 8-manifold $Z \times S^2$, where $Z$ is the twistor space of $X$ equipped with its strict nearly-Kähler structure. As an application, we construct infinitely many topol…

Unstability of pseudoharmonic maps between pseudo-Hermitian manifolds

In this paper, we derive the second variation formula of pseudoharmonic maps into any pseudo-Hermitian manifolds. When the target manifold is an isometric embedded CR manifold in complex Euclidean space or a pseudo-Hermitian immersed submanifold in Heisenberg group, we give some conditions on Weingarten maps to obtain some unstability of pseudoharmonic maps between these pseudo-Hermitian manifolds.

Curvature properties of pseudo-sphere bundles over paraquaternionic manifolds

In this paper we obtain several curvature properties of the twistor and reflector spaces of a paraquaternionic Kähler manifold and prove the existence of both positive and negative mixed 3-Sasakian structures in a principal SO(2,1)-bundle over a paraquaternionic Kähler manifold.

Constancy of $ϕ-$holomorphic scetional curvature in generalized $g.f.f-$manifolds

Tanno [6] provided an algebraic characterization in an almost Hermitian manifold to reduce to a space of constant holomorphic sectional curvature, which he later extended for the Sasakian manifolds as well. In this present paper, we generalize the same characterization in generalized $g.f.f-$manifolds.

Pseudo-Harmonic Maps From Complete Noncompact Pseudo-Hermitian Manifolds To Regular Balls

In this paper, we give an estimate of sub-Laplacian of Riemannian distance functions in pseudo-Hermitian geometry which plays a similar role as Laplacian comparison theorem in Riemannian geometry, and deduce a prior horizontal gradient estimate of pseudo-harmonic maps from pseudo-Hermitian manifolds to regular balls of Riemannian manifolds. As an application, Liouville theorem is established under the conditions of nonnegative pseudo-Hermitian Ricci curvature and vanishing pseudo-Hermitian torsion. Moreover, we obtain the existence of pseudo-harmonic maps from complete noncompact pseudo-Hermit…

Indefinite almost paracontact metric manifolds

In this paper we introduce the concept of $(\varepsilon)$-almost paracontact manifolds, and in particular, of $(\varepsilon)$-para Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of $(\varepsilon)$-para Sasakian manifolds are obtained. We prove that if a semi-Riemannian manifold is one of flat, proper recurrent or proper Ricci-recurrent, then it can not admit an $(\varepsilon)$-para Sasakian structure. We show that, for an $(\varepsilon)$-para Sasakian manifold, the conditions of being symmetric, semi-symmetric or of constant se…

Geometric Realizability of Covariant Derivative Kähler Tensors for almost Pseudo-Hermitian and almost Para-Hermitian Manifolds

The covariant derivative of the Kähler form of an almost pseudo-Hermitian or of an almost para-Hermitian manifold satisfies certain algebraic relations. We show, conversely, that any 3-tensor which satisfies these algebraic relations can be realized geometrically.

On pseudo-Hermitian Einstein spaces

We describe and construct here pseudo-Hermitian structures $θ$ without torsion (i.e. with transversal symmetry) whose Webster-Ricci curvature tensor is a constant multiple of the exterior differential $dθ$. We call these structures pseudo-Hermitian Einstein and our result states that they all can be derived locally from K{ä}hler-Einstein metrics. Moreover, we discuss the corresponding Fefferman metrics of the pseudo-Hermitian Einstein structures. These Fefferman metrics are never Einstein, but they are locally always conformally Einstein.

Almost para-Hermitian and almost paracontact metric structures induced by natural Riemann extensions

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. Sekizawa and O. Kowalski) of the Riemann extension, introduced by E. K. Patterson and A. G. Walker (1952). We construct two almost para-Hermitian structures on $(T^*M,\overline g)$ which are almost para-Kähler or para-Kähler and prove that the defined almost para-complex structures are …

Pseudo-Harmonic Maps From Pseudo-Hermitian Manifolds to Riemannian Manifolds

In this paper, we discuss the heat flow of a pseudo-harmonic map from a closed pseudo-Hermitian manifold to a Riemannian manifold with non-positive sectional curvature, and prove the existence of the pseudo-harmonic map which is a generalization of Eells-Sampson's existence theorem. We also discuss the uniqueness of the pseudo-harmonic representative of its homotopy class which is a generalization of Hartman theorem, provided that the target manifold has negative sectional curvature.

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.

Generalized maximum principles and stochastic completeness for pseudo-Hermitian manifolds

In this paper, we establish a generalized maximum principle for pseudo-Hermitian manifolds. As corollaries, Omori-Yau type maximum principles for pseudo-Hermitian manifolds are deduced. Moreover, we prove that the stochastic completeness for the heat semigroup generated by the sub-Laplacian is equivalent to the validity of a weak form of the generalized maximum principles. Finally, we give some applications of these generalized maximum principles.

具有非负Tanaka-Webster曲率的伪埃尔米特流形上的刘维尔定理

Liouville Theorems on pseudohermitian manifolds with nonnegative Tanaka-Webster curvature

该研究探讨非负曲率非紧Sasakian流形上CR Yamabe方程正解，证明ℍ¹是具非负Tanaka-Webster数量曲率且有非平凡正解的唯一完备Sasakian空间，并将Jerison-Lee结果推广到高维弯曲流形。

Prescribed Webster scalar curvatures on compact pseudo-Hermitian manifolds

In this paper, we investigate the problem of prescribing Webster scalar curvatures on compact pseudo-Hermitian manifolds. In terms of the method of upper and lower solutions and the perturbation theory of self-adjoint operators, we can describe some sets of Webster scalar curvature functions which can be realized through pointwise CR conformal deformations and CR conformally equivalent deformations respectively from a given pseudo-Hermitian structure.

On contact pseudo-metric manifolds satisfying a nullity condition

In this paper, we aim to introduce and study $(κ, μ)$-contact pseudo-metric manifold and prove that if the $φ$-sectional curvature of any point of $M$ is independent of the choice of $φ$-section at the point, then it is constant on $M$ and accordingly the curvature tensor. Also, we introduce generalized $(κ, μ)$-contact pseudo-metric manifold and prove for $n>1$, that a non-Sasakian generalized $(κ, μ)$-contact pseudo-metric manifold is a $(κ, μ)$-contact pseudo-metric manifold.

Pseudo-harmonic Hermitian structures on Weyl manifolds

We find geometric conditions on a Hermitian-Weyl manifold under which the complex structure is a pseudo-harmonic map in the sense of G. Kokarev \cite{K09} from the manifold into its twistor space. This is done under the assumption that the dimension of the manifold is four or the Hermitian-Weyl structure is locally conformally Kähler.

A New Type Warped Product Metric in Contact Geometry

In this study, we show that there is an $α$-Sasakian structure on product manifold $M_{1}\times β(I)$ where $M_{1}$ is a Kaehlerian manifold that has exact 1-form and $β(I)$ is an open curve. After then, we define a new type warped product metric to study the warped product of almost Hermitian manifolds with almost contact metric manifolds.

关于从伪埃尔米特流形到Kähler流形的$\overline{\partial }_{b}$-调和映射

On $\overline{\partial }_{b}$-harmonic maps from pseudo-Hermitian manifolds to Kähler manifolds

本文研究从伪埃尔米特流形到Kähler流形的映射，引入偏能量泛函，定义并探讨了$\overline{\partial }_{b}$-调和映射与$\partial _{b}$-调和映射，推广了叶状Lichnerowicz型结果及Siu的全纯性结论。

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.

Rigidity of pseudo-Hermitian homogeneous spaces of finite volume

Let $M$ be a pseudo-Hermitian homogeneous space of finite volume. We show that $M$ is compact and the identity component $G$ of the group of holomorphic isometries of $M$ is compact. If $M$ is simply connected, then even the full group of holomorphic isometries is compact. These results stem from a careful analysis of the Tits fibration of $M$, which is shown to have a torus as its fiber. The proof builds on foundational results on the automorphisms groups of compact almost pseudo-Hermitian homogeneous spaces. It is known that a compact homogeneous pseudo-Kähler manifold splits as a product of…

Almost hypercomplex pseudo-Hermitian manifolds and a 4-dimensional Lie group with such structure

Almost hypercomplex pseudo-Hermitian manifolds are considered. Isotropic hyper-Kähler manifolds are introduced. A 4-parametric family of 4-dimensional manifolds of this type is constructed on a Lie group. This family is characterized geometrically. The condition a 4-manifold to be isotropic hyper-Kähler is given.

First and second variation formulae for the sub-Riemannian area in three-dimensional pseudo-hermitian manifolds

We calculate the first and the second variation formula for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We consider general variations that can move the singular set of a C^2 surface and non-singular variation for C_H^2 surfaces. These formulas enable us to construct a stability operator for non-singular C^2 surfaces and another one for C2 (eventually singular) surfaces. Then we can obtain a necessary condition for the stability of a non-singular surface in a pseudo-hermitian 3-manifold in term of the pseudo-hermitian torsion and the Webster scalar curvature. Final…

Tangent Bundles with Sasaki Metric and Almost Hypercomplex Pseudo-Hermitian Structure

The tangent bundle as a $4n$-manifold is equipped with an almost hypercomplex pseudo-Hermitian structure and it is characterized with respect to the relevant classifications. A number of 8-dimensional examples of the considered type of manifold are received from the known explicit examples in that manner.

On $(H,\widetilde{H})$-harmonic Maps between pseudo-Hermitian manifolds

In this paper, we investigate critical maps of the horizontal energy functional $E_{H,\widetilde{H}}(f)$ for maps between two pseudo-Hermitian manifolds $(M^{2m+1},H(M),J,θ)$ and $(N^{2n+1},\widetilde{H}(N), \widetilde{J},\widetildeθ)$. These critical maps are referred to as $(H,\widetilde{H})$-harmonic maps. We derive a CR Bochner formula for the horizontal energy density $|df_{H, \widetilde{H}}|^{2}$, and introduce a Paneitz type operator acting on maps to refine the Bochner formula. As a result, we are able to establish some Bochner type theorems for $(H,\widetilde{H})$-harmonic maps. We al…