Sasakian Manifolds Research Articles

Characterization of Almost \(\eta\) -Ricci Solitons With Respect to Schouten-van Kampen Connection on Sasakian Manifolds

In this paper, we investigate Sasakian manifolds that admit almost \(\eta\) -Ricci solitons with respect to the Schouten-van Kampen connection using certain curvature tensors. Concepts of Ricci pseudosymmetry for Sasakian manifolds admitting \(\eta\)-Ricci solitons are introduced based on the selection of specific curvature tensors such as Riemann, concircular, projective, pseudo-projective, M-projective, and W2 tensors. Subsequently, necessary conditions are established for a Sasakian manifold admitting \(\eta\)-Ricci soliton with respect to the Schouten-van Kampen connection to be Ricci semisymmetric, based on the choice of curvature tensors. Characterizations are then derived, and classifications are made under certain conditions.

Characterization of Sasakian manifolds

Weak almost contact manifolds, i.e., the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, defined by the author and Wolak, allowed us to take a new look at the theory of contact manifolds. In this paper, we continue our study of a new structure of this type, called a weak nearly Sasakian structure, and prove theorems characterizing Sasakian manifolds. Our main result generalizes the theorem by Nicola–Dileo–Yudin (2018) and provides a new criterion for a weak almost-contact metric manifold to be Sasakian.

Complex geometry and Hermitian metrics on the product of two Sasakian manifolds

A Sasakian manifold is a Riemannian manifold whose metric cone admits a certain Kähler structure which behaves well under homotheties. We show that the product of two compact Sasakian manifolds admits a family of complex structures indexed by a complex nonreal parameter, none of whose members admits either compatible Kähler metrics, locally conformally Kähler metrics or balanced metrics, if both Sasakian manifolds are of dimension greater than 3. We compare this family with another family of complex structures which has been studied in the literature. We compute the Dolbeault cohomology groups of these products of compact Sasakian manifolds.

Lifts of a Semi-Symmetric Metric Connection from Sasakian Statistical Manifolds to Tangent Bundle

The lifts of Sasakian statistical manifolds associated with a semi-symmetric metric connection in the tangent bundle are characterized in the current research. The relationship between the complete lifts of a statistical manifold with semi-symmetric metric connections and Sasakian statistical manifolds with a semi-symmetric metric connection in the tangent bundle is investigated. We also discuss the classification of Sasakian statistical manifolds with respect to semi-symmetric metric connections in the tangent bundle. Finally, we derive an example of the lifts of Sasakian statistical manifolds to the tangent bundle.

Warped Product Pointwise Semi-slant Submanifolds of Almost Contact Manifolds

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 cosymplectic manifolds and obtain several fundamental results, including a characterization for warped product pointwise semi-slant submanifolds.

Contact CR $ \delta $-invariant: an optimal estimate for Sasakian statistical manifolds

<p>Chen (1993) developed the theory of $ \delta $-invariants to establish novel necessary conditions for a Riemannian manifold to allow a minimal isometric immersion into Euclidean space. Later, Siddiqui et al. (2024) derived optimal inequalities involving the CR $ \delta $-invariant for a generic statistical submanifold in a holomorphic statistical manifold of constant holomorphic sectional curvature. In this work, we extend the study of such optimal inequality to the contact CR $ \delta $-invariant on contact CR-submanifolds in Sasakian statistical manifolds of constant $ \phi $-sectional curvature. This paper concludes with a summary and final remarks.</p>

-Ricci Solitons On Sasakian Manifolds Admitting General Connection

In this paper we study -Ricci soliton on Sasakian manifolds admitting general connection and obtain some conditions for the soliton to be shrinking, steady and expanding. MSC 2020: 53C15, 53C25, 53D15.

Approximation of regular Sasakian manifolds

Approximation of regular Sasakian manifolds

Yamabe solitons in contact geometry

It is shown that the scalar curvature of a Yamabe soliton as a Sasakian manifold is constant and the soliton vector field is Killing. The same conclusion is shown to hold for a Yamabe soliton as a $K$-contact manifold $M^{2n+1}$ if any one of the following conditions hold: (i) its scalar curvature is constant along the soliton vector field $V$, (ii) $V$ is an eigenvector of the Ricci operator with eigenvalue $2n$, (iii) $V$ is gradient.

Immersions of Sasaki–Ricci solitons into homogeneous Sasakian manifolds

Immersions of Sasaki–Ricci solitons into homogeneous Sasakian manifolds

Lifts of hypersurfaces on a Sasakian manifold to tangent bundles

The aim of present paper is to introduce a Sasakian manifold immersed with Quartersymmetric semi-metric connection to tangent bundle. We deduce the relation between
 Riemannian connection and Quarter-symmetric semi-metric connection to the tangent bundle on
 a Sasakian manifold and obtain theorems on the totally geodesic and totally umbilical.

Higgs bundles and flat connections over compact Sasakian manifolds, II: quasi-regular bundles

Higgs bundles and flat connections over compact Sasakian manifolds, II: quasi-regular bundles

Weak Nearly Sasakian and Weak Nearly Cosymplectic Manifolds

Weak contact metric structures on a smooth manifold, introduced by V. Rovenski and R. Wolak in 2022, have provided new insight into the theory of classical structures. In this paper, we define new structures of this kind (called weak nearly Sasakian and weak nearly cosymplectic and nearly Kähler structures), study their geometry and give applications to Killing vector fields. We introduce weak nearly Kähler manifolds (generalizing nearly Kähler manifolds), characterize weak nearly Sasakian and weak nearly cosymplectic hypersurfaces in such Riemannian manifolds and prove that a weak nearly cosymplectic manifold with parallel Reeb vector field is locally the Riemannian product of a real line and a weak nearly Kähler manifold.

On slant Riemannian submersions from conformal Sasakian manifolds

In this paper, we study slant Riemannian submersions from conformal Sasakian manifolds onto Riemannian manifolds. We investigate the geometry of foliations associated with the submersion and give sufficient conditions for the submersion to be harmonic.

Uniformizations of Compact Sasakian Manifolds

Abstract We give a criterion for compact Sasakian manifolds to be deformed to Sasakian manifolds, which are locally isomorphic to circle bundles of anti-canonical bundles over Hermitian symmetric spaces as a Sasakian analogue of Simpson’s uniformization results related to variations of Hodge structure and Higgs bundles.

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

Abstract 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 the Yau uniformization conjecture which states that any complete noncompact Sasakian manifold of positive pseudohermitian bisectional curvature is CR biholomorphic to the standard Heisenberg group. More precisely, we first construct CR-holomorphic functions with controlled growth in a sequence of exhaustion domains in Sasakian manifolds by applying the Cheeger–Colding theory. Secondly, via the CR analogue of a tangent cone at infinity and the three-circle theorem, we are able to take the subsequence to obtain a nonconstant CR-holomorphic function of polynomial growth.

On Quasi Hemi-Slant Submersions

The paper deals with the notion of quasi hemi-slant submersions from Lorentzian para Sasakian manifolds onto Riemannian manifolds. These submersions are generalization of hemi-slant submersions and semi-slant submersions. In this paper, we also study the geometry of leaves of distributions which are involved in the definition of the submersion. Further, we obtain the conditions for such distributions to be integrable and totally geodesic. Moreover, we also give the characterization theorems for proper quasi hemi-slant submersions and provide some examples of it.

Study of Sasakian manifolds admitting $$*$$-Ricci–Bourguignon solitons with Zamkovoy connection

Study of Sasakian manifolds admitting $$*$$-Ricci–Bourguignon solitons with Zamkovoy connection

Solitonic View of Generic Contact CR-Submanifolds of Sasakian Manifolds with Concurrent Vector Fields

This paper mainly devotes to the study of some solitons such as Ricci and Yamabe solitons and also their combination called Ricci-Yamabe solitons. In the geometry of solitons, a fundamental question is to identify the conditions under which these solitons can be trivial. Firstly, in this paper we study some extensive results on generic contact CR-submanifolds of Sasakian manifolds endowed with concurrent vector fields. Then some applications of solitons such as Ricci and Ricci-Yamabe solitons on such submanifolds with concurrent vector fields in the same ambient manifold have been discussed.

A Note on Nearly Sasakian Manifolds

A class of nearly Sasakian manifolds is considered in this paper. We discuss the geometric effects of some symmetries on such manifolds and show, under a certain condition, that the class of Ricci semi-symmetric nearly Sasakian manifolds is a subclass of Einstein manifolds. We prove that a Codazzi-type Ricci nearly Sasakian space form is either a Sasakian manifold with a constant ϕ-holomorphic sectional curvature H=1 or a 5-dimensional proper nearly Sasakian manifold with a constant ϕ-holomorphic sectional curvature H>1. We also prove that the spectrum of the operator H2 generated by the nearly Sasakian space form is a set of a simple eigenvalue of 0 and an eigenvalue of multiplicity 4, and we induce that the underlying space form carries a Sasaki–Einstein structure. We show that there exist integrable distributions with totally geodesic leaves on the same manifolds, and we prove that there are no proper nearly Sasakian space forms with constant sectional curvature.