Sasakian Manifolds Research Articles

On the Pseudohermitian Curvature of Contact Semi-Riemannian Manifolds

Let M be a contact semi-Riemannian manifold, equivalently a non degenerate almost CR manifold. In this paper we study the pseudo-hermitian Ricci curvature, pseudo-Einstein and $$\eta $$-Einstein manifolds. Then, by using the pseudo-Einstein and the $$\eta $$-Einstein conditions, some rigidity theorems are established to characterize Sasakian manifolds among nondegenerate CR manifolds. In particular, if the Webster metric $$g_\theta $$ of nondegenerate CR structure $$({\mathcal {H}},\theta ,J)$$ is pseudo-Einstein with Webster scalar curvature $${\hat{r}}\ne 0$$, then there exists a real constant $$t\ne 0$$ for which the Webster metric associated to $$({\mathcal {H}},t\theta ,J)$$ is Einstein–Sasakian.

Gradient estimate of positive eigenfunctions of sub-Laplacian on complete pseudo-Hermitian manifolds

In this paper, we derive an explicit gradient estimate of positive solutions of Δbu=−λu on complete noncompact pseudo-Hermitian manifolds with bounded geometric conditions. As a consequence, we obtain an estimate of the greatest lower bound for the L2-spectrum of the sub-Laplacian operator. Moreover, we recapture the Liouville theorem of positive pseudo-harmonic functions on complete noncompact Sasakian manifolds with nonnegative pseudo-Hermitian Ricci curvature.

Beta-almost Ricci solitons on Sasakian 3-manifolds

In this paper we characterize the Sasakian 3-manifolds admitting β-almost Ricci solitons whose potential vector field is a contact vector field. Among others we prove that a β-almost Ricci soliton whose potential vector field is a contact vector field on a Sasakian 3-manifold is shrinking, Einstein and non-trivial. Moreover, we prove that this type of manifolds are isometric to a sphere of radius √7.

On a Lorentzian Sasakian manifold endowed with a quarter-symmetric metric connection

Abstract In the present paper, some results on a Lorentzian Sasakian manifold endowed with a quarter-symmetric metric connection have been studied.

Integrability of the distributions of GCR-lightlike submanifolds of (ε)-Sasakian manifolds

We study GCR-lightlike submanifolds of (e)-Sasakian manifolds and derived some important structural characteristics equations for further uses. We also obtain some necessary and sufficient conditions for the integrability of various distributions of GCR-lightlike submanifolds of (e)-Sasakian manifolds.

HOMOGENEOUS SASAKI AND VAISMAN MANIFOLDS OF UNIMODULAR LIE GROUPS

A Vaisman manifold is a special kind of locally conformally Kähler manifold, which is closely related to a Sasaki manifold. In this paper, we show a basic structure theorem of simply connected homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups, up to holomorphic isometry. For the case of unimodular Lie groups, we obtain a complete classification of simply connected Sasaki and Vaisman unimodular Lie groups, up to modification.

Warped product CR-submanifolds of Sasakian manifolds with respect to certain connections

The present paper deals with the study of warped product CR-submanifolds of Sasakian manifolds with respect to semisymmetric metric and semisymmetric non-metric connection. Among others, Ricci solitons of such notions have been investigated.

Lorentzian para-Sasakian Manifolds Admitting a New Type of Quarter-symmetric Non-metric ξ-connection

We define a new type of quarter-symmetric non-metric $\xi$-connection on an $LP$-Sasakian manifold and prove its existence. We provide its application in the general theory of relativity. To validate the existence of the quarter-symmetric non-metric $\xi$-connection on an $LP$-Sasakian manifold, we give a non-trivial example in dimension $4$ and verify our results.

Transverse fully nonlinear equations on Sasakian manifolds and applications

We prove a priori estimates for a class of transverse fully nonlinear equations on Sasakian manifolds and give some geometric applications such as the transversion Calabi-Yau theorem for transverse balanced and (strongly) Gauduchon metrics. We also explain that similar results hold on compact oriented, taut, transverse Hermitian foliated manifold of complex co-dimension n, and give some geometric applications such as the transverse Calabi-Yau theorems for transverse Hermitian and (strongly) Gauduchon metrics.

Some recurrent properties of $LP$-Sasakian nanifolds

The aim of the present paper is to study certain recurrent properties of $LP$-Sasakian manifolds. Here we first describe Ricci $\eta$-recurrent $LP$-Sasakian manifolds. Further we study semi-generalized recurrent and three dimensional locally generalized concircularly $\phi$-recurrent $LP$-Sasakian manifolds and got interesting results.

G-Sasaki Manifolds and K-Energy

In this paper, we introduce a class of Sasaki manifolds with a reductive $G$-group action, called $G$-Sasaki manifolds. By reducing K-energy to a functional defined on a class of convex functions on a moment polytope, we give a criterion for the properness of K-energy. In particular, we deduce a sufficient and necessary condition related to the polytope for the existence of transverse $G$-Sasaki Einstein metrics. A similar result is also obtained for transverse $G$-Sasaki Ricci solitons. As an application, we construct several examples of $G$-Sasaki Ricci solitons by an established openness theorem for transverse $G$-Sasaki Ricci solitons.

On the Sharp Dimension Estimate of CR Holomorphic Functions in Sasakian Manifolds

Abstract This is the very 1st 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.

η-Ricci Solitons in δ-Lorentzian Trans Sasakian Manifolds with a Semi-symmetric Metric Connection

η-Ricci Solitons in δ-Lorentzian Trans Sasakian Manifolds with a Semi-symmetric Metric Connection

A (CHR)3-flat trans-Sasakian manifold

In [4] M. Prvanovic considered several curvaturelike tensors defined for Hermitian manifolds. Developing her ideas in [3], we defined in an almost contact Riemannian manifold another new curvaturelike tensor field, which is called a contact holomorphic Riemannian curvature tensor or briefly (CHR)3-curvature tensor. Then, we mainly researched (CHR)3-curvature tensor in a Sasakian manifold. Also we proved, that a conformally (CHR)3-flat Sasakian manifold does not exist. In the present paper, we consider this tensor field in a trans-Sasakian manifold. We calculate the (CHR)3-curvature tensor in a trans-Sasakian manifold. Also, the (CHR)3-Ricci tensor ρ3 and the (CHR)3-scalar curvature τ3 in a trans-Sasakian manifold have been obtained. Moreover, we define the notion of the (CHR)3-flatness in an almost contact Riemannian manifold. Then, we consider this notion in a trans-Sasakian manifold and determine the curvature tensor, the Ricci tensor and the scalar curvature. We proved that a (CHR)3-flat trans-Sasakian manifold is a generalized ɳ-Einstein manifold. Finally, we obtain the expression of the curvature tensor with respect to the Riemannian metric g of a trans-Sasakian manifold, if the latter is (CHR)3-flat.

On the existence of an almost generalized weakly-symmetric Sasakian manifold admitting quarter symmetric metric connection

The object of the present work is to study an almost generalized weakly symmetric Sasakian manifold admitting quarter symmetric metric connection with a non-trivial example.

On Formality of Some Homogeneous Spaces

Let G / H be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus T H of H is conjugate to a torus T β whose Lie algebra t β is the kernel of the maximal root β of the root system of the complexified Lie algebra g c . We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernández, Muñoz, and Sanchez which contains a full analysis of the formality property of S O ( 3 ) -bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary.

Correction to: Affine connections on 3-Sasakian homogeneous manifolds

In the original publication of this article, the title was incorrectly published as “Affine connections on 3-Sasakian and manifolds”. The correct title should be “Affine connections on 3-Sasakian homogeneous manifolds”.

Some properties of trans Sasakian manifolds admitting a quarter-symmetric non-metric connection

Some properties of trans Sasakian manifolds admitting a quarter-symmetric non-metric connection

Submanifolds with parallel mean curvature vector in η-Einstein Sasaki manifold

In this note, we prove that for a 3-dimensional submanifold in an η-Einstein 5-Sasaki manifold with Einstein constant a≥4, if the Reeb vector field is tangent to the submanifold and it has parallel mean curvature form, then it is a slant submanifold. Furthermore, if a>4, then it is either an invariant submanifold or an anti-invariant submanifold.

On a class of generalized \u03c6-recurrent Sasakian manifold

The object of the present paper was to introduce the notion of hyper generalized φ-recurrent Sasakian manifold and quasi generalized φ-recurrent Sasakian manifold and study its various geometric properties. The existence of hyper generalized φ-recurrent Sasakian manifold and quasi generalized φ-recurrent Sasakian manifold is proved by giving a proper example.