Certain results on tangent bundle endowed with generalized Tanaka Webster connection (GTWC) on Kenmotsu manifolds

1.   Introduction

The study of tangent bundles in differential geometry has long been a focal point, posing new challenges for modern exploration. The use of complete lifts conveniently extends differentiable structures from any manifold to its tangent bundle. Yano and Ishihara ^[25] developed the theory of lifts, encompassing vertical, complete, and horizontal aspects, enabling the extension of various geometric structures and multiple types of connections from a manifold to its tangent bundle. Notable geometers such as Yano and Kobayashi ^[26], Yano and Ishihara ^[25], Tani ^[20], and Khan ^{[7, 8, 9, 10]} have extensively delved into tangent bundle geometry. Several manifolds associated with distinct connections in the tangent bundle were extensively investigated in ^{[12, 13, 14]}. Recently, Kumar et al. ^[15] carried out a comprehensive investigation into the lifts of the semi-symmetric metric connection from Sasakian statistical manifolds, geometric structures combining Sasakian geometry and statistical manifolds, to the tangent bundle.

The canonical affine connection known as the Tanaka-Webster connection was established on a nondegenerate pseudo-Hermitian CR-manifold in ^{[23, 24]}. Tanno ^[22] initially explored the generalized Tanaka-Webster connection (GTWC) for contact metric manifolds by utilizing the canonical connection. This GTWC aligns with the Tanaka-Webster connection when the associated CR-structure is integrable. Several geometers have investigated certain characteristics of real hypersurfaces in complex space forms using the GTWC ^[21]. Kenmotsu manifolds, introduced by Kenmotsu in 1971 ^[6], have recently been the focus of numerous studies on the GTWC connection by various authors ^{[3, 17, 18]}. Ricci solitons, introduced by Hamilton ^[4], represent natural extensions of Einstein metrics and are defined on a Riemannian manifold $(M, g)$.

A Ricci soliton denoted as $(g, V_0, \gamma)$ is defined on a Riemannian manifold $(M, g)$ by the equation

where $\ddot{L}_{V_0}$ represents the Lie derivative, which measures how a tensor field changes along the direction of a vector field, of $g$ along a vector field $V_0$, capturing how $g$ changes along the flow generated by $V_0$. Here, $\gamma$ is a constant, and $X_0, Y_0$ are arbitrary vector fields on $M$. The classification of a Ricci soliton as shrinking, steady, or expanding depends on whether $\gamma$ can take on negative, zero, or positive values, respectively. Extensive research on Ricci solitons has been conducted in the context of contact geometry, as discussed in ^{[5, 16, 19]} and related references.

This paper is organized as follows: Section 1 is devoted to the introduction, and Section 2 concerns the foundational concepts or background information. In Section 3, we investigate the complete lifts of the curvature properties of Kenmotsu manifolds associated with GTWC in the tangent bundle, and several curvature properties and theorems are proved. Subsequently, we investigate the complete lifts of the Ricci soliton of Kenmotsu manifolds associated with the GTWC in the tangent bundle in Section 4. Lastly, in Section 5, we provide an example of the complete lifts of a $5$-dimensional Kenmotsu manifold in the tangent bundle followed by a conclusion section of our proposed paper in Section 6.

2.   Preliminaries

Consider $M$ as a manifold of dimension $(2n+1)$ equipped with an almost contact metric structure $(\Phi, \xi, \eta, g)$. This structure consists of a $(1, 1)$ tensor field $\Phi$ representing a specific linear transformation on the tangent spaces, a vector field $\xi$, a 1-form $\eta$, and Riemannian metric $g$ on $M$ satisfying ^[13]

A manifold with an almost contact metric structure $(\Phi, \xi, \eta, g)$ is classified as a Kenmotsu manifold if and only if it satisfies the following condition: ^[1]

Using the above relations, we have some properties as given below ^[13]:

where $\ddot{R}, \ddot{S}$, and $\ddot{Q}$ refer to the curvature tensor, the Ricci tensor, and the Ricci operator, respectively. These entities are derived from the Levi-Civita connection, which is the unique connection that preserves the metric and is torsion-free.

2.1.   Complete lifts of Kenmotsu manifolds to its tangent bundle

Suppose $T_0M$ is the tangent bundle and $X_0=X_0^i\frac{\partial }{{\partial {x^i}}}$ is a local vector field on $M$. Then its vertical and complete lifts in terms of partial differential equations are

Let $T_0M$ denote the tangent bundle on the Kenmotsu manifolds $M$. Then, applying the complete lifts of the mathematical operators from Eqs (2.1)–(2.13), we obtain ^[13]

The notations $\eta^c$, $\eta^v$, $g^c$, $g^v$, $\Phi^c$, $\Phi^v$, $\ddot{\nabla}^c$, $\ddot{\nabla}^v$, $\ddot{R}^c$, $\ddot{R}^v$, $\ddot{S}^c$, $\ddot{S}^v$ are the complete and vertical lifts of $\eta$, $g$, $\Phi$, $\ddot{\nabla}$, $\ddot{R}$, and $\ddot{S}$, respectively.

3.   Complete lifts of the curvature properties of the Kenmotsu manifolds endowed with the GTWC to its tangent bundle

In a Kenmotsu manifolds $M^{(2n+1)}$, we will use the GTWC $\breve{\nabla}$ given by ^{[3, 11]}

where $\ddot{\nabla}$ is the Levi-Civita connection and $X_0, Y_0$ are vector fields on $M^{(2n+1)}$. Taking the complete lifts of the above equation by using mathematical operators, we get

Employing Eqs (2.21) and (2.22) in the above equation, we get

By setting $Y_0=\xi$ in the above equation and employing Eq (2.21), we get

The complete lifts of the Riemannian curvature tensor in the tangent bundle are given by

By employing Eq (3.3), we get

By setting $Z_0=\xi$ in the above equation and using Eq (2.23), we get

Contracting Eq (3.6), we get

where $\breve{S}^c$ and $\ddot{S}^c$ are the complete lifts of the Ricci tensor associated with the GTWC and Levi-Civita connections, respectively. The complete lifts of the Ricci operator $\breve{Q}^c$ associated with the GTWC are obtained by

where $\ddot{Q}^c$ is the complete lift of the Ricci operator associated with the Levi-Civita connection.

Again, contracting Eq (3.8), we get

where $\breve{r}^c$ and $\ddot{r}^c$ are the complete lifts of the scalar curvature associated with the GTWC and Levi-Civita connections, respectively.

Theorem 3.1. The complete lifts of the GTWC of Kenmotsu manifolds in the tangent bundle are the only affine connection, which are metric and its complete lifts of the torsion are given by

Proof. We have

Employing Eq (3.3) in the above equation, we get

Employing Eq (2.22) in the above equation, we get

Again,

Employing Eq (3.3) in the above equation, we get

This shows that the GTWC is a metric connection. The expression of the complete lifts of the torsion tensor in the tangent bundle are given by

Employing Eq (3.3) in the above equation, we get

Any complete lifts of metric connection can be put in the expression with the help of $\breve{T}^c$ as

Employing Eq (3.18) in the above equation, we get

Contracting the above equation with $Z_0$, we get

□

Proposition 3.1. In the complete lifts of Kenmotsu manifolds associated with the GTWC, $\xi^c, \eta^c,$ and $g^c$ are parallel in the tangent bundle.

Proposition 3.2. The complete lifts of the Kenmotsu manifolds associated with GTWC in the tangent bundle are a metric connections.

Proposition 3.3. In the complete lifts of Kenmotsu manifolds associated with the GTWC, the complete lifts of the integral curves of the vector field $\xi^c$ are geodesic in the tangent bundle.

3.1.   Complete lifts of Ricci semi-symmetry Kenmotsu manifolds endowed with the GTWC to its tangent bundle

The Kenmotsu manifolds associated with the GTWC are said to be Ricci semi-symmetry if ^[3]

Obtaining the complete lifts of the above equation by using mathematical operators, we get

Theorem 3.2. The complete lifts of the Ricci semi-symmetric Kenmotsu manifolds associated with the GTWC and Levi-Civita connections are equal if and only if the manifold is an Einstein manifold, which is associated with the Levi-Civita connection in the tangent bundle.

Proof. Employing Eq (3.6) in the above equation, we get

Suppose

Then, Eq (3.24) becomes

By setting $X_0=V_0=e_i$, $1\leq i \leq (2n+1)$ in the above equation, we get

Again if

Then, from Eq (3.24), we get

□

3.2.   Complete lifts of projective curvature tensor of a Kenmotsu manifold endowed with GTWC to its tangent bundle

The complete lifts of the projective curvature tensor of the Kenmotsu manifolds associated with the GTWC in the tangent bundle are given by

Theorem 3.3. If the complete lifts of the projective curvature tensor of a Kenmotsu manifold $M^{(2n+1)}$ associated with the GTWC to its tangent bundle vanish, then the complete lifts of the curvature tensor of Kenmotsu manifolds associated with the GTWC are also found to vanish to its tangent bundle.

Proof. By setting $\breve{P}^c=0$ in Eq (3.30), we get

Employing Eqs (3.6) and (3.8) in the above equation, we get

By setting $U_0=\xi$ in the above equation, we get

Also, by setting $X_0=\xi$ in the above equation, we get

which gives

Employing Eq (3.34) in Eq (3.31), we get

which shows that the complete lifts of the curvature tensor of Kenmotsu manifolds associated with the GTWC vanish. □

Theorem 3.4. In the complete lifts of the Kenmotsu manifolds $M^{(2n+1)}$ with GTWC to its tangent bundle, the following properties hold:

(i) The complete lifts of the projective curvature tensor are skew-symmetric.

(ii) The complete lifts of the projective curvature tensor are cyclic.

Proof. For (ⅰ), we interchanged $X_0$ and $Y_0$ in Eq (3.30) as

On adding the above equation to Eq (3.30) and employing $\ddot{R}^c(X_0^c, Y_0^c)Z_0^c+\ddot{R}^c(Y_0^c, X_0^c)Z_0^c=0$, we get

For (ⅱ), employing Eqs (3.6) and (3.30) and the complete lifts of the first Bianchi identity $\ddot{R}^c(X_0^c, Y_0^c)Z_0^c+\ddot{R}^c(Y_0^c, Z_0^c)X_0^c+\ddot{R}^c(Z_0^c, X_0^c)Y_0^c=0$, we get

□

Definition 3.1. The Kenmotsu manifold $M^{(2n+1)}$ associated with the GTWC is said to be $\xi$-projectively flat if ^[11]

for any vector field $X_0, Y_0$ on $M^{(2n+1)}$.

Theorem 3.5. The complete lifts of a Kenmotsu manifold $M^{(2n+1)}$ associated with GTWC to its tangent bundle are $\xi$-projectively flat.

Proof. By setting $Z_0=\xi$ in Eq (3.30) and employing Eqs (3.7) and (3.8), we get

□

3.3.   Complete lifts of $\Phi$-projectively semi-symmetric Kenmotsu manifolds endowed with GTWC to its tangent bundle

Definition 3.2. A Kenmotsu manifolds $M^{(2n+1)}$ associated with the GTWC is said to be $\Phi$-projectively semi-symmetric if ^[11]

for any vector field $X_0, Y_0$ on $M^{(2n+1)}$.

Taking a complete lift of Eq (3.41) by using mathematical operators,

The above equation becomes

Employing Eqs (3.30), (3.6), and (3.8) in the above equation, we get

Interchanging $Y_0$ by $\xi$ in the above equation and employing Eqs (2.23) and (2.28), we get

Operating an inner product with $\xi$, interchanging $X_0$ by $\Phi X_0$, and employing Eqs (2.19) and (2.27) in the above equation, we get

Thus, by substituting Eq (3.46) in Eq (3.30), we get

Hence, we can claim the following theorem.

Theorem 3.6. The complete lifts of a Kenmostsu manifold $M^{(2n+1)}$ associated with the GTWC are said to be $\Phi$-projectively semi-symmetric if and only if $\ddot{S}^c(X_0^c, Z_0^c)=-2ng^c(X_0^c, Z_0^c)$ in the tangent bundle. Further, if $\breve{P}^c=0$, then the complete lifts of the manifold are said to be a hyperbolic space $\ddot{H}^{(2n+1)}(-1)$ in the tangent bundle.

By setting $\Big[(\breve{P}(X_0, Y_0).\breve{S})(Z_0, U_0)\Big]^c=0$ on a Kenmotsu manifold $M^{(2n+1)}$, we get

By setting $X_0=\xi$ in the above equation, we get

Employing Eq (3.30) in the above equation, we get

Again employing Eq (3.8) in the above equation, we get

By setting $U_0=\xi$ in the above equation and contracting it with $Y_0$ and $Z_0$, we get

and

Thus, by substituting Eq (3.53) in Eq (3.30), we get

Hence, we can claim the following theorem.

Theorem 3.7. The complete lifts of a Kenmostsu manifold $M^{(2n+1)}$ associated with the GTWC satisfy $\breve{P}^c.\breve{S}^c=0$ if and only if $\ddot{S}^c(Y_0^c, Z_0^c)=-2ng^c(Y_0^c, Z_0^c)$ in the tangent bundle. Further, if $\breve{P}^c=0$, then the complete lifts of the manifold are said to be a hyperbolic space $\ddot{H}^{(2n+1)}(-1)$ in the tangent bundle.

3.4.   Complete lifts of conharmonic curvature tensor of a Kenmotsu manifold endowed with GTWC to its tangent bundle

The complete lifts of the conharmonic curvature tensor associated with the GTWC in the tangent bundle are given as

Theorem 3.8. If the complete lifts of the conharmonic curvature tensor of a Kenmotsu manifolds $M^{(2n+1)}$ with GTWC to its tangent bundle vanishes, then the complete lifts of the curvature tensor of a Kenmotsu manifold with the GTWC are also found to vanish to its tangent bundle.

Proof. By setting $\breve{K}(X_0, Y_0)Z_0=0$, Eq (3.56) becomes

Employing Eqs (3.6), (3.8), and (3.9) in the above equation, we get

By setting $U_0=\xi$ in the above equation, we get

By setting $X_0=\xi$ in the above equation, we get

Contracting the above equation gives

Employing Eq (3.60) in (3.57), we get

□

3.5.   Complete lifts of concircular curvature tensor of a Kenmotsu manifold endowed with GTWC to its tangent bundle

The complete lifts of the concircular curvature tensor of the Riemannian manifold in the tangent bundle are

where $X_0, Y_0, Z_0$ are vector fields on the Riemannian manifold, $\ddot{R}$ is the Riemmanian curvature tensor, and $\ddot{r}$ is the scalar curvature associated with the Levi-Civita connection.

Similarly, the complete lifts of the concircular curvature tensor of Kenmotsu manifolds in the tangent bundle are given as

where $X_0, Y_0, Z_0$ are vector fields on Kenmotsu manifolds, $\breve{R}$ is the Riemmanian curvature tensor, and $\breve{r}$ is the scalar curvature associated with the GTWC.

Theorem 3.9. The complete lifts of concircular curvature tensors of Kenmotsu manifolds associated with the GTWC and Levi-Civita connections are equal in the tangent bundle.

Proof. Employing Eqs (3.6) and (3.10) in the above equation, we get

□

Theorem 3.10. If the complete lifts of the Kenmotsu manifolds endowed with GTWC to its tangent bundle satisfy the condition $\breve{K_0}^c(X_0^c, Y_0^c).\breve{S}^c=0$, then the manifold is classified as an Einstein manifold with respect to the Levi-Civita connection in the tangent bundle.

Proof. By setting $\breve{K_0}^c(X_0^c, Y_0^c).\breve{S}^c=0$, we have

for all $X_0, Y_0, V_0, U_0 \in \chi(M^{(2n+1)})$. Replacing $X_0$ with $\xi$ in the above equation, we get

Employing Eqs (3.6), (3.10), and (3.64) in the above equation, we get

By setting $V_0=\xi$ in the above equation, we get $\breve{S}^c(Y_0^c, U_0^c)=0.$ Then, from Eq (3.8) it follows that

□

3.6.   Complete lifts of Weyl conformal curvature tensor of a Kenmotsu manifold endowed with GTWC to its tangent bundle

In a Riemannian manifold, the complete lifts of the Weyl conformal curvature tensor associated with the GTWC in the tangent bundle is

Theorem 3.11. In a Kenmotsu manifolds $M^{(2n+1)}$ with the GTWC, the complete lifts of the Weyl conformal curvature tensor with the Levi-Civita connection and the complete lifts of the Weyl conformal curvature tensor with the GTWC are equivalent in the tangent bundle.

Proof. Employing Eqs (3.6), (3.8)–(3.10) in Eq (3.70), we get

for all $X_0, Y_0, Z_0$ on $M^{(2n+1)}$. □

Definition 3.3. A Kenmotsu manifold associated with the GTWC is called recurrent if its curvature tensor $\breve{R}$ satisfies the condition ^[11]

where $\breve{R}$ is the curvature tensor associated with the GTWC and $D$ is a 1-form associated with vector field $\rho$ such that $D(X_0)=g(X_0, \rho)$.

Taking the complete lifts of Eq (3.72) by mathematical operators, we get

We can write the above equation as

Employing Eqs (3.3), (3.6), and (3.8) in the above equation, we get

Interchanging $Z_0$ by $\xi$ in the above equation and employing Eqs (2.16), (2.17), (2.23), (2.24), and (2.26) in the above equation, we get

Obtaining an inner product with $V_0$ in the above equation, we get

Suppose $\{e_1, e_2, ...e_{2n+1}\}$ is a local orthonormal basis of vector fields in $M$. Then, by setting $X_0=V_0=e_i$ in Eq (3.77) and summing up with respect to $i$, $1\leq i \leq2n+1$, we get

We suppose that the vector field $\rho$ is in co-direction with vector field $\xi$. Then their associated 1-forms will be equal, i.e. $D=\eta$, and so the above equation becomes

Hence, we can claim the following theorem.

Theorem 3.12. If the complete lifts of the Kenmotsu manifolds associated with the GTWC are recurrent and the associated 1-form $D$ is equal to the associated 1-form $\eta$, then the manifold is an $\eta$-Einstein manifold in the tangent bundle.

4.   Complete lifts of the Ricci soliton on Kenmotsu manifolds endowed with the GTWC in the tangent bundle

The Ricci soliton of Kenmotsu manifolds associated with the GTWC is given by ^[11]

Obtaining the complete lift of Eq (4.1) by using mathematical operators, we get

If the complete lift of the potential vector field $V_0^c$ is the structure of the vector field $\xi^c$ in the tangent bundle, and since the complete lift of $\xi^c$ is a parallel vector field with respect to the GTWC (from (3.4)) in the tangent bundle causing the first term of Eq (4.2) to vanish, then the manifold reduces to an Einstein manifold. If the complete lift of $V_0^c$ is pointwise collinear with the structure vector field $\xi^c$ in the tangent bundle such that $V_0^c=a\xi^c$, where $a$ is a function on $M^{(2n+1)}$, then Eq (4.2) implies that

Employing Eqs (3.4) and (3.8) in the above equation, we get

By setting $Y_0=\xi$ in the above equation and employing Eq (2.28), we get

Interchanging $X_0$ with $\xi$ in the above equation, we get

Employing the above equation in Eq (4.5), we get

By differentiating the above equation, we get

Since, $d\eta \neq 0$ from the above equation, we get

Employing the above equation in Eq (4.7), we can say that $a$ is a constant. Thus, it is verified from Eq (4.4) that

Thus, we state the following theorem.

Theorem 4.1. If the complete lifts of the vector field $X_0^c$ on the Ricci soliton of Kenmotsu manifolds associated with the GTWC are pointwise collinear with the complete lifts of the vector field $\xi^c$ in the tangent bundle, then the manifold is an $\eta$-Einstein manifold and the data provided by the complete lifts of the Ricci soliton is found to be steady in the tangent bundle.

5.   Example

This section is devoted to a comprehensive examination of a particular example. It will provide a detailed analysis of the manifold's structural properties and the relevant mathematical framework. Let $M$ be a five-dimensional manifold defined as

where $\mathbb{R}$ represents the set of real numbers. Consider $\{E_1, E_2, E_3, E_4, E_5\}$ as a linearly independent global frame on $M$ where

Let the 1-form $\eta$ be given by

The Riemannian metric $g$ is defined by

Let $\Phi$ be the $(1, 1)$ tensor field defined by

Using the linearity of $\Phi$ and $g$, we acquire $\eta(E_5)=1$, $\Phi^2X_0=-X_0+\eta(X_0)E_5$ and $g(\Phi X_0, \Phi Y_0)=g(X_0, Y_0)-\eta(X_0)\eta(Y_0).$ Thus, for $E_5=\xi$, the structure $(\Phi, \xi, \eta, g)$ is then an almost contact metric structure on $M$, and $M$ is called an almost contact metric manifold. In addition, $M$ satisfies

Here, for $E_5=\xi$, $M$ is a Kenmotsu manifold. Let the complete and vertical lifts of $E_1, E_2, E_3, E_4, E_5$ be $E_1^c, E_2^c, E_3^c, E_4^c, E_5^c$ and $E_1^v, E_2^v, E_3^v, E_4^v, E_5^v$, respectively, in the tangent bundle $T_0M$ of manifold $M$, and let the complete lift of the Riemannian metric $g$ be $g^c$ on $T_0M$ such that

and so on. Let the complete and vertical lifts of the $(1, 1)$ tensor field $\Phi_0$ be $\Phi_0^c$ and $\Phi_0^v$, respectively, and be defined by

Using the linearity of $\Phi$ and $g$, we infer that

Thus, for $E_5=\xi$ in (5.6)–(5.8) and (5.14)–(5.15), the structure $(\Phi^c, \xi^c, \eta^c, g^c)$ is an almost contact metric structure on $T_0M$ and satisfies the relation

Thus, $(\Phi^c, \xi^c, \eta^c, g^c, T_0M)$ is a Kenmotsu manifold.

6.   Conclusions

In this study, we delve into the properties of the complete lifts of Kenmotsu manifolds within the tangent bundle, particularly those endowed with the generalized Tanaka-Webster connection (GTWC). Our examination covers a broad spectrum of curvature aspects associated with these lifts, including the Ricci semi-symmetric tensor, projective curvature tensor, conharmonic curvature tensor, concircular curvature tensor, and the Weyl projective curvature tensor. Additionally, we explore the recurrent conditions pertinent to these complete lifts and present proofs for several theorems proposed in this framework.

Moreover, we extend our investigation to the complete lifts of Ricci solitons within Kenmotsu manifolds associated with the GTWC in the tangent bundle. Our results reveal that the data derived from the Ricci solitons exhibit stability within the tangent bundle, and that the manifold is an $\eta$-Einstein manifold when the complete lifts of the vector field $V_0^c$ are pointwise collinear with $\xi^c$ in the tangent bundle. An example of the complete lifts of a $5$-dimensional Kenmotsu manifold in the tangent bundle is provided.

Author contributions

Rajesh Kumar: Conceptualization, Formal analysis, Investigation, Methodology, Supervision, Visualization, Writing – review & editing. Sameh Shenawy: Conceptualization, Formal analysis, Investigation, Methodology, Supervision, Visualization, Writing – review & editing. Lalnunenga Colney: Conceptualization, Formal analysis, Investigation, Methodology, Writing – original draft, Writing – review & editing. Nasser Bin Turki: Conceptualization, Formal analysis, Investigation, Methodology, Supervision, Visualization, Funding acquisition, Project administration.

Acknowledgments

This project was supported by the Researchers Supporting Project number (RSP2024R413), King Saud University, Riyadh, Saudi Arabia.

The third author is thankful to the Government of India, Ministry of Tribal Affairs (Scholarship Division), New Delhi, India for financial support in the form of NFST award no. 202122-NFST-MIZ-00686.

Conflict of interest

The authors declare that they have no conflict of interest.