Sasakian Manifolds Research Articles

Weak Quasi-Contact Metric Manifolds and New Characteristics of K-Contact and Sasakian Manifolds

Quasi-contact metric manifolds (introduced by Y. Tashiro and then studied by several authors) are a natural extension of contact metric manifolds. Weak almost-contact metric manifolds, i.e., where the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, have been defined by the author and R. Wolak. In this paper, we study a weak analogue of quasi-contact metric manifolds. Our main results generalize some well-known theorems and provide new criterions for K-contact and Sasakian manifolds in terms of conditions on the curvature tensor and other geometric objects associated with the weak quasi-contact metric structure.

Continuity Equation of Transverse Kähler Metrics on Sasakian Manifolds

We introduce the continuity equation of transverse Kähler metrics on Sasakian manifolds and establish its interval of maximal existence. When the first basic Chern class is null (resp. negative), we prove that the solution of the (resp. normalized) continuity equation converges smoothly to the unique η-Einstein metric in the basic Bott–Chern cohomological class of the initial transverse Kähler metric (resp. first basic Chern class). These results are the transverse version of the continuity equation of the Kähler metrics studied by La Nave and Tian, and also counterparts of the Sasaki–Ricci flow studied by Smoczyk, Wang, and Zhang.

Semi-invariant Riemannian maps from Sasakian manifolds endowed with Ricci soliton structure

In this paper, we investigate the behavior of semi-invariant Riemannian maps taking Sasakian structure as total manifolds satisfying Ricci soliton equation, to Riemannian manifolds. We establish necessary and sufficient conditions for the cases when fibers and rangeF⁎ are Einstein. Further, we calculate scalar curvature for rangeF⁎, fibers and total manifolds. Also, we derive some inequalities for semi-invariant Riemannian maps from Sasakian space forms satisfying Ricci soliton equation, to Riemannian manifolds. We construct some examples in support of assumed maps.

η-Einstein soliton on α-Sasakian manifold with respect to Zamkovoy connection

The purpose of this paper is to study some properties of [Formula: see text]-Sasakian manifolds with respect to Zamkovoy connection. We study [Formula: see text]-Einstein soliton on pseudo-projectively flat [Formula: see text]-Sasakian with respect to Zamkovoy connection. Further, we discuss [Formula: see text]-Einstein soliton on this manifold under various curvature conditions.

SCREEN PSEUDO-SLANT LIGHTLIKE SUBMERSIONS FROM INDEFINITE SASAKIAN MANIFOLDS ONTO LIGHTLIKE MANIFOLDS

As a generalization of screen slant lightlike submersions, we introduce the notion of screen pseudo-slant lightlike submersions from indefinite Sasakian manifolds onto lightlike manifolds. We give examples and prove a characterization theorem for the existence of such lightlike submersions. We also obtain integrability conditions of distributions involved in the definition of this class of lightlike submersions. Further, we find necessary and sufficient conditions for foliations determined by these distributions to be totally geodesic.

Radical transversal screen pseudo-slant lightlike submersions from indefinite Sasakian manifolds onto lightlike manifolds

As a generalization of transversal and radical transversal screen slant lightlike submersions, we introduce the notion of radical transversal screen pseudo-slant lightlike submersions from indefinite Sasakian manifolds onto lightlike manifolds with examples. We obtain an existence theorem for such lightlike submersions and study the geometry of distributions involved in the definition.

ON QUASI BI-SLANT RIEMANNIAN MAPS FROM LORENTZIAN PARA SASAKIAN MANIFOLDS

We first introduce quasi bi-slant Riemannian maps and study such Riemannian maps from Lorentzian para Sasakian manifolds into Riemannian manifolds. We give necessary and sufficient conditions for the integrability of the distributions which are involved in the definition of the quasi bi-slant Riemannian map and investigate their leaves. We also obtain totally geodesic conditions for such maps. Moreover, we provide some examples for this notion.

Projectively induced Kähler cones over regular Sasakian manifolds

Motivated by a conjecture in Loi et al. (Math Zeit 290:599–613, 2018) we prove that the Kähler cone over a regular complete Sasakian manifold is Ricci-flat and projectively induced if and only if it is flat. We also obtain that, up to Da\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal D_a$$\\end{document}—homothetic transformations, Kähler cones over homogeneous compact Sasakian manifolds are projectively induced. As main tool we provide a relation between the Kähler potentials of the transverse Kähler metric and of the cone metric.

Immersions into Sasakian space forms

We study immersions of Sasakian manifolds into finite and infinite dimensional Sasakian space forms. After proving Calabi’s rigidity results in the Sasakian setting, we characterise all homogeneous Sasakian manifolds which admit a (local) Sasakian immersion into a nonelliptic Sasakian space form. Moreover, we give a characterisation of homogeneous Sasakian manifolds which can be embedded into the standard sphere both in the compact and noncompact case.

Nilpotent Aspherical Sasakian Manifolds

Abstract We show that every compact aspherical Sasakian manifold with nilpotent fundamental group is diffeomorphic to a Heisenberg nilmanifold.

$\mathcal{Z^\ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure

The main goal of this manuscript is to investigate the properties of $N(k)$-contact metric manifolds admitting a $\mathcal{Z^\ast}$-tensor. We prove the necessary conditions for which $N(k)$-contact metric manifolds endowed with a $\mathcal{Z^\ast}$-tensor are Einstein manifolds. In this sequel, we accomplish that an $N(k)$-contact metric manifold endowed with a $\mathcal{Z^\ast}$-tensor satisfying $\mathcal{Z^\ast}(\mathcal{G}_{1},\hat{\zeta})\cdot \mathcal{\overset{\star}R}=0$ is either locally isometric to the Riemannian product $E^{n+1}(0)\times S^{n}(4)$ or an Einstein manifold. We also prove the condition for which an $N(k)$-contact metric manifold endowed with a $\mathcal{Z^\ast}$-tensor is a Sasakian manifold. To validate some of our results, we construct a non-trivial example of an $N(k)$-contact metric manifold.

Conformal bi-slant ξ⊥-Riemannian submersion: a note in contact geometry

This study explores conformal bi-slant ξ⊥-Riemannian submer- sion where the total manifold is a contact metric manifold, more specifically a Sasakian manifold. To illustrate this study, few non-trivial examples are discussed. Meanwhile, we addressed the conditions for integrability of anti- invariant and slant distributions and determine the conditions for the map that must be met in order to be totally geodesic. Furthermore, some decomposition theorems for the fibres as well as for the total space are discussed.

Gradient Estimates for the CR Heat Equation on Closed Sasakian Manifolds

Gradient Estimates for the CR Heat Equation on Closed Sasakian Manifolds

Coisotropic Warped Product Submanifolds of a Mixed 3-Sasakian Statistical Manifold

Coisotropic Warped Product Submanifolds of a Mixed 3-Sasakian Statistical Manifold

On submanifolds of a Sasakian manifold

In this paper, we study submanifolds of a Sasakian manifolds provide with a torqued vector field and also accepting a Ricci-Yamabe soliton of both Sasakian manifold and Sasakian spaceform. Further, we obtain some important results which categorize the submanifolds admitting a Ricci-Yamabe soliton of Sasakian spaceform.

Foliated structure of weak nearly Sasakian manifolds

Foliated structure of weak nearly Sasakian manifolds

A New Representation for Slant Curves in Sasakian 3-Manifolds

In this paper, we define a new kind of curve called $N$-slant curve whose principal normal vector field makes a constant angle with the Reeb vector field $\xi$ in Sasakian $3$-manifolds. Then, we give some characterizations of $N$-slant curves in Sasakian $3$-manifolds and we obtain some properties of the curves in $\mathbb{R}^{3}(-3)$. Moreover, we investigate the conditions of $C$-parallel and $C$-proper mean curvature vector fields along $N$-slant curves in Sasakian $3$-manifolds. Finally, we study $N$-slant curves of type $AW(k)$ where k=1,2 or 3.

Harmonic Complex Structures and Special Hermitian Metrics on Products of Sasakian Manifolds

Harmonic Complex Structures and Special Hermitian Metrics on Products of Sasakian Manifolds

Characterizations of warped product pointwise semi-slant submanifolds of Sasakian manifold

A warped product submanifold, whose tangent bundle can be decomposed to two orthogonal distributions; invariant and pointwise slant, is called a warped product pointwise semi-slant submanifold. The objective of this paper is to classify warped product pointwise semi-slant submanifolds, isometrically immersed into a Sasakian manifold. Park [25] provided the (non)-existence of a warped product pointwise semi-slant submanifold in a Sasakian manifold such that the structure vector field is tangential to fibers. In contrast, we provide intriguing theorems on warped product pointwise semi-slant submanifolds in a Sasakian manifold in terms of the shape operator and tensor fields such that the structure vector field is tangential to a base manifold and each fiber is a pointwise slant submanifold.

On statistical submersions from 3-Sasakian statistical manifolds

In this paper, we define and characterize 3-Sasakian statistical manifolds and then investigate statistical submersions from 3-Sasakian statistical manifolds. We prove that invariant statistical submersions from 3-Sasakian statistical manifolds with vertical structure vector fields have 3-Sasakian statistical totally geodesic fibers. Moreover, the base space admits a quaternionic Kähler statistical structure. We construct non-trivial examples to illustrate some results of the paper.