Soliton on Sasakian manifold endowed with quarter-symmetric non-metric connection on the tangent bundle

Colney, Lalnunenga; Kumar, Rajesh

doi:10.1108/AJMS-02-2025-0032

Purpose

The purpose of this paper is to study the properties of the solitons on Sasakian manifold on the tangent bundle with respect to quarter symmetric non metric connection.

Design/methodology/approach

We used the vertical and complete lifts, Ricci solitons, tangent bundle, Einstein manifold, partial differential equation, Einstein semi-symmetric.

Findings

In the proposed paper, we study the complete lift of the Ricci soliton on Sasakian manifold endowed with quarter-symmetric non-metric connection (QSNMC) on the tangent bundle and discuss its curvature properties. The complete lift of the Ricci soliton on Sasakian manifold with QSNMC is investigated and it was found to be η-Einstein manifold on the tangent bundle. We study several properties of the complete lift of the Ricci soliton on Sasakian manifold with QSNMC satisfying certain conditions. Also the complete lifts of the Ricci soliton on Φ-recurrent and Einstein semi-symmetric Sasakian manifolds on the tangent bundle are studied and some theorems are also proved.

Originality/value

The present author carried out this research in extension of studying the properties of differentiable manifold using lifting theory.

1. Introduction

The study of the geometry of the tangent bundle has been a topic of enduring fascination in the field of differential geometry, presenting distinctive challenges. The lift function facilitates extending differentiable structures from any manifold M to its tangent bundle. Yano and Ishihara [1] developed the theory of lifts of geometric structures and connections to tangent bundles. Researchers like Yano and Kobayashi [2], Tani [3], Pandey and Chaturvedi [4] and Khan [5–16] explored various connections and structures on different manifolds on the tangent bundles. In 2023, Kumar et al. [17, 18] studied the complete lifts of LP-Sasakian manifold endowed with quarter-symmetric non-metric connection (QSNMC) and Sasakian statistical manifolds endowed with semi-symmetric metric connection on the tangent bundle. Li et al. in Ref. [19] studied Ricci and gradient Ricci solitons of pseudo-Riemannian manifold associated with lift of Ricci quarter-symmetric metric connection (RQSMC) on the tangent bundle. Murat in Ref. [20] studied conformal Yamabe solitons on tangent bundles with respect to the complete lifts of a semi-symmetric metric connection and a projective semi-symmetric connection. Recently the same author Murat in Ref. [21] studied the Ricci solitons on tangent bundles with respect to the complete lift of a projective semi-symmetric connection. Motivated by their studies, we will investigate the complete lift of Ricci soliton on Sasakian manifold to tangent bundle with associated quarter-symmetric non-metric connection (QSNMC).

In a differentiable manifold M, we define T₀M = ⋃_p _∈ _MT_0pM as the tangent bundle, with T_0pM being the tangent space at point p ∈ M and π: T₀M → M is the natural bundle structure of T₀M over M. For any coordinate system (Q, x^h) in M, where (x^h) is a local coordinate system in the neighborhood Q, (π⁻¹(Q), x^h, y^h) is coordinate system in T₀M, where (x^h, y^h) is an induced coordinate system in π⁻¹(Q) from (x^h) [1]. We define τ₁ as a vector field, F₀ as a (1,1) tensor field, f₀ as a function, ω₀ as a 1-form and $\dot{\nabla}$ as an affine connection in M, its vertical and complete lifts are denoted by subscripts v and c, respectively. The subsequent equations for complete and vertical lifts are established by [1, 22]

{(f_{0} τ_{1})}^{v} = f_{0}^{v} τ_{1}^{v}, {(f_{0} τ_{1})}^{c} = f_{0}^{c} τ_{1}^{v} + f_{0}^{v} τ_{1}^{c},

(1.1)

τ_{1}^{v} f_{0}^{v} = 0, τ_{1}^{v} f_{0}^{c} = τ_{1}^{c} f_{0}^{v} = {(τ_{1} f_{0})}^{v}, τ_{1}^{c} f_{0}^{c} = {(τ_{1} f_{0})}^{c},

(1.2)

ω_{0} (f_{0}^{v}) = 0, ω_{0}^{v} (τ_{1}^{c}) = ω_{0}^{c} (τ_{1}^{v}) = ω_{0} {(τ_{1})}^{v}, ω_{0}^{c} (τ_{1}^{c}) = ω_{0} {(τ_{1})}^{c},

(1.3)

F_{0}^{v} τ_{1}^{c} = {(F_{0} τ_{1})}^{v}, F_{0}^{c} τ_{1}^{c} = {(F_{0} τ_{1})}^{c},

(1.4)

{[τ_{1}, τ_{2}]}^{v} = [τ_{1}^{c}, τ_{2}^{v}] = [τ_{1}^{v}, τ_{2}^{c}], {[τ_{1}, τ_{2}]}^{c} = [τ_{1}^{c}, τ_{2}^{c}],

(1.5)

{\dot{\nabla}}^{c}_{τ_{1}^{c}} τ_{2}^{c} = {({\dot{\nabla}}_{τ_{1}} τ_{2})}^{c}, {\dot{\nabla}}^{c}_{τ_{1}^{c}} τ_{2}^{v} = {({\dot{\nabla}}_{τ_{1}} τ_{2})}^{v} .

(1.6)

The idea of a Ricci soliton can be seen as a natural extension of Einstein manifolds. Ricci flow, initially introduced by Hamilton in Ref. [23] to establish a standard metric on a smooth manifold, has become a powerful tool in analyzing Riemannian manifolds, particularly those with positive curvature. It can be described as a differential equation governing how metrics on a Riemannian manifold evolve, expressed as $\frac{\partial g}{\partial t} = - 2 \dot{S}$ ⁠. Essentially, a Ricci soliton is a form of Ricci flow that undergoes transformations solely through a one-parameter group of diffeomorphisms and scaling. Perelman in Refs. [24, 25] used Ricci flow to prove the Poincaré conjecture. A Ricci soliton (g, V, γ) on a Riemannian manifold (M⁽²ⁿ⁺¹⁾, g) is defined by the equation in Ref. [26] as:

{\dot{L}}_{V} g + 2 \dot{S} + 2 γ g = 0,

(1.7)

where ${\dot{L}}_{V}$ is the Lie derivative along V on M, $\dot{S}$ denotes the Ricci tensor, and γ is a constant. Ricci solitons are classified as shrinking, steady, or expanding based on whether γ is less than zero, equal to zero or greater than zero respectively. Sharma in Ref. [27] extensively studied Ricci solitons in contact geometry, while Ghosh et al. in Ref. [28] explored gradient Ricci solitons on non-Sasakian (κ, μ)-contact manifolds. The term “Sasakian manifold” was coined in 1960 by Shigeo Sasaki in Ref. [29], and since then, many geometers have investigated the Ricci soliton of Sasakian manifolds, as can be seen in Refs. [26, 30–33]. Recently, Singh et al. in Ref. [26] examined the Ricci soliton on Sasakian manifolds with QSNMCs.

Let $\overset{ˇ}{\nabla}$ be a linear connection on a (2n + 1)-dimensional differentiable manifold. If the torsion tensor $\overset{ˇ}{T}$ of $\overset{ˇ}{\nabla}$ defined by

\overset{ˇ}{T} (τ_{1}, τ_{2}) = {\overset{ˇ}{\nabla}}_{τ_{1}} τ_{2} - {\overset{ˇ}{\nabla}}_{τ_{2}} τ_{1} - [τ_{1}, τ_{2}],

(1.8)

satisfies

\overset{ˇ}{T} (τ_{1}, τ_{2}) = η (τ_{2}) Φ τ_{1} + η (τ_{1}) Φ τ_{2},

(1.9)

for any vector fields τ₁, τ₂ ∈ Γ(M²ⁿ⁺¹), where η is a 1-form and Φ is a (1, 1) tensor field, then the connection $\overset{ˇ}{\nabla}$ is called a quarter-symmetric connection [34]. Additionally, if $\overset{ˇ}{\nabla}$ satisfy the condition

({\overset{ˇ}{\nabla}}_{τ_{1}} g) (τ_{2}, τ_{3}) \neq 0,

(1.10)

then it is called QSNMC [35, 36].

The proposed paper begins with introduction in section 1. Section 2 is concerned with the preliminaries. In section 3, we investigate the complete lift of the QSNMC on Sasakian manifold to the tangent bundle and proved some theorems. The complete lift of the Ricci soliton on Sasakian manifold associated with QSNMC and its properties satisfying some conditions on the tangent bundle are studied on sections 4. In section 5, we investigate the complete lifts of the Ricci soliton on Ricci-recurrent and Φ-recurrent Sasakian manifold with QSNMC on the tangent bundle. In the last section, we study the complete lift of Ricci solitons on Einstein semi-symmetric Sasakian manifold admitting QSNMC on the tangent bundle.

2. Preliminaries

Let M be a (2n + 1) dimensional differentiable manifold equipped with an almost contact structure (Φ, B, η, g) satisfying [26].

Φ^{2} τ_{1} = - τ_{1} + η (τ_{1}) B, η (B) = 1, Φ B = 0, η (Φ τ_{1}) = 0,

(2.1)

g (Φ τ_{1}, τ_{2}) = - g (τ_{1}, Φ τ_{2}), g (τ_{1}, B) = η (τ_{1}),

(2.2)

g (Φ τ_{1}, Φ τ_{2}) = g (τ_{1}, τ_{2}) - η (τ_{1}) η (τ_{2}),

(2.3)

for any vector field τ₁, τ₂ ∈ Γ(M), where Φ is (1,1)-tensor field, B is a vector field, η is 1-form and g is the Riemannian metric.

An almost contact metric manifold M of dimension (2n + 1) is said to be Sasakian manifold if

({\dot{\nabla}}_{τ_{1}} Φ) τ_{2} = g (τ_{1}, τ_{2}) B - η (τ_{2}) τ_{1},

(2.4)

where $\dot{\nabla}$ is a Levi-Civita connection on Riemannian metric g. From equation (2.3), we get

{\dot{\nabla}}_{τ_{1}} B = - Φ τ_{1},

(2.5)

({\dot{\nabla}}_{τ_{1}} η) (τ_{2}) = g (τ_{1}, Φ τ_{2}) .

(2.6)

The following relations are also hold by Sasakian manifold M⁽²ⁿ⁺¹⁾:

\dot{R} (τ_{1}, τ_{2}) B = η (τ_{2}) τ_{1} - η (τ_{1}) τ_{2},

(2.7)

\dot{R} (B, τ_{2}) τ_{1} = g (τ_{1}, τ_{2}) B - η (τ_{1}) τ_{2},

(2.8)

\dot{R} (B, τ_{1}) B = η (τ_{1}) B - τ_{1},

(2.9)

\dot{S} (τ_{1}, B) = 2 n η (τ_{1}),

(2.10)

\dot{Q} B = 2 n B,

(2.11)

\dot{S} (Φ τ_{1}, Φ τ_{2}) = S (τ_{1}, τ_{2}) - 2 n η (τ_{1}) η (τ_{2}) .

(2.12)

From now on, we will use the notation M to denote the Sasakian manifold of dimension (2n + 1).

Definition 2.1.

An almost contact metric manifold M is known as η-Einstein manifolds if there exists the real valued function γ₁, γ₂, such that [26]

\dot{S} (Φ τ_{1}, τ_{2}) = γ_{1} g (τ_{1}, τ_{2}) + γ_{2} η (τ_{1}) η (τ_{2}) .

(2.13)

For γ₂ = 0, the manifold M is said to be an Einstein manifold.

Definition 2.2.

In a Riemannian manifold (M, g), the projective curvature tensor is defined as [26]

\dot{P} (τ_{1}, τ_{2}) τ_{3} = \dot{R} (τ_{1}, τ_{2}) τ_{3} - \frac{1}{2 n} [\dot{S} (τ_{2}, τ_{3}) τ_{1} - \dot{S} (τ_{1}, τ_{3}) τ_{2}] .

(2.14)

2.1 Sasakian manifold on the tangent bundle

We suppose that T₀M is the tangent bundle, and $τ_{1} = τ_{1}^{i} \frac{\partial}{\partial x^{i}}$ is a local vector field on M; then, its vertical and complete lifts in terms of partial differential equations are [15–18].

τ_{1}^{v} = τ_{1}^{i} \frac{\partial}{\partial y^{i}},

(2.15)

τ_{1}^{c} = τ_{1}^{i} \frac{\partial}{\partial x^{i}} + \frac{\partial τ_{1}^{i}}{\partial x^{j}} y^{j} \frac{\partial}{\partial y^{i}} .

(2.16)

Obtaining the complete lifts on equations (2.1–2.14) by mathematical operator we get

Φ^{2} τ_{1}^{c} = - τ_{1}^{c} + η^{c} (τ_{1}^{c}) B^{v} + η^{v} (τ_{1}^{c}) B^{c}, η^{c} (B^{c}) = 1,

(2.17)

{(Φ B)}^{v} = 0, η^{c} {(Φ τ_{1})}^{c} = 0,

(2.18)

g^{c} ({(Φ τ_{1})}^{c}, τ_{2}^{c}) = - g^{c} (τ_{1}^{c}, {(Φ τ_{2})}^{c}),

(2.19)

g^{c} (τ_{1}^{c}, B^{c}) = η^{c} (τ_{1}^{c}),

(2.20)

g^{c} ({(Φ τ_{1})}^{c}, {(Φ τ_{2})}^{c}) = g^{c} (τ_{1}^{c}, τ_{2}^{c}) - η^{c} (τ_{1}^{c}) η^{v} (τ_{2}^{c}) - η^{v} (τ_{1}^{c}) η^{c} (τ_{2}^{c}),

(2.21)

({\dot{\nabla}}_{τ_{1}^{c}}^{c} Φ^{c}) τ_{2}^{c} = g^{c} (τ_{1}^{c}, τ_{2}^{c}) B^{v} + g^{c} (τ_{1}^{v}, τ_{2}^{c}) B^{c} - η^{c} (τ_{2}^{c}) τ_{1}^{v} - η^{v} (τ_{2}^{c}) τ_{1}^{c},

(2.22)

{\dot{\nabla}}_{τ_{1}^{c}}^{c} B^{c} = - {(Φ τ_{1})}^{c},

(2.23)

({\dot{\nabla}}_{τ_{1}^{c}}^{c} η^{c}) (τ_{2}^{c}) = g^{c} (τ_{1}^{c}, {(Φ τ_{2})}^{c}) .

(2.24)

{\dot{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) B^{c} = η^{c} (τ_{2}^{c}) τ_{1}^{v} + η^{v} (τ_{2}^{c}) τ_{1}^{c} - η^{c} (τ_{1}^{c}) τ_{2}^{v} - η^{v} (τ_{1}^{c}) τ_{2}^{c},

(2.25)

{\dot{R}}^{c} (B^{c}, τ_{2}^{c}) τ_{1}^{c} = g^{c} (τ_{1}^{c}, τ_{2}^{c}) B^{v} + g^{c} (τ_{1}^{v}, τ_{2}^{c}) B^{c} - η^{c} (τ_{1}^{c}) τ_{2}^{v} - η^{v} (τ_{1}^{c}) τ_{2}^{c},

(2.26)

{\dot{R}}^{c} (B^{c}, τ_{1}^{c}) B^{c} = η^{c} (τ_{1}^{c}) B^{v} + η^{v} (τ_{1}^{c}) B^{c} - τ_{1}^{c},

(2.27)

{\dot{S}}^{c} (τ_{1}^{c}, B^{c}) = 2 n η^{c} (τ_{1}^{c}),

(2.28)

{\dot{Q}}^{c} B^{c} = 2 n B^{c},

(2.29)

{\dot{S}}^{c} ({(Φ τ_{1})}^{c}, {(Φ τ_{2})}^{c}) = {\dot{S}}^{c} (τ_{1}^{c}, τ_{2}^{c}) - 2 n η^{c} (τ_{1}^{c}) η^{v} (τ_{2}^{c}) - 2 n η^{v} (τ_{1}^{c}) η^{c} (τ_{2}^{c}),

(2.30)

{\dot{S}}^{c} (Φ τ_{1}^{c}, τ_{2}^{c}) = η_{1} g^{c} (τ_{1}^{c}, τ_{2}^{c}) + η_{2} (η^{c} (τ_{1}^{c}) η^{v} (τ_{2}^{c}) + η^{v} (τ_{1}^{c}) η^{c} (τ_{2}^{c})),

(2.31)

\begin{aligned} {\dot{P}}^{c} (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c} & = {\dot{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c} - \frac{1}{2 n} [{\dot{S}}^{c} (τ_{2}^{c}, τ_{3}^{c}) τ_{1}^{v} + {\dot{S}}^{v} (τ_{2}^{c}, τ_{3}^{c}) τ_{1}^{c} \\ - {\dot{S}}^{c} (τ_{1}^{c}, τ_{3}^{c}) τ_{2}^{v} - {\dot{S}}^{v} (τ_{1}^{c}, τ_{3}^{c}) τ_{2}^{c}], \end{aligned}

(2.32)

where the notations ${\dot{R}}^{c}, {\dot{S}}^{c}, {\dot{Q}}^{c}, B^{c}$ and ${\dot{P}}^{c}$ denote the complete lifts of $\dot{R}, \dot{S}, \dot{Q}, \dot{B}$ and $\dot{P}$ respectively for all $τ_{1}^{c}, τ_{2}^{c}, τ_{1}^{v}, τ_{2}^{v} \in Γ (T_{0} M)$ ⁠.

3. Quarter-symmetric non-metric connection (QSNMC) on the tangent bundle

In a (2n + 1)-dimensional Sasakian manifold M, let $\dot{\nabla}$ be the Levi-Civita connection. The QSNMC is given by [36].

{\overset{ˇ}{\nabla}}_{τ_{1}} τ_{2} = {\dot{\nabla}}_{τ_{1}} τ_{2} + \frac{1}{2} η (τ_{2}) Φ τ_{1} - \frac{1}{2} η (τ_{1}) Φ τ_{2},

(3.1)

satisfying the torsion tensor as

\overset{ˇ}{T} (τ_{1}, τ_{2}) = η (τ_{2}) Φ τ_{1} - η (τ_{1}) Φ τ_{2},

(3.2)

and

\begin{aligned} ({\overset{ˇ}{\nabla}}_{τ_{1}} g) (τ_{2}, τ_{3}) = \frac{1}{2} [η (τ_{2}) g (Φ τ_{1}, τ_{3}) + η (τ_{3}) g (Φ τ_{1}, τ_{2})] . \end{aligned}

(3.3)

Taking the complete lifts on equations (3.1–3.3) by mathematical operator we get

\begin{aligned} {\overset{ˇ}{\nabla}}_{τ_{1}^{c}}^{c} τ_{2}^{c} & = {\dot{\nabla}}_{τ_{1}^{c}}^{c} τ_{2}^{c} + \frac{1}{2} [η^{c} (τ_{2}^{c}) {(Φ τ_{1})}^{v} + η^{v} (τ_{2}^{c}) {(Φ τ_{1})}^{c} \\ - η^{c} (τ_{1}^{c}) {(Φ τ_{2})}^{v} - η^{v} (τ_{1}^{c}) {(Φ τ_{2})}^{c}], \end{aligned}

(3.4)

\begin{aligned} {\overset{ˇ}{T}}^{c} (τ_{1}^{c}, τ_{2}^{c}) & = η^{c} (τ_{2}^{c}) {(Φ τ_{1})}^{v} + η^{v} (τ_{2}^{c}) {(Φ τ_{1})}^{c} - η^{c} (τ_{1}^{c}) {(Φ τ_{2})}^{v} \\ - η^{v} (τ_{1}^{c}) {(Φ τ_{2})}^{c}, \end{aligned}

(3.5)

\begin{aligned} ({\overset{ˇ}{\nabla}}_{τ_{1}^{c}}^{c} g^{c}) (τ_{2}^{c}, τ_{3}^{c}) & = \frac{1}{2} [η^{c} (τ_{2}^{c}) g^{c} ({(Φ τ_{1})}^{v}, τ_{3}^{c}) + η^{v} (τ_{2}^{c}) g^{c} ({(Φ τ_{1})}^{c}, τ_{3}^{c}) \\ + η^{c} (τ_{3}^{c}) g^{c} ({(Φ τ_{1})}^{v}, τ_{2}^{c}) + η^{v} (τ_{3}^{c}) g^{c} ({(Φ τ_{1})}^{c}, τ_{2}^{c})] . \end{aligned}

(3.6)

Also, we obtain

({\overset{ˇ}{\nabla}}_{τ_{1}^{c}}^{c} Φ^{c}) τ_{2}^{c} = ({\dot{\nabla}}_{τ_{1}^{c}}^{c} Φ^{c}) τ_{2}^{c} + \frac{1}{2} [η^{c} (τ_{2}^{c}) τ_{1}^{v} + η^{v} (τ_{2}^{c}) τ_{1}^{c} - η^{c} (τ_{1}^{c}) η^{c} (τ_{2}^{c}) B^{v} - η^{c} (τ_{1}^{c}) η^{v} (τ_{2}^{c}) B^{c} - η^{v} (τ_{1}^{c}) η^{c} (τ_{2}^{c}) B^{c}],

(3.7)

{\overset{ˇ}{\nabla}}_{τ_{1}^{c}}^{c} B^{c} = {\dot{\nabla}}_{τ_{1}^{c}}^{c} B^{c} + \frac{1}{2} {(Φ τ_{1})}^{c},

(3.8)

({\overset{ˇ}{\nabla}}_{τ_{1}^{c}}^{c} η^{c}) τ_{2}^{c} = ({\dot{\nabla}}_{τ_{1}^{c}}^{c} η^{c}) τ_{2}^{c} .

(3.9)

Now, setting τ₁ = B in equation (3.6), we get

({\overset{ˇ}{\nabla}}_{B^{c}}^{c} g^{c}) (τ_{2}^{c}, τ_{3}^{c}) = 0 .

(3.10)

Thus, we can state the following theorem.

Theorem 3.1.

In a contact metric manifold associated with the lift of QSNMC on the tangent bundle, the lift of the covariant differentiation of the Riemannian metric g^c associated with B^c on the tangent bundle vanishes identically.

The lift of the curvature tensor R^c associated with the QSNMC is defined as

{\overset{ˇ}{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c} = {\overset{ˇ}{\nabla}}_{τ_{1}^{c}}^{c} {\overset{ˇ}{\nabla}}_{τ_{2}^{c}}^{c} τ_{3}^{c} - {\overset{ˇ}{\nabla}}_{τ_{2}^{c}}^{c} {\overset{ˇ}{\nabla}}_{τ_{1}^{c}}^{c} τ_{3}^{c} - {\overset{ˇ}{\nabla}}_{[τ_{1}^{c}, τ_{2}^{c}]} τ_{3}^{c} .

(3.11)

Using equations (2.4), (2.6) and (3.4) in equation (3.11), we obtain

\begin{aligned} {\overset{ˇ}{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c} & = {\dot{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c} + \frac{1}{2} [2 g^{c} (τ_{2}^{c}, {(Φ τ_{1})}^{c}) {(Φ Z)}^{v} \\ + 2 g^{c} (τ_{2}^{v}, {(Φ τ_{1})}^{c}) {(Φ Z)}^{c} + g^{c} (τ_{1}^{c}, {(Φ τ_{3})}^{c}) {(Φ τ_{2})}^{v} \\ + g^{c} (τ_{1}^{v}, {(Φ τ_{3})}^{c}) {(Φ τ_{2})}^{c} - g^{c} (τ_{2}^{c}, {(Φ τ_{3})}^{c}) {(Φ τ_{1})}^{v} \\ - g^{c} (τ_{2}^{v}, {(Φ τ_{3})}^{c}) {(Φ τ_{1})}^{c} + η^{c} (τ_{1}^{c}) g^{c} (τ_{2}^{c}, τ_{3}^{c}) B^{v} \\ + η^{c} (τ_{1}^{c}) g^{c} (τ_{2}^{v}, τ_{3}^{c}) B^{c} + η^{v} (τ_{1}^{c}) g^{c} (τ_{2}^{c}, τ_{3}^{c}) B^{c} \\ - η^{c} (τ_{2}^{c}) g^{c} (τ_{1}^{c}, τ_{3}^{c}) B^{v} - η^{c} (τ_{2}^{c}) g^{c} (τ_{1}^{v}, τ_{3}^{c}) B^{c} \\ - η^{v} (τ_{2}^{c}) g^{c} (τ_{1}^{c}, τ_{3}^{c}) B^{c}] + \frac{1}{4} [η^{c} (τ_{1}^{c}) η^{c} (τ_{3}^{c}) τ_{2}^{v} \\ + η^{c} (τ_{1}^{c}) η^{v} (τ_{3}^{c}) τ_{2}^{c} + η^{v} (τ_{1}^{c}) η^{c} (τ_{3}^{c}) τ_{2}^{c} \\ - η^{c} (τ_{2}^{c}) η^{c} (τ_{3}^{c}) τ_{1}^{v} - η^{c} (τ_{2}^{c}) η^{v} (τ_{3}^{c}) τ_{1}^{c} \\ - η^{v} (τ_{2}^{c}) η^{c} (τ_{3}^{c}) τ_{1}^{c}] . \end{aligned}

(3.12)

Contracting equation (3.12) along τ₁, we get

{\overset{ˇ}{S}}^{c} (τ_{2}^{c}, τ_{3}^{c}) = {\dot{S}}^{c} (τ_{2}^{c}, τ_{3}^{c}) - \frac{n}{2} [η^{c} (τ_{2}^{c}) η^{v} (τ_{3}^{c}) + η^{v} (τ_{2}^{c}) η^{c} (τ_{3}^{c})] .

(3.13)

Which yields

{\overset{ˇ}{Q}}^{c} τ_{2}^{c} = {\dot{Q}}^{c} τ_{2}^{c} - \frac{n}{2} η^{c} (τ_{2}^{c}) .

(3.14)

Again we contract equation (3.14), and get

{\overset{ˇ}{r}}^{c} = {\dot{r}}^{c} - \frac{n}{2},

(3.15)

where, ${\dot{R}}^{c}, {\overset{ˇ}{R}}^{c}, {\dot{S}}^{c}, {\overset{ˇ}{S}}^{c}, {\dot{Q}}^{c}, {\overset{ˇ}{Q}}^{c}, {\dot{r}}^{c}$ ⁠, and ${\overset{ˇ}{r}}^{c}$ are the complete lifts of the curvature tensor, Ricci tensor, Ricci operator and scalar curvature associated with the Levi-Civita connection ${\dot{\nabla}}^{c}$ and QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle respectively.

Theorem 3.2.

In a Sasakian manifold equipped with a QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle, the following properties are satisfied:

${\overset{ˇ}{R}}^{c} (B^{c}, τ_{2}^{c}) τ_{3}^{c} = \frac{3}{2} [g^{c} (τ_{2}^{c}, τ_{3}^{c}) B^{v} + g^{c} (τ_{2}^{v}, τ_{3}^{c}) B^{c}] - \frac{3}{4} [η^{c} (τ_{3}^{c}) τ_{2}^{v} + η^{v} (τ_{3}^{c}) τ_{2}^{c} + η^{c} (τ_{2}^{c}) η^{c} (τ_{3}^{c}) B^{v} + η^{c} (τ_{2}^{c}) η^{v} (τ_{3}^{c}) B^{c} + η^{v} (τ_{2}^{c}) η^{c} (τ_{3}^{c}) B^{c}]$ ⁠.
${\overset{ˇ}{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) B^{c} = \frac{3}{4} [η^{c} (τ_{2}^{c}) τ_{1}^{v} + η^{v} (τ_{2}^{c}) τ_{1}^{c} - η^{c} (τ_{1}^{c}) τ_{2}^{v} - η^{v} (τ_{1}^{c}) τ_{2}^{c}]$ ⁠.
$η^{c} ({\overset{ˇ}{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c}) = \frac{3}{4} [g^{c} (τ_{2}^{c}, τ_{3}^{c}) η^{v} (τ_{1}^{c}) + g^{c} (τ_{2}^{v}, τ_{3}^{c}) η^{c} (τ_{1}^{c}) - g^{c} (τ_{1}^{c}, τ_{3}^{c}) η^{v} (τ_{2}^{c}) - g^{c} (τ_{1}^{v}, τ_{3}^{c}) η^{c} (τ_{2}^{c})]$ ⁠.
${\overset{ˇ}{S}}^{c} (τ_{2}^{c}, B^{c}) = \frac{3 n}{2} η^{c} (τ_{2}^{c})$ ⁠.

Proof. Setting τ₁ = B in equation (3.12) and employing (2.21), we get 1.

Using (2.21) and setting τ₃ = B in (3.12) we get 2.

For 3, we use (2.21),(3.12) and $g^{c} ({\dot{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}, τ_{3}^{c}), U_{0}) = - g^{c} ({\dot{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}, U_{0}^{c}), τ_{3}^{c})$ to get 3. For 4 we take contraction on 2 and we obtain the result. □

Theorem 3.3.

In a Sasakian manifold equipped with a QSNMC on the tangent bundle, if the lift of the Riemannian curvature tensor associated with ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle vanishes identically. Then, the lift of the scalar curvature is found to be constant on the tangent bundle.

Proof. Since the lift of the Riemannian curvature tensor associated with ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle vanishes identically, we can say its Ricci tensor $\overset{ˇ}{S} = 0$ in equation (3.13), we get

{\dot{S}}^{c} (τ_{2}^{c}, τ_{3}^{c}) = \frac{n}{2} [η^{c} (τ_{2}^{c}) η^{v} (τ_{3}^{c}) + η^{v} (τ_{2}^{c}) η^{c} (τ_{3}^{c})] .

(3.16)

Contracting the above equation, we get

{\dot{r}}^{c} = \frac{n}{2} .

(3.17)

□

4. Complete lift of Ricci soliton with QSNMC on the tangent bundle

The complete lift of the Ricci soliton (g^c, B^c, γ^c) with respect to ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle is given as

\begin{aligned} ({\overset{ˇ}{L}}_{B^{c}}^{c} g^{c}) (τ_{1}^{c}, τ_{2}^{c}) + 2 {\overset{ˇ}{S}}^{c} (τ_{1}^{c}, τ_{2}^{c}) + 2 γ g^{c} (τ_{1}^{c}, τ_{2}^{c}) = 0 . \end{aligned}

(4.1)

Employing equations (2.21), (2.23) and (3.4), we get

\begin{aligned} ({\overset{ˇ}{L}}_{B^{c}}^{c} g^{c}) (τ_{1}^{c}, τ_{2}^{c}) = 0 . \end{aligned}

(4.2)

Again employing equation (4.2) in equation (4.1), we get

{\overset{ˇ}{S}}^{c} (τ_{1}^{c}, τ_{2}^{c}) = - γ g^{c} (τ_{1}^{c}, τ_{2}^{c}) .

(4.3)

and

{\overset{ˇ}{Q}}^{c} τ_{1}^{c} = - γ τ_{1}^{c} .

(4.4)

Setting τ₂ = B in equation (4.3), we get

{\overset{ˇ}{S}}^{c} (τ_{1}^{c}, B^{c}) = - γ η^{c} (τ_{1}^{c}) .

(4.5)

Also contracting equation (4.3), we get

{\overset{ˇ}{r}}^{c} = - γ (2 n + 1) .

(4.6)

From equations (3.13) and (4.3) we can say that

\begin{aligned} {\dot{S}}^{c} (τ_{1}^{c}, τ_{2}^{c}) = - γ g^{c} (τ_{1}^{c}, τ_{2}^{c}) + \frac{n}{2} [η^{c} (τ_{1}^{c}) η^{v} (τ_{2}^{c}) + η^{v} (τ_{1}^{c}) η^{c} (τ_{2}^{c})] . \end{aligned}

(4.7)

Theorem 4.1.

Let (g^c, B^c, γ) be the complete lifts of the Ricci soliton on a Sasakian manifold M associated with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle. Then, the manifold M is said to be an η-Einstein manifold.

Corollary 4.1.

The complete lifts of the Ricci soliton (g^c, B^c, γ) on a Sasakian manifold M associated with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle is found to be always shrinking.

Proof. Setting τ₂ = B in equation (4.7), we have

{\dot{S}}^{c} (τ_{1}^{c}, B^{c}) = - (γ - \frac{n}{2}) η^{c} (τ_{1}^{c}) .

(4.8)

Using equations (2.28) and (4.8), we obtain

γ = - \frac{3 n}{2} < 0 .

(4.9)

□

Theorem 4.2.

The complete lifts of the Ricci soliton (g^c, D^c, γ) on a Sasakian manifold M associated with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle such that D is pointwise collinear with B. Then, D^c is found to be constant multiplier of B^c and the soliton (g^c, D^c, γ) is shrinking.

Proof. Let (g^c, D^c, γ) be the complete lifts of the Ricci soliton on a Sasakian manifold M associated with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle such that D^c is pointwise collinear with B^c such that D^c = eB^c, where e is a function, then

\begin{aligned} e^{c} g^{c} ({({\overset{ˇ}{\nabla}}_{τ_{1}} B)}^{v}, τ_{2}^{c}) + e^{v} g^{c} ({({\overset{ˇ}{\nabla}}_{τ_{1}} B)}^{c}, τ_{2}^{c}) + {(τ_{1} e)}^{c} η^{v} (τ_{2}^{c}) + {(τ_{1} e)}^{v} η^{c} (τ_{2}^{c}) \\ + e^{v} g^{c} (τ_{1}^{c}, {({\overset{ˇ}{\nabla}}_{τ_{2}} B)}^{c}) + e^{c} g^{c} (τ_{1}^{v}, {({\overset{ˇ}{\nabla}}_{τ_{2}} B)}^{c}) + {(τ_{2} e)}^{c} η^{v} (τ_{1}^{c}) + {(τ_{2} e)}^{v} η^{c} (τ_{1}^{c}) \\ + 2 {\overset{ˇ}{S}}^{c} (τ_{1}^{c}, τ_{2}^{c}) + 2 γ g^{c} (τ_{1}^{c}, τ_{2}^{c}) = 0 . \end{aligned}

(4.10)

Setting τ₂ = B in the above equation and employing equations (2.21), (2.23), (3.4) and 4 of Theorem 3.2, we have

{(τ_{1} e)}^{c} + {(B e)}^{c} η^{v} (τ_{1}^{c}) + {(B e)}^{v} η^{c} (τ_{1}^{c}) + (2 γ + 3 n) η^{c} (τ_{1}^{c}) = 0 .

(4.11)

Setting τ₁ = B in the above equation, we get

{(B e)}^{c} + γ + \frac{3 n}{2} = 0 .

(4.12)

From equations (4.12) and (4.11), we get

{(τ_{1} e)}^{c} = - [γ + \frac{3 n}{2}] η^{c} (τ_{1}^{c}) .

(4.13)

Employing d on equation (4.13), we get

- [γ + \frac{3 n}{2}] d η^{c} = 0 .

(4.14)

Since dη^c ≠ 0, we get

γ = - \frac{3 n}{2} < 0 .

(4.15)

From equations (4.13), and (4.15) we get

{(τ_{1} e)}^{c} = 0 .

(4.16)

Hence, e is constant. □

4.1 Complete lift of Ricci soliton on Sasakian manifold satisfying ${\overset{ˇ}{P}}^{c} \cdot {\overset{ˇ}{S}}^{c} = 0$ with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle

We consider that the complete lift of the Sasakian manifold of dimension (2n + 1) associated with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle satisfy the condition

({\overset{ˇ}{P}}^{c} (τ_{1}^{c}, τ_{2}^{c}) \cdot {\overset{ˇ}{S}}^{c}) (τ_{3}^{c}, U_{0}^{c}) = 0,

(4.17)

then, we get

{\overset{ˇ}{S}}^{c} ({\overset{ˇ}{P}}^{c} (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c}, U_{0}^{c}) + {\overset{ˇ}{S}}^{c} (τ_{3}^{c}, {\overset{ˇ}{P}}^{c} (τ_{1}^{c}, τ_{2}^{c}) U_{0}^{c}) = 0 .

(4.18)

Setting τ₁ = B in the above equation we get

{\overset{ˇ}{S}}^{c} ({\overset{ˇ}{P}}^{c} (B^{c}, τ_{2}^{c}) τ_{3}^{c}, U_{0}^{c}) + {\overset{ˇ}{S}}^{c} (τ_{3}^{c}, {\overset{ˇ}{P}}^{c} (B^{c}, τ_{2}^{c}) U_{0}^{c}) = 0 .

(4.19)

From equation (2.32), we write the complete lift of the projective curvature tensor of the Riemannian manifold with respect to QSNMC on the tangent bundle and setting τ₁ = B, we get

\begin{aligned} {\overset{ˇ}{P}}^{c} (B^{c}, τ_{2}^{c}) τ_{3}^{c} & = {\overset{ˇ}{R}}^{c} (B^{c}, τ_{2}^{c}) τ_{3}^{c} - \frac{1}{2 n} [{\overset{ˇ}{S}}^{c} (τ_{2}^{c}, τ_{3}^{c}) B^{v} + {\overset{ˇ}{S}}^{v} (τ_{2}^{c}, τ_{3}^{c}) B^{c} \\ - {\overset{ˇ}{S}}^{c} (B^{c}, τ_{3}^{c}) τ_{2}^{v} - {\overset{ˇ}{S}}^{v} (B^{c}, τ_{3}^{c}) τ_{2}^{c}] . \end{aligned}

(4.20)

Employing 1, 4 of Theorem 3.2 and equation (4.20) in equation (4.19), we get

\begin{aligned} \frac{9 n}{4} [g^{c} (τ_{2}^{c}, τ_{3}^{c}) η^{v} (U_{0}^{c}) + g^{c} (τ_{2}^{v}, τ_{3}^{c}) η^{c} (U_{0}^{c})] + \frac{9 n}{4} [g^{c} (τ_{2}^{c}, U_{0}^{c}) η^{v} (τ_{3}^{c}) \\ + g^{c} (τ_{2}^{v}, U_{0}^{c}) η^{c} (τ_{3}^{c})] - \frac{3}{4} [η^{c} (τ_{3}^{c}) {\overset{ˇ}{S}}^{v} (τ_{2}^{c}, U_{0}^{c}) + η^{v} (τ_{3}^{c}) {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, U_{0}^{c})] \\ - \frac{3}{4} [η^{c} (U_{0}^{c}) {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, τ_{3}^{c} + η^{v} (U_{0}^{c}) {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, τ_{3}^{c}))] - \frac{9 n}{4} [η^{c} (τ_{2}^{c}) η^{c} (τ_{3}^{c}) η^{v} (U_{0}^{c}) \\ + η^{c} (τ_{2}^{c}) η^{v} (τ_{3}^{c}) η^{c} (U_{0}^{c}) + η^{v} (τ_{2}^{c}) η^{c} (τ_{3}^{c}) η^{c} (U_{0}^{c})] = 0 . \end{aligned}

(4.21)

Now, setting U₀ = B in the above equation and employing equation (2.21) and 4 of Theorem 3.2, we get

\begin{aligned} {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, τ_{3}^{c}) = 3 n g^{c} (τ_{2}^{c}, τ_{3}^{c}) - \frac{3 n}{2} [η^{c} (τ_{2}^{c}) η^{v} (τ_{3}^{c}) + η^{v} (τ_{2}^{c}) η^{c} (τ_{3}^{c})] . \end{aligned}

(4.22)

Employing equations (4.1), (2.24) and (4.22), we get

\begin{aligned} g^{c} (({\overset{ˇ}{\nabla}}_{τ_{1}^{c}}^{c} B^{c}), τ_{2}^{c}) + g^{c} (τ_{1}^{c}, ({\overset{ˇ}{\nabla}}_{τ_{2}^{c}}^{c} B^{c})) + 2 [(3 n + γ) g^{c} (τ_{1}^{c}, τ_{2}^{c})] \\ - 3 n [η^{c} (τ_{1}^{c}) η^{v} (τ_{2}^{c}) + η^{v} (τ_{1}^{c}) η^{c} (τ_{2}^{c})] = 0 . \end{aligned}

(4.23)

Setting τ₁ = τ₂ = B in equation (4.23) and employing equations (2.21), (3.8) we get $g^{c} (({\dot{\nabla}}_{B^{c}}^{c} B^{c}), B^{c}) = - (γ + \frac{3 n}{2})$ ⁠. Since for any vector field τ₁ on M, we have $g^{c} (({\dot{\nabla}}_{τ_{1}^{c}}^{c} B^{c}), B^{c}) = 0$ and B has a constant term, we get

γ = - \frac{3 n}{2} < 0 .

(4.24)

Consequently equation (4.23) reduces to

\begin{aligned} g^{c} (({\overset{ˇ}{\nabla}}_{τ_{1}^{c}}^{c} B^{c}), τ_{2}^{c}) + g^{c} (τ_{1}^{c}, ({\overset{ˇ}{\nabla}}_{τ_{2}^{c}}^{c} B^{c})) + 3 n [g^{c} (τ_{1}^{c}, τ_{2}^{c}) \\ - η^{c} (τ_{1}^{c}) η^{v} (τ_{2}^{c}) - η^{v} (τ_{1}^{c}) η^{c} (τ_{2}^{c})] = 0 . \end{aligned}

(4.25)

Setting τ₁ = B in the above equation and employing equations (2.21), (2.23) and (3.8), we get

g^{c} (({\dot{\nabla}}_{B^{c}}^{c} B^{c}), τ_{2}^{c}) = 0,

(4.26)

for all τ₂ on M. Hence, we can say

{\dot{\nabla}}_{B^{c}} B^{c} = 0,

(4.27)

which shows that B^c is a geodesic vector field. Hence we can state the following theorem:

Theorem 4.3.

If the complete lifts of a Ricci soliton on Sasakian manifold associated with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle satisfy ${\overset{ˇ}{P}}^{c} \cdot {\overset{ˇ}{S}}^{c} = 0$ , then

The complete lift of the manifold (M, g^c) is an η-Einstein manifold on the tangent bundle.
The complete lift of the Ricci soliton is shrinking on the tangent bundle.
B^c is a geodesic vector field on the tangent bundle.

4.2 Complete lift of Ricci soliton on Sasakian manifold satisfying ${\overset{ˇ}{R}}^{c} \cdot {\overset{ˇ}{S}}^{c} = 0$ and ${\overset{ˇ}{S}}^{c} \cdot {\overset{ˇ}{R}}^{c} = 0$ with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle

Here, we consider that the complete lift of the Sasakian manifold of dimension (2n + 1) associated with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle satisfy the condition

{\overset{ˇ}{R}}^{c} \cdot {\overset{ˇ}{S}}^{c} = 0,

(4.28)

then

\begin{aligned} {\overset{ˇ}{S}}^{c} ({\overset{ˇ}{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c}, U_{0}^{c}) + {\overset{ˇ}{S}}^{c} (τ_{3}^{c}, {\overset{ˇ}{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) U_{0}^{c}) = 0 . \end{aligned}

(4.29)

Setting τ₁ = B in equation (4.29), we get

\begin{aligned} {\overset{ˇ}{S}}^{c} ({\overset{ˇ}{R}}^{c} (B^{c}, τ_{2}^{c}) τ_{3}^{c}, U_{0}^{c}) + {\overset{ˇ}{S}}^{c} (τ_{3}^{c}, {\overset{ˇ}{R}}^{c} (B^{c}, τ_{2}^{c}) U_{0}^{c}) = 0 . \end{aligned}

(4.30)

Employing 1 and 4 of Theorem 3.2 in the above equation, we get

\begin{aligned} 3 n [g^{c} (τ_{2}^{c}, τ_{3}^{c}) η^{v} (U_{0}^{c}) + g^{c} (τ_{2}^{v}, τ_{3}^{c}) η^{c} (U_{0}^{c})] + 3 n [g^{c} (τ_{2}^{c}, U_{0}^{c}) η^{v} (τ_{3}^{c}) \\ + g^{c} (τ_{2}^{v}, U_{0}^{c}) η^{c} (τ_{3}^{c})] - η^{c} (τ_{3}^{c}) {\overset{ˇ}{S}}^{v} (τ_{2}^{c}, U_{0}^{c}) - η^{v} (τ_{3}^{c}) {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, U_{0}^{c}) \\ - η^{c} (U_{0}^{c}) {\overset{ˇ}{S}}^{v} (τ_{2}^{c}, τ_{3}^{c}) - η^{v} (U_{0}^{c}) {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, τ_{3}^{c}) - 3 n [η^{c} (τ_{2}^{c}) η^{c} (τ_{3}^{c}) η^{v} (U_{0}^{c}) \\ + η^{c} (τ_{2}^{c}) η^{v} (τ_{3}^{c}) η^{c} (U_{0}^{c}) + η^{v} (τ_{2}^{c}) η^{c} (τ_{3}^{c}) η^{c} (U_{0}^{c})] = 0 . \end{aligned}

(4.31)

Again setting τ₃ = B in the above equation and employing equation (2.21) and 4 of Theorem 3.2, we get

\begin{aligned} {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, U_{0}^{c}) = 3 n g^{c} (τ_{2}^{c}, U_{0}^{c}) - \frac{3 n}{2} [η^{c} (τ_{2}^{c}) η^{v} (U_{0}^{c}) + η^{v} (τ_{2}^{c}) η^{c} (U_{0}^{c})] . \end{aligned}

(4.32)

Hence, we can state the following theorem:

Theorem 4.4.

The complete lift of a (2n + 1)-dimensional Sasakian manifold M associated with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ which satisfy ${\overset{ˇ}{R}}^{c} \cdot {\overset{ˇ}{S}}^{c} = 0$ on the tangent bundle is found to be an η-Einstein manifold.

Setting τ₂ = U₀ = B in equation (4.32) and employing equation (4.5), then we can state the following corollary:

Corollary 4.2.

Let the complete lift of Ricci soliton (g^c, B^c, γ) on Sasakian manifold endowed with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ satisfied ${\overset{ˇ}{R}}^{c} \cdot {\overset{ˇ}{S}}^{c} = 0$ on the tangent bundle, then the complete lift of the Ricci soliton is found to be shrinking on the tangent bundle.

Now, employing the condition ${\overset{ˇ}{S}}^{c} \cdot {\overset{ˇ}{R}}^{c} = 0$ ⁠, we have

\begin{aligned} {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, {\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) τ_{3}^{c}) τ_{1}^{v} + {\overset{ˇ}{S}}^{v} (τ_{2}^{c}, {\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) τ_{3}^{c}) τ_{1}^{c} \\ - {\overset{ˇ}{S}}^{c} (τ_{1}^{c}, {\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) τ_{3}^{c}) τ_{2}^{v} - {\overset{ˇ}{S}}^{v} (τ_{1}^{c}, {\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) τ_{3}^{c}) τ_{2}^{c} \\ + {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, U_{0}^{c}) {\overset{ˇ}{R}}^{v} (τ_{1}^{c}, V_{0}^{c}) τ_{3}^{c} + {\overset{ˇ}{S}}^{v} (τ_{2}^{c}, U_{0}^{c}) {\overset{ˇ}{R}}^{c} (τ_{1}^{c}, V_{0}^{c}) τ_{3}^{c} \\ - {\overset{ˇ}{S}}^{c} (τ_{1}^{c}, U_{0}^{c}) {\overset{ˇ}{R}}^{v} (τ_{2}^{c}, V_{0}^{c}) τ_{3}^{c} - {\overset{ˇ}{S}}^{v} (τ_{1}^{c}, U_{0}^{c}) {\overset{ˇ}{R}}^{c} (τ_{2}^{c}, V_{0}^{c}) τ_{3}^{c} \\ + {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, V_{0}^{c}) {\overset{ˇ}{R}}^{v} (U_{0}^{c}, τ_{1}^{c}) τ_{3}^{c} + {\overset{ˇ}{S}}^{V} (τ_{2}^{c}, V_{0}^{c}) {\overset{ˇ}{R}}^{c} (U_{0}^{c}, τ_{1}^{c}) τ_{3}^{c} \\ - {\overset{ˇ}{S}}^{c} (τ_{1}^{c}, V_{0}^{c}) {\overset{ˇ}{R}}^{v} (U_{0}^{c}, τ_{2}^{c}) - {\overset{ˇ}{S}}^{v} (τ_{1}^{c}, V_{0}^{c}) {\overset{ˇ}{R}}^{c} (U_{0}^{c}, τ_{2}^{c}) \\ + {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, τ_{3}^{c}) {\overset{ˇ}{R}}^{v} (U_{0}^{c}, V_{0}^{c}) τ_{1}^{c} + {\overset{ˇ}{S}}^{v} (τ_{2}^{c}, τ_{3}^{c}) {\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) τ_{1}^{c} \\ - {\overset{ˇ}{S}}^{c} (τ_{1}^{c}, τ_{3}^{c}) {\overset{ˇ}{R}}^{v} (U_{0}^{c}, V_{0}^{c}) τ_{2}^{c} - {\overset{ˇ}{S}}^{v} (τ_{1}^{c}, τ_{3}^{c}) {\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) τ_{2}^{c} = 0 . \end{aligned}

(4.33)

Setting τ₂ = B in the above equation and employing 4 of Theorem 3.2 we get

\begin{aligned} \frac{3 n}{2} [η^{c} ({\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) τ_{3}^{c}) τ_{1}^{v} + η^{v} ({\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) τ_{3}^{c}) τ_{1}^{c} \\ + η^{c} (U_{0}^{c}) {\overset{ˇ}{R}}^{v} (τ_{1}^{c}, V_{0}^{c}) τ_{3}^{c} + η^{(} v) {\overset{ˇ}{R}}^{c} (τ_{1}^{c}, V_{0}^{c}) τ_{3}^{c} \\ + η^{c} (V_{0}^{c}) {\overset{ˇ}{R}}^{v} (U_{0}^{c}, τ_{1}^{c}) τ_{3}^{c} + η^{v} (V_{0}^{c}) {\overset{ˇ}{R}}^{c} (U_{0}^{c}, τ_{1}^{c}) τ_{3}^{c} \\ + η^{c} (τ_{3}^{c}) {\overset{ˇ}{R}}^{v} (U_{0}^{c}, V_{0}^{c}) τ_{1}^{c} + η^{v} (τ_{3}^{c}) {\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) τ_{1}^{c}] \\ - {\overset{ˇ}{S}}^{c} (τ_{1}^{c}, {\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) τ_{3}^{c}) B^{v} - {\overset{ˇ}{S}}^{v} (τ_{1}^{c}, {\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) τ_{3}^{c}) B^{c} \\ - {\overset{ˇ}{S}}^{c} (τ_{1}^{c}, U_{0}^{c}) {\overset{ˇ}{R}}^{v} (B^{c}, V_{0}^{c}) τ_{3}^{c} - {\overset{ˇ}{S}}^{v} (τ_{1}^{c}, U_{0}^{c}) {\overset{ˇ}{R}}^{c} (B^{c}, V_{0}^{c}) τ_{3}^{c} \\ - {\overset{ˇ}{S}}^{c} (τ_{1}^{c}, V_{0}^{c}) {\overset{ˇ}{R}}^{v} (U_{0}^{c}, B^{c}) τ_{3}^{c} - {\overset{ˇ}{S}}^{v} (τ_{1}^{c}, V_{0}^{c}) {\overset{ˇ}{R}}^{c} (U_{0}^{c}, B^{c}) τ_{3}^{c} \\ - {\overset{ˇ}{S}}^{c} (τ_{1}^{c}, τ_{3}^{c}) {\overset{ˇ}{R}}^{v} (U_{0}^{c}, V_{0}^{c}) B^{c} - {\overset{ˇ}{S}}^{v} (τ_{1}^{c}, τ_{3}^{c}) {\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) B^{c} = 0 . \end{aligned}

(4.34)

Taking inner product with B in the above equation, we get

\begin{aligned} \frac{3 n}{2} [η^{c} ({\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) τ_{3}^{c}) η^{v} (τ_{1}^{c}) + η^{v} ({\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) τ_{3}^{c}) η^{c} (τ_{1}^{c}) \\ + η^{c} (U_{0}^{c}) η^{v} ({\overset{ˇ}{R}}^{c} (τ_{1}^{c}, V_{0}^{c}) τ_{3}^{c}) + η^{v} (U_{0}^{c}) η^{c} ({\overset{ˇ}{R}}^{c} (τ_{1}^{c}, V_{0}^{c}) τ_{3}^{c}) \\ + η^{c} (V_{0}^{c}) η^{v} ({\overset{ˇ}{R}}^{c} (U_{0}^{c}, τ_{1}^{c}) τ_{3}^{c}) + η^{v} (V_{0}^{c}) η^{c} ({\overset{ˇ}{R}}^{c} (U_{0}^{c}, τ_{1}^{c}) τ_{3}^{c}) \\ + η^{c} (τ_{3}^{c}) η^{v} ({\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) τ_{1}^{c}) + η^{v} (τ_{3}^{c}) η^{c} ({\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) τ_{1}^{c})] \\ - {\overset{ˇ}{S}}^{c} (τ_{1}^{c}, {\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) τ_{3}^{c}) - {\overset{ˇ}{S}}^{v} (τ_{1}^{c}, {\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) τ_{3}^{c}) \\ - {\overset{ˇ}{S}}^{c} (τ_{1}^{c}, U_{0}^{c}) η^{v} ({\overset{ˇ}{R}}^{c} (B^{c}, V_{0}^{c}) τ_{3}^{c}) - {\overset{ˇ}{S}}^{v} (τ_{1}^{c}, U_{0}^{c}) η^{c} ({\overset{ˇ}{R}}^{c} (B^{c}, V_{0}^{c}) τ_{3}^{c}) \\ - {\overset{ˇ}{S}}^{c} (τ_{1}^{c}, V_{0}^{c}) η^{v} ({\overset{ˇ}{R}}^{v} (U_{0}^{c}, B^{c}) τ_{3}^{c}) - {\overset{ˇ}{S}}^{v} (τ_{1}^{c}, V_{0}^{c}) η^{c} ({\overset{ˇ}{R}}^{v} (U_{0}^{c}, B^{c}) τ_{3}^{c}) \\ - {\overset{ˇ}{S}}^{c} (τ_{1}^{c}, τ_{3}^{c}) η^{v} ({\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) B^{c}) - {\overset{ˇ}{S}}^{v} (τ_{1}^{c}, τ_{3}^{c}) η^{c} ({\overset{ˇ}{R}}^{c} (U_{0}^{c}, V_{0}^{c}) B^{c}) = 0 . \end{aligned}

(4.35)

Setting U₀ = τ₃ = B in the above equation and employing 1,2,3 and 4 of Theorem 3.2, we get

\begin{aligned} {\overset{ˇ}{S}}^{c} (τ_{1}^{c}, V_{0}^{c}) = - \frac{3 n}{2} g^{c} (τ_{1}^{c}, V_{0}^{c}) + \frac{9 n}{2} [η^{c} (τ_{1}^{c}) η^{v} (V_{0}^{c}) + η^{v} (τ_{1}^{c}) η^{c} (V_{0}^{c})] . \end{aligned}

(4.36)

Hence, we can state the following theorem:

Theorem 4.5.

The complete lift of a Sasakian manifold associated with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ which satisfy ${\overset{ˇ}{S}}^{c} \cdot {\overset{ˇ}{R}}^{c} = 0$ on the tangent bundle is an η-Einstein manifold.

Theorem 4.6.

In the complete lift of the Ricci soliton (g^c, B^c, γ) on Sasakian manifold endowed with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ which satisfy ${\overset{ˇ}{S}}^{c} \cdot {\overset{ˇ}{R}}^{c} = 0$ , then the complete lift of the Ricci soliton is found to be shrinking.

Proof. Setting V₀ = B in equation (4.36) and employing equation (4.5), we get

γ = - 3 n < 0 .

(4.37)

□

5. Complete lift of Ricci soliton on Ricci-recurrent and Φ-recurrent Sasakian manifold associated with QSNMC on the tangent bundle

We consider that the Sasakian manifold M associated with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ is Ricci-recurrent with respect to QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle such that M satisfies

({\overset{ˇ}{\nabla}}_{U_{0}^{c}}^{c} {\overset{ˇ}{S}}^{c}) (τ_{2}^{c}, τ_{3}^{c}) = A_{0}^{c} (U_{0}^{c}) {\overset{ˇ}{S}}^{v} (τ_{2}^{c}, τ_{3}^{c}) + A_{0}^{v} (U_{0}^{c}) {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, τ_{3}^{c}) .

(5.1)

Setting τ₃ = B in the above equation, we get

({\overset{ˇ}{\nabla}}_{U_{0}^{c}}^{c} {\overset{ˇ}{S}}^{c}) (τ_{2}^{c}, B^{c}) = A_{0}^{c} (U_{0}^{c}) {\overset{ˇ}{S}}^{v} (τ_{2}^{c}, B^{c}) + A_{0}^{v} (U_{0}^{c}) {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, B^{c}) .

(5.2)

Employing 4 of Theorem 3.2 in the above equation, we get

({\overset{ˇ}{\nabla}}_{U_{0}^{c}}^{c} {\overset{ˇ}{S}}^{c}) (τ_{2}^{c}, τ_{3}^{c}) = \frac{3 n}{2} [η^{c} (τ_{2}^{c}) A_{0}^{v} (U_{0}^{c}) + η^{v} (τ_{2}^{c}) A_{0}^{c} (U_{0}^{c})] .

(5.3)

We recall that

\begin{aligned} ({\overset{ˇ}{\nabla}}_{U_{0}^{c}}^{c} {\overset{ˇ}{S}}^{c}) (τ_{2}^{c}, B^{c}) = {\overset{ˇ}{\nabla}}_{U_{0}^{c}}^{c} {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, B^{c}) - {\overset{ˇ}{S}}^{c} ({\overset{ˇ}{\nabla}}_{U_{0}^{c}}^{c} τ_{2}^{c}, B^{c}) - {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, {\overset{ˇ}{\nabla}}_{U_{0}^{c}}^{c} B^{c}) . \end{aligned}

(5.4)

Employing equations (2.23), (3.9) and 4 of Theorem 3.2 in equation (5.3), we get

\begin{aligned} ({\overset{ˇ}{\nabla}}_{U_{0}^{c}}^{c} {\overset{ˇ}{S}}^{c}) (τ_{2}^{c}, B^{c}) = \frac{3 n}{2} g^{c} (U_{0}^{c}, {(Φ τ_{2})}^{c}) + \frac{1}{2} {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, {(Φ U_{0})}^{c}) . \end{aligned}

(5.5)

From equations (5.3) and (5.5), we get

\begin{aligned} {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, {(Φ U_{0})}^{c}) = 3 n g^{c} (τ_{2}^{c}, U_{0}^{c}) - \frac{n}{2} [η^{c} (τ_{2}^{c}) A_{0}^{v} (U_{0}^{c}) + η^{v} (τ_{2}^{c}) A_{0}^{c} (U_{0}^{c})] . \end{aligned}

(5.6)

Setting τ₂ = Φτ₂ in the above equation, we get

\begin{aligned} {\dot{S}}^{c} (τ_{2}^{c}, U_{0}^{c}) = 3 n g^{c} (τ_{2}^{c}, U_{0}^{c}) - n [η^{c} (τ_{2}^{c}) η^{v} (U_{0}^{c}) + η^{v} (τ_{2}^{c}) η^{c} (U_{0}^{c})] . \end{aligned}

(5.7)

Again setting U₀ = B in the above equation and employing equations (2.21) and (4.7) we get

\begin{aligned} - (γ - \frac{n}{2}) η^{c} (τ_{2}^{c}) = 3 n η^{c} (τ_{2}^{c}) - n η^{c} (τ_{2}^{c}) . \end{aligned}

(5.8)

Hence, we get

γ = - \frac{3}{2} n < 0 .

(5.9)

Thus, we can state the following theorem:

Theorem 5.1.

The complete lift of the Ricci soliton (g^c, B^c, γ) on Ricci-recurrent Sasakian manifold associate with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle is found to be shrinking and the manifold is an η-Einstein manifold.

Theorem 5.2.

The complete lift of a Φ-recurrent Sasakian manifold M associated with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle is an η-Einstein manifold and the Ricci soliton is found to be shrinking.

Proof. The complete lift of Φ-recurrent Sasakian manifold M associated with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle satisfies

\begin{aligned} {[Φ^{2} (({\overset{ˇ}{\nabla}}_{U_{0}^{c}}^{c} {\overset{ˇ}{R}}^{c}) (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c})]}^{c} & = A_{0}^{c} (U_{0}^{c}) {\overset{ˇ}{R}}^{v} (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c} \\ + A_{0}^{v} (U_{0}^{c}) {\overset{ˇ}{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c} . \end{aligned}

(5.10)

Employing equation (2.21) and the above equation we get

\begin{aligned} - g^{c} (({\overset{ˇ}{\nabla}}_{U_{0}^{c}}^{c} {\overset{ˇ}{R}}^{c}) (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c}, τ_{3}^{c}) + η^{c} (({\overset{ˇ}{\nabla}}_{U_{0}^{c}}^{c} {\overset{ˇ}{R}}^{c}) (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c}) η^{v} (τ_{3}^{c}) \\ + η^{v} (({\overset{ˇ}{\nabla}}_{U_{0}^{c}}^{c} {\overset{ˇ}{R}}^{c}) (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c}) η^{c} (τ_{3}^{c}) \\ = A_{0}^{c} (U_{0}^{c}) g^{c} (τ_{3}^{v}, {\overset{ˇ}{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c}) + A_{0}^{v} (U_{0}^{c}) g^{c} (τ_{3}^{c}, {\overset{ˇ}{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c}) . \end{aligned}

(5.11)

Setting τ₃ = B in the above equation we get

\begin{aligned} - ({\overset{ˇ}{\nabla}}_{U_{0}^{c}}^{c} {\overset{ˇ}{S}}^{c}) (τ_{2}^{c}, B^{c}) = A_{0}^{c} (U_{0}^{c}) {\overset{ˇ}{S}}^{v} (τ_{2}^{c}, B^{c}) + A_{0}^{v} (U_{0}^{c}) {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, B^{c}) . \end{aligned}

(5.12)

From 4 of Theorem 3.2 and equations (5.5),(5.12) we get

\begin{aligned} {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, {(Φ U_{0})}^{c}) = 3 n [g^{c} (τ_{2}^{c}, {(Φ U_{0})}^{c}) - η^{c} (τ_{2}^{c}) A_{0}^{v} (U_{0}^{c}) - η^{v} (τ_{2}^{c}) A_{0}^{c} (U_{0}^{c})] . \end{aligned}

(5.13)

Setting τ₂ = Φτ₂ in the above equation, we get

\begin{aligned} {\dot{S}}^{c} (τ_{2}^{c}, U_{0}^{c}) = 3 n g^{c} (τ_{2}^{c}, U_{0}^{c}) - n [η^{c} (τ_{2}^{c}) η^{v} (U_{0}^{c}) - η^{v} (τ_{2}^{c}) η^{c} (U_{0}^{c})] . \end{aligned}

(5.14)

Again setting U₀ = B in the above equation and employing equations (2.21) and (4.7) we get

γ = - \frac{3}{2} n < 0 .

(5.15)

□

6. Complete lift of Ricci soliton on Einstein semi-symmetric Sasakian manifold associated with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle

We consider that the complete lift of the Einstein semi-symmetric Sasakian manifold M associated with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ satisfies

\begin{aligned} ({\overset{ˇ}{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}), {\overset{ˇ}{E}}^{c}) (τ_{3}^{c}, U_{0}^{c}) = 0 . \end{aligned}

(6.1)

we rewrite the above equation as

{\overset{ˇ}{E}}^{c} ({\overset{ˇ}{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c}, U_{0}^{c}) + {\overset{ˇ}{E}}^{c} (τ_{3}^{c}, {\overset{ˇ}{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) U_{0}^{c}) = 0 .

(6.2)

In light of

\begin{aligned} {\overset{ˇ}{E}}^{c} (τ_{2}^{c}, τ_{3}^{c}) = {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, τ_{3}^{c}) - \frac{{\overset{ˇ}{r}}^{c}}{n} g^{c} (τ_{2}^{c}, τ_{3}^{c}) . \end{aligned}

(6.3)

Equation (6.2) can be rewritten as

\begin{aligned} {\overset{ˇ}{S}}^{c} ({\overset{ˇ}{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c}, U_{0}^{c}) + {\overset{ˇ}{S}}^{c} (τ_{3}^{c}, {\overset{ˇ}{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) U_{0}^{c}) \\ = \frac{{\overset{ˇ}{r}}^{c}}{n} [g^{c} ({\overset{ˇ}{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) τ_{3}^{c}, U_{0}^{c}) + g^{c} (τ_{3}^{c}, {\overset{ˇ}{R}}^{c} (τ_{1}^{c}, τ_{2}^{c}) U_{0}^{c})] . \end{aligned}

(6.4)

Setting τ₁ = B in the above equation, we get

\begin{aligned} {\overset{ˇ}{S}}^{c} ({\overset{ˇ}{R}}^{c} (B^{c}, τ_{2}^{c}) τ_{3}^{c}, U_{0}^{c}) + {\overset{ˇ}{S}}^{c} (τ_{3}^{c}, {\overset{ˇ}{R}}^{c} (B^{c}, τ_{2}^{c}) U_{0}^{c}) \\ = \frac{{\overset{ˇ}{r}}^{c}}{n} [g^{c} ({\overset{ˇ}{R}}^{c} (B^{c}, τ_{2}^{c}) τ_{3}^{c}, U_{0}^{c}) + g^{c} (τ_{3}^{c}, {\overset{ˇ}{R}}^{c} (B^{c}, τ_{2}^{c}) U_{0}^{c})] . \end{aligned}

(6.5)

From equations 1 and 4 of Theorem 3.2, we can write the above equation as

\begin{aligned} \frac{1}{4} [9 n (g^{c} (τ_{2}^{c}, τ_{3}^{c}) η^{v} (U_{0}^{c}) + g^{c} (τ_{2}^{v}, τ_{3}^{c}) η^{c} (U_{0}^{c})) - 3 (η^{c} (τ_{3}^{c}) {\overset{ˇ}{S}}^{v} (τ_{2}^{c}, U_{0}^{c}) \\ + η^{v} (τ_{3}^{c}) {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, U_{0}^{c})) - 4 (η^{c} (τ_{2}^{c}) η^{c} (U_{0}^{c}) η^{v} (τ_{3}^{c}) + η^{c} (τ_{2}^{c}) η^{v} (U_{0}^{c}) η^{c} (τ_{3}^{c}) \\ + η^{v} (τ_{2}^{c}) η^{c} (U_{0}^{c}) η^{c} (τ_{3}^{c})) + 9 n (g^{c} (τ_{2}^{c}, U_{0}^{c}) η^{v} (τ_{3}^{c}) + g^{c} (τ_{2}^{v}, U_{0}^{c}) η^{c} (τ_{3}^{c})) \\ - 3 (η^{c} (U_{0}^{c}) {\overset{ˇ}{S}}^{v} (τ_{2}^{c}, τ_{3}^{c}) + η^{v} (U_{0}^{c}) {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, τ_{3}^{c}))] \\ = \frac{{\overset{ˇ}{r}}^{c}}{n} [\frac{3}{4} (g^{c} (τ_{2}^{c}, τ_{3}^{c}) η^{v} (U_{0}^{c}) + g^{c} (τ_{2}^{v}, τ_{3}^{c}) η^{c} (U_{0}^{c})) + \frac{3}{4} (g^{c} (τ_{2}^{c}, U_{0}^{c}) η^{v} (τ_{3}^{c}) \\ + g^{c} (τ_{2}^{v}, U_{0}^{c}) η^{c} (τ_{3}^{c})) - \frac{3}{2} (η^{c} (τ_{2}^{c}) η^{c} (τ_{3}^{c}) η^{v} (U_{0}^{c}) + η^{c} (τ_{2}^{c}) η^{v} (τ_{3}^{c}) η^{c} (U_{0}^{c}) \\ + η^{v} (τ_{2}^{c}) η^{c} (τ_{3}^{c}) η^{c} (U_{0}^{c}))] . \end{aligned}

(6.6)

Setting τ₃ = B in the above equation and employing equations (3.15), (3.17) and 4 of Theorem 3.2, we get

\begin{aligned} {\overset{ˇ}{S}}^{c} (τ_{2}^{c}, U_{0}^{c}) = 3 n g^{c} (τ_{2}^{c}, U_{0}^{c}) - \frac{3 n}{2} [η^{c} (τ_{2}^{c}) η^{v} (U_{0}^{c}) + η^{v} (τ_{2}^{c}) η^{c} (U_{0}^{c})] . \end{aligned}

(6.7)

Hence, we can state the following theorem:

Theorem 6.1.

The complete lift of every Einstein semi-symmetric Sasakian manifold is an η-Einstein manifold on the tangent bundle.

Theorem 6.2.

The complete lift of the Ricci soliton (g^c, B^c, γ) of an Einstein semi-symmetric Sasakian manifold is always shrinking on the tangent bundle.

Proof. setting U₀ = B in the above equation and using equation (4.5), we get

γ = - \frac{3 n}{2} < 0 .

(6.8)

□

7. Conclusion

In the proposed work, we investigate the complete lift of the Ricci soliton on Sasakian manifold endowed with QSNMC on the tangent bundle. Firstly, the property of the Sasakian manifold and its complete lift to the tangent bundle are established. The properties of the lift of its curvature tensor on the tangent bundle are studied, and some theorems are also shown. Next, the condition of the complete lift of the Ricci soliton on the Sasakian manifold with respect to QSNMC on the tangent bundle is studied, as are the conditions of the complete lifts of the Ricci solitons satisfying ${\overset{ˇ}{P}}^{c} . {\overset{ˇ}{S}}^{c} = 0$ ⁠, ${\overset{ˇ}{R}}^{c} . {\overset{ˇ}{S}}^{c} = 0$ and ${\overset{ˇ}{S}}^{c} . {\overset{ˇ}{R}}^{c} = 0$ are investigated, and it was observed that the manifolds are η-Einstein manifolds and the Ricci solitons are shrinking. Also, the data on the complete lifts of the Ricci solitons on Ricci-recurrent, ϕ-recurrent, and Einstein semi-symmetric Sasakian manifolds are studied and found to be shrinking, and we also proved some theorems about it.

Author contribution

Conceptualization – L.Colney; Formal analysis – L.Colney; Investigation – L.Colney; Methodology – L.Colney; Software – L.Colney; Supervision – R.Kumar; Validation – L.Colney; Visualization – L.Colney, R.Kumar; Roles/Writing – original draft L.Colney; and Writing – review and editing – L.Colney.

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions for improving the quality of this paper.

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Lalnunenga Colney and Rajesh Kumar

Published in the Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at Link to the terms of the CC BY 4.0 licence.

Soliton on Sasakian manifold endowed with quarter-symmetric non-metric connection on the tangent bundle

1. Introduction

2. Preliminaries

2.1 Sasakian manifold on the tangent bundle

3. Quarter-symmetric non-metric connection (QSNMC) on the tangent bundle

4. Complete lift of Ricci soliton with QSNMC on the tangent bundle

4.1 Complete lift of Ricci soliton on Sasakian manifold satisfying ${\overset{ˇ}{P}}^{c} \cdot {\overset{ˇ}{S}}^{c} = 0$ with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle

4.2 Complete lift of Ricci soliton on Sasakian manifold satisfying ${\overset{ˇ}{R}}^{c} \cdot {\overset{ˇ}{S}}^{c} = 0$ and ${\overset{ˇ}{S}}^{c} \cdot {\overset{ˇ}{R}}^{c} = 0$ with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle

5. Complete lift of Ricci soliton on Ricci-recurrent and Φ-recurrent Sasakian manifold associated with QSNMC on the tangent bundle

6. Complete lift of Ricci soliton on Einstein semi-symmetric Sasakian manifold associated with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle

7. Conclusion

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Soliton on Sasakian manifold endowed with quarter-symmetric non-metric connection on the tangent bundle Open Access

1. Introduction

2. Preliminaries

2.1 Sasakian manifold on the tangent bundle

3. Quarter-symmetric non-metric connection (QSNMC) on the tangent bundle

4. Complete lift of Ricci soliton with QSNMC on the tangent bundle

4.1 Complete lift of Ricci soliton on Sasakian manifold satisfying Pˇc⋅Sˇc=0 with QSNMC ∇ˇc on the tangent bundle

4.2 Complete lift of Ricci soliton on Sasakian manifold satisfying Rˇc⋅Sˇc=0 and Sˇc⋅Rˇc=0 with QSNMC ∇ˇc on the tangent bundle

5. Complete lift of Ricci soliton on Ricci-recurrent and Φ-recurrent Sasakian manifold associated with QSNMC on the tangent bundle

6. Complete lift of Ricci soliton on Einstein semi-symmetric Sasakian manifold associated with QSNMC ∇ˇc on the tangent bundle

7. Conclusion

Author contribution

References

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Soliton on Sasakian manifold endowed with quarter-symmetric non-metric connection on the tangent bundle

4.1 Complete lift of Ricci soliton on Sasakian manifold satisfying ${\overset{ˇ}{P}}^{c} \cdot {\overset{ˇ}{S}}^{c} = 0$ with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle

4.2 Complete lift of Ricci soliton on Sasakian manifold satisfying ${\overset{ˇ}{R}}^{c} \cdot {\overset{ˇ}{S}}^{c} = 0$ and ${\overset{ˇ}{S}}^{c} \cdot {\overset{ˇ}{R}}^{c} = 0$ with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle

6. Complete lift of Ricci soliton on Einstein semi-symmetric Sasakian manifold associated with QSNMC ${\overset{ˇ}{\nabla}}^{c}$ on the tangent bundle