Slant Submersions in Generalized Sasakian Space Forms and Some Optimal Inequalities

Abstract

This research examines key inequalities associated with the scalar and Ricci curvatures of slant submersions within generalized Sasakian space forms (GSSFs). We establish significant geometric constraints and conduct a detailed analysis of the conditions that lead to equality in these bounds. By expanding the existing framework of curvature inequalities, our results provide new insights into the geometric characteristics of slant submersions in contact structures.

1. Introduction

The study of the intricate relationship between intrinsic and extrinsic geometric properties has been a fundamental aspect of differential geometry. In particular, the connection between the squared mean curvature and Ricci curvature of submanifolds in real space forms $R^n\left(c\right)$ has drawn considerable interest. This relationship, initially proposed by B.-Y. Chen in 1999 [1] and later extended in 2005 [2], laid the groundwork for a series of curvature inequalities that reveal deep geometric properties of submanifolds.

Subsequent research expanded these ideas in various directions. For example, the classification of ideal Lagrangian submanifolds and studies of Casorati curvature were carried out in [3,4,5]. Investigations into Ricci curvature behavior for slant and other types of submanifolds in complex space forms were undertaken in [6,7,8]. Related work on Chen-type inequalities for Riemannian maps and submersions is found in [9,10,11], and extensions to quaternionic and conformal settings appear in [12,13,14]. Recent contributions also include explorations of projective Ricci curvature in Finsler geometry, Ricci solitons on specific Lie groups and the behavior of $\lambda$-wave fronts in relation to Gauss maps, as in [15,16,17]. In differential geometry, smooth submersions $\psi$ between (semi)-Riemannian manifolds $(\overline{M},g)$ and $(N,g_2)$ serve as an essential tool for analyzing geometric structures and interactions. These mappings, which project the total space onto the base space while maintaining critical geometric properties, have facilitated the development of various specialized submersion types. Notable examples include Riemannian submersions [18,19,20], with further structure and geometric properties explored in [21,22,23]. Almost Hermitian submersions [24], quaternionic submersions [25], and slant submersions [26,27] have also gained attention. Moreover, other classifications such as anti-invariant [28], conformal anti-invariant [29], and semi-invariant submersions [30] have enriched the field. Foundational studies by B. O’Neill [22] and A. Gray [20] provided the theoretical basis for understanding the geometric and algebraic aspects of Riemannian submersions.

Holomorphic submersions, introduced by Watson [24], highlight the deep connection between almost complex structures and submersion geometry. Watson demonstrated that when the total space of a holomorphic submersion is a Kähler manifold, the base space inherits a compatible structure. Expanding upon this idea, Şahin [31] introduced slant submersions, which generalize holomorphic submersions by incorporating a constant angle condition between vertical and horizontal distributions.

Let M be an almost Hermitian manifold equipped with a complex structure $\mathsf{ϝ}$ of type $(1,1)$. Submanifolds within Kähler manifolds are classified based on how their tangent spaces interact with the complex structure of the surrounding space. Holomorphic submanifolds are characterized by the condition $\mathsf{ϝ}\left(T_{x}M\right)\subseteq T_{x}M$, while totally real submanifolds satisfy $\mathsf{ϝ}\left(T_{x}M\right)\subseteq T_x{M}^{\perp }$. To unify and extend these classifications, Chen [32] introduced the concept of slant submanifolds.

A submanifold is defined as slant if the angle $\theta \left(I\right)$ between the tangent space $T_{x}M$ and $\mathsf{ϝ}I$ for any vector $I\in T_{x}M$ remains constant and falls within the interval $\left[0,{\displaystyle \frac{\pi }{2}}\right]$. If $\theta =0$, the submanifold is holomorphic, whereas if $\theta ={\displaystyle \frac{\pi }{2}}$, it is classified as completely real. When the angle $\theta$ assumes an intermediate constant value, the submanifold is termed a proper slant submanifold. In parallel, recent developments have also addressed broader theoretical contexts, such as weakly Ricci-symmetric spacetimes and f(R,G) gravity theories [33], the behavior of spacelike hypersurfaces in de Sitter space [34], and convergence properties related to quantization techniques on the sphere [35], all contributing to a richer geometric framework.

Recent studies have extensively examined Chen–Ricci inequalities in the context of submersions within real, complex space forms, and contact space forms, especially for anti-invariant and semi-invariant cases [36,37,38]. This research aims to extend these inequalities to slant Riemannian submersions in GSSFs. The organization of this paper is as follows: the first section reviews fundamental definitions and concepts. The subsequent sections explore inequalities for Ricci and scalar curvatures, specifically analyzing their behavior within vertical $\left({\mathcal{V}}_p\left(M\right)\right)$ and horizontal $\left({\mathcal{H}}_p\left(M\right)\right)$ distributions. Finally, we present generalized Chen–Ricci inequalities that are uniquely tailored to slant Riemannian submersions.

2. Fundamentals of Riemannian Submersions and Generalized Sasakian Manifolds

Let $\pi :\overline{M}\to M$ be a Riemannian submersion, where $\overline{M}$ is a $(2m+1)$-dimensional manifold and M is a b-dimensional Riemannian manifold. For each point $x\in M$, the fiber ${\pi }^{-1}\left(x\right)$ inherits a Riemannian structure from $\overline{M}$, and is treated as a Riemannian submanifold, denoted by ${\overline{M}}_x$.

A vector field on $\overline{M}$ is termed vertical if it is tangent to the fibers, whereas it is called horizontal if it is orthogonal to the fibers at every point. The dimension of the vertical distribution is $r=(2m+1)-b$, while the horizontal distribution has dimension $b=(2m+1)-r$. Within the tangent bundle $T\overline{M}$, the vertical and horizontal distributions are denoted by $\mathcal{V}\left(\overline{M}\right)$ and $\mathcal{H}\left(\overline{M}\right)$, respectively.

A vector field I on $\overline{M}$ is said to be projectable if there exists a vector field $I_B$ on B such that ${\pi }_{*}\left(I\right)\left(p\right)=I_B\left(\pi \left(p\right)\right)$ for all $p\in \overline{M}$. In this case, I and $I_B$ are referred to as $\pi$-related. A basic vector field is one that is both horizontal and projectable [9,38]. The tangent space $T_p\overline{M}$ at any point $p\in \overline{M}$ splits as a direct sum of vertical and horizontal components ${\mathcal{V}}_p\left(\overline{M}\right)$ and ${\mathcal{H}}_p\left(\overline{M}\right)$, respectively.

The tensor fields T and $\mathsf{\Lambda }$ of type $(1,2)$ associated with $\pi$ are defined by

$$T_{E}G=h\left({\nabla }_{vE}vG\right)+v\left({\nabla }_{vE}hG\right),$$

$${\mathsf{\Lambda }}_{E}G=h\left({\nabla }_{hE}vG\right)+v\left({\nabla }_{hE}hG\right),$$

where ∇ is the Levi-Civita connection on $(\overline{M},\varrho )$, and h and v are the projections onto $\mathcal{H}\left(\overline{M}\right)$ and $\mathcal{V}\left(\overline{M}\right)$, respectively.

Let ℜ, ${\Re }^{\prime }$, and ${\Re }^{*}$ denote the Riemannian curvature tensors of $\overline{M}$, M, and the horizontal distribution $\mathcal{H}\left(\overline{M}\right)$, respectively. Then, the Gauss–Codazzi-type relations are given by

$$\begin{array}{cc}\hfill \Re (L,D,G,B)& ={\Re }^{\prime }(L,D,G,B)+\varrho (T_{L}B,T_{D}G)-\varrho (T_{D}B,T_{L}G),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \Re (I,Y,Z,H)& ={\Re }^{*}(I,Y,Z,H)-2\varrho ({\mathsf{\Lambda }}_{I}Y,{\mathsf{\Lambda }}_{Z}H)\hfill \\ \hfill & +\varrho ({\mathsf{\Lambda }}_{Y}Z,{\mathsf{\Lambda }}_{I}H)-\varrho ({\mathsf{\Lambda }}_{Z}I,{\mathsf{\Lambda }}_{Y}H),\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \Re (I,D,Y,B)& =\varrho (\left({\nabla }_{I}T\right)(D,B),Y)+\varrho (\left({\nabla }_D\mathsf{\Lambda }\right)(I,Y),B)\hfill \\ \hfill & -\varrho (T_{D}I,T_{B}Y)+\varrho ({\mathsf{\Lambda }}_{Y}B,{\mathsf{\Lambda }}_{I}D),\hfill \end{array}$$

(3)

where the relation between ${\Re }^{*}$ and ${\Re }^{\prime }$ satisfies

$${\pi }_{*}\left({\Re }^{*}(I,Y)Z\right)={\Re }^{\prime }({\pi }_{*}I,{\pi }_{*}Y){\pi }_{*}Z,$$

for all horizontal vector fields $I,Y,Z\in \mathsf{\Gamma }\left(\mathcal{H}\left(\overline{M}\right)\right)$ and vertical vector fields $L,D,G,B\in \mathsf{\Gamma }\left(\mathcal{V}\left(\overline{M}\right)\right)$ ([38]).

The mean curvature vector field H of a fiber in a Riemannian submersion $\pi$ is given by

$$H=rN,N=\sum _{j=1}^r{T}_{L_j}{L}_j,$$

where $\{L_1,\dots ,L_r\}$ is an orthonormal basis for $\mathcal{V}\left(\overline{M}\right)$. It follows that the fibers are completely geodesic if and only if the tensor T vanishes identically ([38]).

We now state two important lemmas.

Lemma 1 

([26]). Let $(\overline{M},\varrho )$ and $(M,{\varrho }^{\prime })$ be Riemannian manifolds connected by a Riemannian submersion $\pi :\overline{M}\to M$. Then, for all $E,F,G\in TM$, the tensors T and $\mathsf{\Lambda }$ satisfy

$$\varrho (T_{E}G,G)=-\varrho (G,T_{E}G),$$

$$\varrho ({\mathsf{\Lambda }}_{E}G,G)=-\varrho (G,{\mathsf{\Lambda }}_{E}G),$$

meaning that both T and $\mathsf{\Lambda }$ are skew-symmetric with respect to the metric ϱ.

Lemma 2 

([26]). Let $\pi :\overline{M}\to M$ be a Riemannian submersion. Then, the following apply:

(i) 

For all $L,D\in \mathcal{V}\left(M\right)$,

$$T_{L}D=TD\left(L\right).$$

(ii) 

For all $I,Y\in \mathcal{H}\left(M\right)$,

$${\mathsf{\Lambda }}_{I}Y=-{\mathsf{\Lambda }}_{Y}I.$$

For a broader treatment of Riemannian submersions, the reader may consult [8].

Consider now a $(2m+1)$-dimensional contact metric manifold $(\overline{M},\varphi ,\xi ,\eta ,\varrho )$. Then, for tensor field $\varphi$, the structure vector field $\xi$, the 1-form $\eta$, and the Riemannian metric $\varrho$ on $\overline{M}$,

$$\begin{array}{c}\hfill {\varphi }^2=-\iota +\eta \otimes \xi ,\eta \left(\xi \right)=1,\varrho (\varphi X,\varphi Y)=\varrho (X,Y)-\eta \left(X\right)\eta \left(Y\right),\end{array}$$

which implies that

$$\begin{array}{c}\hfill \varphi \xi =0,\eta \left(\varphi \xi \right)=0,\eta \left(X\right)=\varrho (X,\xi )\varrho (\varphi X,Y)+\varrho (X,\varphi Y)=0.\end{array}$$

A contact metric manifold $(\overline{M},\varphi ,\xi ,\eta ,\varrho )$ whose curvature tensor ℜ can be expressed as

$$\begin{array}{cc}\hfill \Re (I,Y)Z& =f_1\left(\varrho (Y,Z)I-\varrho (I,Z)Y\right)\hfill \\ \hfill & +f_2\left(\varrho (I,\varphi Z)\varphi Y-\varrho (Y,\varphi Z)\varphi I+2\varrho (Y,\varphi Y)\varphi Z\right)\hfill \\ \hfill & +f_3\left(\eta \left(I\right)\eta \left(Z\right)Y-\eta \left(Y\right)\eta \left(Z\right)I+\varrho (I,Z)\eta \left(Y\right)\xi -\varrho (Y,Z)\eta \left(I\right)\xi \right).\hfill \end{array}$$

(4)

For $I,Y,Z\in \overline{M}$ and the smooth functions $f_1,f_2,f_3$ on $\overline{M}$, the expression is called a GSSF. It is worth emphasizing that GSSFs encompass several classical geometric structures as special cases. It is a Sasakian space form for $f_1={\displaystyle \frac{c+3}{4}},f_2=f_3={\displaystyle \frac{c-1}{4}}$, a Kenmotsu space form for $f_1={\displaystyle \frac{c-3}{4}},f_2=f_3={\displaystyle \frac{c+1}{4}}$, and a cosymplactic space form for $f_1=f_2=f_3={\displaystyle \frac{c}{4}}$.

3. The Geometry of Slant Riemannian Submersions (SRSs)

In this section, we develop the concept of SRSs from Sasakian manifolds onto Riemannian manifolds. We also illustrate this notion with examples and establish several characterization equations that will be instrumental for later discussions.

Definition 1. 

Let $\left(\overline{M},\varphi ,\xi ,\eta ,\varrho \right)$ denote a contact manifold, and let $\left(M,{\varrho }_2\right)$ be a Riemannian manifold. A Riemannian submersion $\mathsf{\Psi }:\left(\overline{M},\varphi ,\xi ,\eta ,\varrho \right)\to (M,{\varrho }_2)$ is called a slant submersion if, for each nonzero vector field $I\in \mathsf{\Gamma }\left({(ker{\mathsf{\Psi }}_{*})}^{\perp }\right)$, the angle $\theta \left(I\right)$ between $\varphi I$ and $ker{\mathsf{\Psi }}_{*}$ remains constant throughout $\overline{M}$, independent of the choice of point $p\in \overline{M}$ and the specific $I\in \mathsf{\Gamma }({(ker{\mathsf{\Psi }}_{*})}^{\perp }\setminus \left\{\xi \right\})$. This fixed angle θ is referred to as the slant angle of the submersion.

Assume that $\mathsf{\Psi }:\left(\overline{M},\varrho ,\varphi ,\xi ,\eta \right)\to \left(M,{\varrho }_2\right)$ is an SRS. Then, for every $L\in \mathsf{\Gamma }(ker{\mathsf{\Psi }}_{*})$, the tensor field $\varphi L$ can be decomposed as

$$\begin{array}{c}\hfill \varphi L=\psi L+\omega L,\end{array}$$

(5)

where $\psi L$ and $\omega L$ represent, respectively, the vertical and horizontal components of $\varphi L$. Similarly, for any $I\in \mathsf{\Gamma }\left({(ker{\mathsf{\Psi }}_{*})}^{\perp }\right)$, we can write

$$\begin{array}{c}\hfill \varphi I=BI+CI,\end{array}$$

(6)

where $BI$ and $CI$ denote the vertical and horizontal parts of $\varphi I$, respectively.

Let us now consider $\left(\overline{M}(f_1,f_2,f_3),\varrho \right)$ as a generalized Sasakian space form and $\left(M,{\varrho }_2\right)$ as a Riemannian manifold. Suppose that $\mathsf{\Psi }:\overline{M}(f_1,f_2,f_3)\to M$ is an SRS. Let $\{L_1,\dots ,L_r,I_1,\dots ,I_n\}$ be an orthonormal basis of the tangent space $T_p\overline{M}\left(c\right)$ such that the vertical distribution $\mathcal{D}$ is given by $\mathcal{D}=span\{L_1,\dots ,L_r,\xi \}$ and the horizontal distribution $\mathcal{H}$ is given by $\mathcal{H}=span\{I_1,\dots ,I_n\}$.

Based on this, we can construct a slant orthonormal frame as

$$L_1,L_2={\displaystyle \frac{1}{cos\theta }}\psi L_1,\dots ,L_{2k}={\displaystyle \frac{1}{cos\theta }}\psi L_k,L_r=\xi .$$

Furthermore, we have the following relations:

$$\begin{array}{cc}\hfill \varrho (\varphi L_1,L_2)& =\varrho \left(\varphi L_1,{\displaystyle \frac{1}{cos\theta }}\psi L_1\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{cos\theta }}\varrho (\varphi L_1,\psi L_1)\hfill \\ \hfill & ={\displaystyle \frac{1}{cos\theta }}\varrho (\psi L_1,\psi L_1)\hfill \\ \hfill & =cos\theta ,\hfill \end{array}$$

and similarly, for the remaining frame vectors,

$${\varrho }^2(\varphi L_i,L_{i+1})={cos}^2\theta ,$$

which leads to

$${\mathsf{\Xi }}_{i=1}^{r-1}{\varrho }^2(\varphi L_i,L_{i+1})=(r-1){cos}^2\theta .$$

In the special case where the Reeb vector field $\xi$ lies entirely in the horizontal distribution, we obtain

$${\varrho }^2(\varphi L_i,L_{i+1})={cos}^2\theta$$

for each i, and thus,

$${\mathsf{\Xi }}_{i=1}^r{\varrho }^2(\varphi L_i,L_{i+1})=r{cos}^2\theta .$$

4. Optimal Curvature Estimates

Let $\left(\overline{M}(f_1,f_2,f_3),\varrho \right)$ and $\left(M,{\varrho }_2\right)$ be a GSSF and a Riemannian manifold, respectively, and let $\mathsf{\Psi }:\overline{M}(f_1,f_2,f_3)\to M$ be an SRS. Furthermore, let $\{L_1,\dots ,L_r,I_1,\dots ,I_n\}$ be an orthonormal basis of $T_p\overline{M}(f_1,f_2,f_3)$ such that $\mathcal{V}=span\{L_1,\dots ,L_r\}$ and $\mathcal{H}=span\{I_1,\dots ,I_n\}$. Then, using (1), (2), and (4), we have

$$\begin{array}{cc}\hfill \widehat{\Re }(L,D,G,B)& =f_1\left\{\varrho (D,G)\varrho (L,B)-\varrho (L,G)\varrho (D,B)\right\}\hfill \\ \hfill & +f_2\{\varrho (\varphi D,G)\varrho (\varphi L,B)-\varrho (\varphi L,G)\varrho (\varphi D,B)\hfill \\ \hfill & -2\varrho (\varphi L,D)\varrho (B,\varphi G)\}\hfill \\ \hfill & +f_3\{\eta \left(L\right)\eta \left(G\right)\varrho (D,B)-\eta \left(D\right)\eta \left(G\right)\varrho (L,B)\hfill \\ \hfill & +\eta \left(D\right)\eta \left(B\right)\varrho (L,G)-\eta \left(L\right)\eta \left(B\right)\varrho (D,G)\}\hfill \\ \hfill & -\varrho (T_{L}B,T_{D}G)+\varrho (T_{D}B,T_{L}G),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\Re }^{*}(I,Y,Z,H)& =f_1\left\{\varrho (Y,Z)\varrho (I,H)-\varrho (I,Z)\varrho (Y,H)\right\}\hfill \\ \hfill & -f_2\{\varrho (\varphi Y,Z)\varrho (\varphi I,H)-\varrho (\varphi Y,H)\varrho (\varphi I,Z)\hfill \\ \hfill & -2\varrho (\varphi I,Y)\varrho (\varphi Z,H)\}\hfill \\ \hfill & +f_3\{\eta \left(I\right)\eta \left(Z\right)\varrho (Y,H)-\eta \left(Y\right)\eta \left(Z\right)\varrho (I,H)\hfill \\ \hfill & +\eta \left(Y\right)\eta \left(H\right)\varrho (I,Z)-\eta \left(I\right)\eta \left(H\right)\varrho (Y,Z)\}\hfill \\ \hfill & +2\varrho ({\mathsf{\Lambda }}_{I}Y,{\mathsf{\Lambda }}_{I}H)-\varrho ({\mathsf{\Lambda }}_{Y}Z,{\mathsf{\Lambda }}_{I}H)+\varrho ({\mathsf{\Lambda }}_{I}Z,{\mathsf{\Lambda }}_{Y}H).\hfill \end{array}$$

(8)

Theorem 1. 

Let $\mathsf{\Psi }$ be an SRS mapping the GSSF $\left(\overline{M}(f_1,f_2,f_3),\varrho \right)$ onto a Riemannian manifold $\left(M,{\varrho }_2\right)$, with ξ entirely contained in the vertical distribution. Then,

$$\begin{array}{cc}\hfill \widehat{Ric}\left(L\right)& \ge f_1(r-1)+3f_2{cos}^2\theta \left\{1-{\left(\eta \left(L\right)\right)}^2\right\}\hfill \\ \hfill & +f_3{\{(2-r)\left(\eta \left(L\right)\right)}^2-1\}-r\varrho (T_{L}L,H).\hfill \end{array}$$

(9)

The necessary and sufficient condition for the equality case in (9) is if each fiber is completely geodesic for a unit vertical vector $L\in {\mathcal{V}}_p\left(\overline{M}\left(c\right)\right)$.

Proof. 

Making use of (7), we derive

$$\begin{array}{cc}\hfill \widehat{Ric}\left(L\right)& =f_1(r-1)\varrho (L,L)+3f_2{\mathsf{\Xi }}_{i=1}^r{\varrho }^2(\varphi L,L_i)\hfill \\ \hfill & +f_3\left\{(2-r){\left(\eta \left(L\right)\right)}^2-\varrho (L,L)\right\}\hfill \\ \hfill & -r\varrho (T_{L}L,H)+{\parallel T_{L}L\parallel }^2,\hfill \end{array}$$

(10)

where

$$\widehat{Ric}\left(L\right)={\mathsf{\Xi }}_{i=1}^r\widehat{\Re }(L,L_i,L_i,L).$$

Since

$$\begin{array}{c}\hfill {\mathsf{\Xi }}_{i=1}^r{\varrho }^2(\varphi L,L_i)={cos}^2\theta \left(\varrho (L,L)-{\left(\eta \left(L\right)\right)}^2\right),\end{array}$$

(11)

by applying the last relation in (10), we derive (9). □

Theorem 2. 

Let $\mathsf{\Psi }$ be an SRS mapping the GSSF $\left(\overline{M}(f_1,f_2,f_3),\varrho \right)$ onto a Riemannian manifold $\left(M,{\varrho }_2\right)$, with ξ entirely contained in the vertical distribution. Then,

$$\begin{array}{c}\hfill 2\tau \ge f_{1}r(r-1)+3f_2{cos}^2\theta (r-1)+2f_3(1-r)-r^2{\parallel H\parallel }^2.\end{array}$$

(12)

The necessary and sufficient condition for the equality case in (12) is if each fiber is totally geodesic.

Proof. 

By employing the symmetry property of T in (7), we derive

$$\begin{array}{cc}\hfill 2\tau & =f_{1}r(r-1)+3f_2(r-1){cos}^2\theta +f_3(2-2r)\hfill \\ \hfill & -r^2{\parallel H\parallel }^2+{\mathsf{\Xi }}_{i,j=1}^r\varrho (T_{L_i}{L}_j,T_{L_i}{L}_j),\hfill \end{array}$$

(13)

which implies (12), where

$$\tau ={\mathsf{\Xi }}_{1\le i<j\le r}\widehat{\Re }(L_i,L_j,L_j,L_i).$$

□

Considering the horizontal distribution, and in view of (8) and noting that $\mathsf{\Psi }$ is an SRS with vertical $\xi$, the anti-symmetry of $\mathsf{\Lambda }$ leads to

$$\begin{array}{cc}\hfill 2{\tau }^{*}& =f_{1}n(n-1)+3f_2{\mathsf{\Xi }}_{i,j=1}^n\varrho (C{I}_i,I_j)\varrho (C{I}_i,I_j)\hfill \\ \hfill & -3{\mathsf{\Xi }}_{i,j=1}^{n}g({\mathsf{\Lambda }}_{I_i}{I}_j,{\mathsf{\Lambda }}_{I_i}{I}_j),\hfill \end{array}$$

(14)

where

$${\tau }^{*}={\mathsf{\Xi }}_{1\le i<j\le r}\widehat{\Re }(I_i,I_j,I_j,I_i).$$

Now, we define

$$\begin{array}{c}\hfill {\parallel C\parallel }^2={\mathsf{\Xi }}_{i=1}^n{\varrho }^2(C{I}_i,I_j),\end{array}$$

(15)

and then, from (14) and (15), we obtain

$$\begin{array}{c}\hfill 2{\tau }^{*}=f_{1}n(n-1)+3f_2{\parallel C\parallel }^2-3{\mathsf{\Xi }}_{i,j=1}^n\varrho (\mathsf{\Lambda }{I}_i,I_j,\mathsf{\Lambda }{I}_i,I_j).\end{array}$$

(16)

From the relation given in (16), the next result follows.

Theorem 3. 

Let $\mathsf{\Psi }$ be an SRS mapping the GSSF $\left(\overline{M}(f_1,f_2,f_3),\varrho \right)$ onto a Riemannian manifold $\left(M,{\varrho }_2\right)$, with ξ entirely contained in the vertical distribution. Then,

$$\begin{array}{c}\hfill 2{\tau }^{*}\le f_{1}n(n-1)+3f_2{\parallel C\parallel }^2.\end{array}$$

(17)

The necessary and sufficient condition for the equality case in (17) is if $\mathcal{H}\left(\overline{M}\right)$ is integrable.

In what follows, we assume that $\xi$ is a horizontal vector field.

Theorem 4. 

Let $\mathsf{\Psi }$ be an SRS mapping the GSSF $\left(\overline{M}(f_1,f_2,f_3),\varrho \right)$ onto a Riemannian manifold $\left(M,{\varrho }_2\right)$, with ξ entirely contained in the horizontal distribution. Then,

$$\begin{array}{c}\hfill 2\tau \ge f_{1}r(r-1)+3f_{2}r{cos}^2\theta -r^2{\parallel H\parallel }^2.\end{array}$$

(18)

The necessary and sufficient condition for the equality case in (18) is if each fiber is totally geodesic.

Proof. 

By employing the symmetry property of T in (7), we derive

$$\begin{array}{c}\hfill 2\tau =f_{1}r(r-1)+3f_{2}r{cos}^2\theta -r^2{\parallel H\parallel }^2+{\mathsf{\Xi }}_{i,j=1}^r\varrho (T_{L_i}{L}_j,T_{L_i}{L}_j),\end{array}$$

(19)

which implies (18). □

With $\xi$ assumed to be horizontal and $\mathsf{\Lambda }$ anti-symmetric, and by applying (8), a series of computations leads to

$$\begin{array}{c}\hfill 2{\tau }^{*}=f_{1}n(n-1)+3f_2{\parallel C\parallel }^2+f_3(2-2n)-3{\mathsf{\Xi }}_{i,j=1}^n\varrho ({\mathsf{\Lambda }}_{I_i}{I}_j,{\mathsf{\Lambda }}_{I_i}{I}_j).\end{array}$$

(20)

Employing (20), we are led to the next result.

Theorem 5. 

Let $\mathsf{\Psi }$ be an SRS mapping the GSSF $\left(\overline{M}(f_1,f_2,f_3),\varrho \right)$ onto a Riemannian manifold $\left(M,{\varrho }_2\right)$, with ξ entirely contained in the horizontal distribution. Then,

$$\begin{array}{c}\hfill 2{\tau }^{*}\le f_{1}n(n-1)+3f_2{\parallel C\parallel }^2+2f_3(1-n).\end{array}$$

(21)

The necessary and sufficient condition for the equality case in (21) is $\mathcal{H}\left(\overline{M}\right)$ is integrable.

Let $\left(\overline{M}(f_1,f_2,f_3),\varrho \right)$ be a GSSF, and let $\left(M,{\varrho }_2\right)$ be a Riemannian manifold. Suppose that $\mathsf{\Psi }:\overline{M}(f_1,f_2,f_3)\to M$ is an SRS. Consider an orthonormal basis $\{L_1,\dots ,L_r,I_1,\dots ,I_n\}$ of the tangent space $T_p\overline{M}\left(c\right)$ such that the vertical distribution $\mathcal{V}\left(\overline{M}\right)$ is spanned by $\{L_1,\dots ,L_r\}$ and the horizontal distribution $\mathcal{H}\left(\overline{M}\right)$ is spanned by $\{I_1,\dots ,I_n\}$.

We define the components $T_{ij}^s$ by

$$\begin{array}{c}\hfill T_{ij}^s=\varrho (T_{L_i}{L}_j,I_s),\end{array}$$

(22)

for indices $1\le i,j\le r$ and $1\le s\le n$ (see [39]).

Likewise, the components ${\mathsf{\Lambda }}_{ij}^{\alpha }$ are defined as

$$\begin{array}{c}\hfill {\mathsf{\Lambda }}_{ij}^{\alpha }=\varrho ({\mathsf{\Lambda }}_{I_i}{I}_j,L_{\alpha }),\end{array}$$

(23)

where $1\le i,j\le n$ and $1\le \alpha \le r$.

Following [39], we also make use of the following expression:

$$\begin{array}{c}\hfill \delta \left(N\right)={\mathsf{\Xi }}_{i=1}^n{\mathsf{\Xi }}_{k=1}^r\varrho (\left({\nabla }_{I_i}T\right)(L_k,L_k),I_i).\end{array}$$

(24)

From the binomial theorem, we derive the following equation involving the tensor fields $\mathcal{T}$:

$$\begin{array}{cc}\hfill {\mathsf{\Xi }}_{s=1}^n{\mathsf{\Xi }}_{i,j=1}^r{\left(T_{ij}^s\right)}^2& ={\displaystyle \frac{1}{2}}{r}^2{\parallel H\parallel }^2+{\displaystyle \frac{1}{2}}{\left(T_{11}^s-T_{22}^s-\cdots -T_{rr}^s\right)}^2\hfill \\ \hfill & +2{\mathsf{\Xi }}_{s=1}^n{\mathsf{\Xi }}_{j=2}^r{\left(T_{1j}^s\right)}^2-2{\mathsf{\Xi }}_{s=1}^n{\mathsf{\Xi }}_{1\le i<j\le r}\left(T_{ii}^s{T}_{jj}^s-{\left(T_{ij}^s\right)}^2\right).\hfill \end{array}$$

(25)

Theorem 6. 

Let $\mathsf{\Psi }$ be an SRS mapping the GSSF $\left(\overline{M}(f_1,f_2,f_3),\varrho \right)$ onto a Riemannian manifold $\left(M,{\varrho }_2\right)$, with ξ entirely contained in the vertical distribution. Then,

$$\begin{array}{cc}\hfill \widehat{Ric}\left(L_1\right)& \ge f_1(r-1)+3f_2{cos}^2\theta (1-{\left(\eta \left(L_1\right)\right)}^2)\hfill \\ \hfill & +f_3\left[(2-r){\left(\eta \left(L_1\right)\right)}^2-1)\right]-{\displaystyle \frac{1}{4}}{r}^2{\parallel H\parallel }^2.\hfill \end{array}$$

(26)

The necessary and sufficient condition for the equality case in (26) is

$$T_{11}^s=T_{22}^s=\cdots =T_{rr}^s,T_{1j}^s=0,j=2,\dots ,r.$$

Proof. 

By applying (22) to (13) and utilizing the symmetry of $\mathcal{T}$, we can write

$$\begin{array}{cc}\hfill 2\tau & =f_{1}r(r-1)+3f_2(r-1){cos}^2\theta +f_3(2-2r)\hfill \\ \hfill & -r^2{\parallel H\parallel }^2+{\mathsf{\Xi }}_{s=1}^n{\mathsf{\Xi }}_{i,j=1}^r{\left(T_{ij}^s\right)}^2.\hfill \end{array}$$

(27)

Consequently, substituting (25) into (27), we find

$$\begin{array}{cc}\hfill 2\tau & =f_{1}r(r-1)+3f_2(r-1){cos}^2\theta +f_3(2-2r)-{\displaystyle \frac{1}{2}}{r}^2{\parallel H\parallel }^2\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\left(T_{11}^s-T_{22}^s-\cdots -T_{rr}^s\right)}^2+2{\mathsf{\Xi }}_{s=1}^n{\mathsf{\Xi }}_{j=2}^r{\left(T_{1j}^s\right)}^2\hfill \\ \hfill & -2{\mathsf{\Xi }}_{s=1}^n{\mathsf{\Xi }}_{1\le i<j\le r}\left(T_{ii}^s{T}_{jj}^s-{\left(T_{ij}^s\right)}^2\right).\hfill \end{array}$$

(28)

Based on (28), we derive

$$\begin{array}{cc}\hfill 2\tau & \ge f_{1}r(r-1)+3f_2(r-1){cos}^2\theta +f_3(2-2r)-{\displaystyle \frac{1}{2}}{r}^2{\parallel H\parallel }^2\hfill \\ \hfill & -2{\mathsf{\Xi }}_{s=1}^n{\mathsf{\Xi }}_{1\le i<j\le r}\left(T_{ii}^s{T}_{jj}^s-{\left(T_{ij}^s\right)}^2\right).\hfill \end{array}$$

(29)

Proceeding by setting $L=B=L_i$ and $D=G=L_j$ in (1), and making use of (22), we find

$$\begin{array}{cc}\hfill 2{\mathsf{\Xi }}_{2\le i<j\le r}\tilde{\Re }(L_i,L_j,L_j,L_i)& =2{\mathsf{\Xi }}_{2\le i<j\le r}\Re (L_i,L_j,L_j,L_i)\hfill \\ \hfill & +2{\mathsf{\Xi }}_{s=1}^n{\mathsf{\Xi }}_{2\le i<j\le r}\left(T_{ii}^s{T}_{jj}^s-{\left(T_{ij}^s\right)}^2\right).\hfill \end{array}$$

(30)

Taking (30) into account with (29), we find

$$\begin{array}{cc}\hfill 2\tau & \ge f_{1}r(r-1)+3f_2(r-1){cos}^2\theta +f_3(2-2r)-{\displaystyle \frac{1}{2}}{r}^2{\parallel H\parallel }^2\hfill \\ \hfill & +2{\mathsf{\Xi }}_{2\le i<j\le r}\Re (L_i,L_j,L_j,L_i)-2{\mathsf{\Xi }}_{2\le i<j\le r}\tilde{\Re }(L_i,L_j,L_j,L_i).\hfill \end{array}$$

(31)

Continuing in this direction, we derive

$$\begin{array}{c}\hfill 2\tau =2{\mathsf{\Xi }}_{2\le i<j\le r}\tilde{\Re }(L_i,L_j,L_j,L_i)+2{\mathsf{\Xi }}_{j=1}^r\Re (L_1,L_j,L_j,L_1).\end{array}$$

(32)

Incorporating (32) into (31), we arrive at

$$\begin{array}{cc}\hfill 2Ric\left(L_1\right)& \ge f_{1}r(r-1)+3f_2(r-1){cos}^2\theta +f_3(2-2r)-{\displaystyle \frac{1}{2}}{r}^2{\parallel H\parallel }^2\hfill \\ \hfill & -2{\mathsf{\Xi }}_{2\le i<j\le r}\tilde{\Re }(L_i,L_j,L_j,L_i).\hfill \end{array}$$

(33)

Owing to the fact that $\overline{M}(f_1,f_2,f_3)$ is a GSSF, the curvature tensor $\tilde{\Re }$ satisfies the equality specified by (4). Hence, we find

$$\begin{array}{cc}\hfill {\mathsf{\Xi }}_{2\le i<j\le r}\tilde{\Re }(L_i,L_j,L_j,L_i)& ={\displaystyle \frac{f_1}{2}}(r-2)(r-1)+{\displaystyle \frac{3f_2}{2}}{cos}^2\theta \left[(r-3)+2{\left(\eta \left(L_1\right)\right)}^2\right]\hfill \\ \hfill & +f_3[(2-r)+(r-2){\left(\eta \left(L_1\right)\right)}^2].\hfill \end{array}$$

(34)

From (33) and (34), we obtain (26). □

Theorem 7. 

Let $\mathsf{\Psi }$ be an SRS mapping the GSSF $\left(\overline{M}(f_1,f_2,f_3),\varrho \right)$ onto a Riemannian manifold $\left(M,{\varrho }_2\right)$, with ξ entirely contained in the vertical distribution. Then,

$$\begin{array}{c}\hfill {Ric}^{*}\left(I_1\right)\le f_1(n-1)+3f_2{\parallel C{I}_1\parallel }^2.\end{array}$$

(35)

The necessary and sufficient condition for the equality case in (35) is

$${\mathsf{\Lambda }}_{ij}^{\alpha }=0,j=2,\dots ,n.$$

Proof. 

From (16), we have

$$\begin{array}{cc}\hfill 2{\tau }^{*}& =f_{1}n(n-1)+3f_2{\parallel C\parallel }^2-3{\mathsf{\Xi }}_{\alpha =1}^r{\mathsf{\Xi }}_{i,j=1}^n{\left({\mathsf{\Lambda }}_{ij}^{\alpha }\right)}^2.\hfill \end{array}$$

(36)

Given that $\mathsf{\Lambda }$ is anti-symmetric on $\mathcal{H}\left(\overline{M}\left(c\right)\right)$, we can express (36) as

$$\begin{array}{cc}\hfill 2{\tau }^{*}& =f_{1}n(n-1)+f_2{\parallel C\parallel }^2-6{\mathsf{\Xi }}_{\alpha =1}^r{\mathsf{\Xi }}_{1\le i<j\le n}{\left({\mathsf{\Lambda }}_{ij}^{\alpha }\right)}^2.\hfill \end{array}$$

(37)

Taking $I=H=I_i$ and $Y=Z=I_j$ in (2), and subsequently applying (23), leads to

$$\begin{array}{cc}\hfill 2{\mathsf{\Xi }}_{2\le i<j\le n}\tilde{\Re }(I_i,I_j,I_j,I_i)& =2{\mathsf{\Xi }}_{2\le i<j\le n}{\Re }^{*}(I_i,I_j,I_j,I_i)\hfill \\ \hfill & +6{\mathsf{\Xi }}_{\alpha =1}^r{\mathsf{\Xi }}_{1\le i<j\le n}{\left({\mathsf{\Lambda }}_{ij}^{\alpha }\right)}^2.\hfill \end{array}$$

(38)

Making use of (38) in (37), we find

$$\begin{array}{cc}\hfill 2{\tau }^{*}& =f_{1}n(n-1)+f_2{\parallel C\parallel }^2-6{\mathsf{\Xi }}_{\alpha =1}^r{\mathsf{\Xi }}_{1\le i<j\le n}{\left({\mathsf{\Lambda }}_{ij}^{\alpha }\right)}^2\hfill \\ \hfill & +2{\mathsf{\Xi }}_{2\le i<j\le n}{\Re }^{*}(I_i,I_j,I_j,I_i)-2{\mathsf{\Xi }}_{2\le i<j\le n}\tilde{\Re }(I_i,I_j,I_j,I_i).\hfill \end{array}$$

(39)

Furthermore, from (4), we obtain

$$\begin{array}{cc}\hfill {\mathsf{\Xi }}_{2\le i<j\le n}\tilde{\Re }(I_i,I_j,I_j,I_i)& ={\displaystyle \frac{f_1}{2}}(n-2)(n-1)\hfill \\ \hfill & +3f_2{\mathsf{\Xi }}_{2\le i<j\le n}{\varrho }^2(C{X}_i,I_j).\hfill \end{array}$$

(40)

Then, from (39) and (40), we obtain (35). □

We proceed to compute the Chen–Ricci inequality between the vertical and horizontal distributions, considering the case where $\xi$ lies in the vertical distribution. Consequently, for the scalar curvature ${\tau }^{*}$ of $\tilde{M}\left(c\right)$, we have

$$\begin{array}{cc}\hfill 2\tilde{\tau }& ={\mathsf{\Xi }}_{s=1}^{n}Ric(I_s,I_s)+{\mathsf{\Xi }}_{k=1}^{r}Ric(L_k,L_k),\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill 2\tilde{\tau }& ={\mathsf{\Xi }}_{j,k=1}^r\tilde{\Re }(L_j,L_k,L_k,L_j)+{\mathsf{\Xi }}_{i=1}^n{\mathsf{\Xi }}_{k=1}^r\tilde{\Re }(I_i,L_k,L_k,I_i)\hfill \\ \hfill & +{\mathsf{\Xi }}_{i,s=1}^n\tilde{\Re }(I_i,I_s,I_s,I_i)+{\mathsf{\Xi }}_{s=1}^n{\mathsf{\Xi }}_{j=1}^r\tilde{\Re }(L_j,I_s,I_s,L_j).\hfill \end{array}$$

(42)

Let us denote the following:

$$\begin{array}{cc}\hfill \parallel {\mathcal{T}}^{\mathcal{V}}{\parallel }^2& ={\mathsf{\Xi }}_{i=1}^n{\mathsf{\Xi }}_{k=1}^r\varrho (T_{L_k}{I}_i,T_{L_k}{I}_i),\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill \parallel {\mathcal{T}}^{\mathcal{H}}{\parallel }^2& ={\mathsf{\Xi }}_{j,k=1}^r\varrho (T_{L_j}{L}_k,T_{L_j}{L}_k),\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill \parallel {\mathcal{A}}^{\mathcal{V}}{\parallel }^2& ={\mathsf{\Xi }}_{i,j=1}^n\varrho ({\mathsf{\Lambda }}_{I_i}{I}_j,{\mathsf{\Lambda }}_{I_i}{I}_j),\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill \parallel {\mathcal{A}}^{\mathcal{H}}{\parallel }^2& ={\mathsf{\Xi }}_{i=1}^n{\mathsf{\Xi }}_{k=1}^r\varrho ({\mathsf{\Lambda }}_{I_i}{L}_k,{\mathsf{\Lambda }}_{I_i}{L}_k).\hfill \end{array}$$

(46)

Theorem 8. 

Let $\mathsf{\Psi }$ be an SRS mapping the GSSF $\left(\overline{M}(f_1,f_2,f_3),\varrho \right)$ onto a Riemannian manifold $\left(M,{\varrho }_2\right)$, with ξ entirely contained in the vertical distribution. Then,

$$\begin{array}{cc}\hfill & f_1(nr+n+r-2)+3f_2\left[(1-{\left(\eta \left(L_1\right)\right)}^2){cos}^2\theta +{\parallel \mathcal{B}\parallel }^2+\parallel C{L}_1{\parallel }^2)\right]\hfill \\ \hfill & +f_3\left[-1-n+(2-r){\left(\eta \left(L_1\right)\right)}^2\right]\hfill \\ \hfill & \le Ric\left(L_1\right)+{Ric}^{*}\left(I_1\right)+{\displaystyle \frac{1}{4}}{r}^2{\parallel H\parallel }^2+3{\mathsf{\Xi }}_{\alpha =1}^r{\mathsf{\Xi }}_{s=2}^n{\left({\mathsf{\Lambda }}_{ij}^{\alpha }\right)}^2-\delta \left(N\right)\hfill \\ \hfill & +\parallel {\mathcal{T}}^{\mathcal{V}}{\parallel }^2-{\parallel {\mathcal{A}}^{\mathcal{H}}\parallel }^2.\hfill \end{array}$$

(47)

The necessary and sufficient condition for the equality case in (47) is

$$T_{11}^s=T_{22}^s=\cdots =T_{rr}^s,T_{1j}^s=0,j=2,\dots ,r.$$

Proof. 

Since $\tilde{M}\left(c\right)$ is a Sasakian space form, from (42), we obtain

$$\begin{array}{cc}\hfill 2\tilde{\tau }& =f_1(n+r)(n+r-1)\hfill \\ \hfill & +3f_2\{(r-1){cos}^2\theta +{\parallel \mathcal{C}\parallel }^2+2{\mathsf{\Xi }}_{i=1}^n{\mathsf{\Xi }}_{k=1}^r{\varrho }^2(B{L}_i,L_k)\}\hfill \\ \hfill & +2f_3(1-r-n).\hfill \end{array}$$

(48)

Now, we define

$$\begin{array}{c}\hfill {\parallel \mathcal{B}\parallel }^2={\mathsf{\Xi }}_{i=1}^n{\mathsf{\Xi }}_{k=1}^r{\varrho }^2(B{X}_i,L_k).\end{array}$$

(49)

Through the application of the Gauss–Codazzi-type identities (1), (2), and (3), the following results can be established:

$$\begin{array}{cc}\hfill 2\tilde{\tau }& =2{\tau }^{*}+2{\tau }^r+r^2{\parallel H\parallel }^2-{\mathsf{\Xi }}_{k,j=1}^r\varrho (T_{L_k}{L}_j,T_{L_k}{L}_j)\hfill \\ \hfill & +3{\mathsf{\Xi }}_{i,s=1}^n\varrho ({\mathsf{\Lambda }}_{I_i}{I}_s,{\mathsf{\Lambda }}_{I_i}{I}_s)-{\mathsf{\Xi }}_{k=1}^r{\mathsf{\Xi }}_{i=1}^n\varrho (\left({\nabla }_{I_i}T\right)L_k,I_i)\hfill \\ \hfill & +{\mathsf{\Xi }}_{i=1}^n{\mathsf{\Xi }}_{k=1}^r\varrho (T_{L_k}{I}_i,T_{L_k}{I}_i)-\varrho ({\mathsf{\Lambda }}_{I_i}{L}_k,{\mathsf{\Lambda }}_{I_i}{L}_k)\hfill \\ \hfill & -{\mathsf{\Xi }}_{s=1}^n{\mathsf{\Xi }}_{j=1}^r\varrho (\left({\nabla }_{I_s}T\right)L_j,I_s)+{\mathsf{\Xi }}_{s=1}^n{\mathsf{\Xi }}_{j=1}^r\varrho (T_{L_j}{I}_s,T_{L_j}{I}_s)\hfill \\ \hfill & -\varrho ({\mathsf{\Lambda }}_{I_s}{L}_j,{\mathsf{\Lambda }}_{I_s}{L}_j).\hfill \end{array}$$

(50)

Thus, from (25) and (50), we derive

$$\begin{array}{cc}\hfill 2\tilde{\tau }& =2{\tau }^{*}+2{\tau }^r+{\displaystyle \frac{1}{2}}{r}^2{\parallel H\parallel }^2-{\displaystyle \frac{1}{2}}{(T_{11}^s-T_{22}^s-\cdots -T_{rr}^s)}^2\hfill \\ \hfill & -2{\mathsf{\Xi }}_{s=1}^r{\mathsf{\Xi }}_{j=2}^n{\left(T_{1j}^s\right)}^2+2{\mathsf{\Xi }}_{s=1}^r{\mathsf{\Xi }}_{2\le i<j\le r}(T_{ii}^s{T}_{jj}^s-{\left(T_{ij}^s\right)}^2)\hfill \\ \hfill & +6{\mathsf{\Xi }}_{\alpha =1}^r{\mathsf{\Xi }}_{s=2}^n{\left({\mathsf{\Lambda }}_{1s}^{\alpha }\right)}^2+6{\mathsf{\Xi }}_{\alpha =1}^r{\mathsf{\Xi }}_{2\le i<s\le n}{\left({\mathsf{\Lambda }}_{is}^{\alpha }\right)}^2\hfill \\ \hfill & +{\mathsf{\Xi }}_{i=1}^n{\mathsf{\Xi }}_{k=1}^r\left(\varrho (T_{L_k}{I}_i,T_{L_k}{I}_i)-\varrho ({\mathsf{\Lambda }}_{I_k},{\mathsf{\Lambda }}_{I_k})\right)\hfill \\ \hfill & -2\delta \left(N\right)+{\mathsf{\Xi }}_{s=1}^n{\mathsf{\Xi }}_{j=1}^r\varrho (T_{L_j}{I}_s,T_{L_j}{I}_s)-\varrho ({\mathsf{\Lambda }}_{I_j}{L}_k,{\mathsf{\Lambda }}_{I_j}{L}_k).\hfill \end{array}$$

(51)

Incorporating (30), (38), (48), and (49) into (51), we arrive at

$$\begin{array}{cc}\hfill & f_1(n+r)(n+r-1)\hfill \\ \hfill & +3f_2\{(r-1){cos}^2\theta +{\parallel \mathcal{C}\parallel }^2+{2\parallel \mathcal{B}\parallel }^2\}+2f_3(1-r-n)\hfill \\ \hfill & =2Ric\left(L_1\right)+2{Ric}^{*}\left(I_1\right)+{\displaystyle \frac{1}{2}}{r}^2{\parallel H\parallel }^2-{\displaystyle \frac{1}{2}}{(T_{11}^s-T_{22}^s-\cdots -T_{rr}^s)}^2\hfill \\ \hfill & -2{\mathsf{\Xi }}_{s=1}^n{\mathsf{\Xi }}_{j=2}^r{\left(T_{1j}^s\right)}^2+6{\mathsf{\Xi }}_{\alpha =1}^r{\mathsf{\Xi }}_{s=2}^n{\left({\mathsf{\Lambda }}_{1s}^{\alpha }\right)}^2\hfill \\ \hfill & +{\mathsf{\Xi }}_{i=1}^n{\mathsf{\Xi }}_{k=1}^r\{\varrho (T_{L_k}{I}_i,T_{L_k}{I}_i)-\varrho ({\mathsf{\Lambda }}_{I_i}{L}_k,{\mathsf{\Lambda }}_{I_i}{L}_k)\}\hfill \\ \hfill & -2\delta \left(N\right)+{\mathsf{\Xi }}_{s=1}^n{\mathsf{\Xi }}_{j=1}^r\{\varrho (T_{L_j}{I}_s,T_{L_j}{I}_s)-\varrho ({\mathsf{\Lambda }}_{I_s}{L}_j,{\mathsf{\Lambda }}_{I_s}{L}_j)\}\hfill \\ \hfill & +2{\mathsf{\Xi }}_{2\le i<j\le r}\tilde{\Re }(L_i,L_j,L_j,L_i)+2{\mathsf{\Xi }}_{2\le i<j\le n}\tilde{\Re }(I_i,I_j,I_j,I_i).\hfill \end{array}$$

(52)

Moreover, using (34) and (40) in (52), we obtain (47). □

Theorem 9. 

Let $\mathsf{\Psi }$ be an SRS mapping the GSSF $\left(\overline{M}(f_1,f_2,f_3),\varrho \right)$ onto a Riemannian manifold $\left(M,{\varrho }_2\right)$, where ξ is entirely contained in the horizontal distribution. Then,

$$\begin{array}{cc}\hfill \widehat{Ric}\left(L_1\right)& \ge f_1(r-1)+3f_2{cos}^2\theta -{\displaystyle \frac{1}{4}}{r}^2{\parallel H\parallel }^2.\hfill \end{array}$$

(53)

The necessary and sufficient condition for the equality case in (53) is

$$T_{11}^s=T_{22}^s=\cdots =T_{rr}^s,T_{1j}^s=0,j=2,\dots ,r.$$

Proof. 

By employing arguments analogous to those in the proof of Theorem 6, the result follows. □

Theorem 10. 

Let $\mathsf{\Psi }$ be an SRS mapping the GSSF $\left(\overline{M}(f_1,f_2,f_3),\varrho \right)$ onto a Riemannian manifold $\left(M,{\varrho }_2\right)$, with ξ entirely contained in the horizontal distribution. Then,

$$\begin{array}{cc}\hfill {Ric}^{*}\left(I_1\right)& \le f_1(n-1)+3f_2{\parallel C{L}_1\parallel }^2+f_3((2-n)\eta {\left(I_1\right)}^2-1).\hfill \end{array}$$

(54)

The necessary and sufficient condition for the equality case in (54) is

$${\mathsf{\Lambda }}_{ij}^{\alpha }=0,j=2,\dots ,n.$$

Proof. 

By employing arguments analogous to those in the proof of Theorem 7, we obtain the result. □

Theorem 11. 

Let $\mathsf{\Psi }$ be an SRS mapping the GSSF $\left(\overline{M}(f_1,f_2,f_3),\varrho \right)$ onto a Riemannian manifold $\left(M,{\varrho }_2\right)$, with ξ entirely contained in the vertical distribution. Then,

$$\begin{array}{cc}\hfill 2\tilde{\tau }+2{\tau }^{*}& \le f_1(n+r)(n+r-1)\hfill \\ \hfill & +3f_2\{(r-1){cos}^2\theta +{2\parallel \mathcal{B}\parallel }^2+{\parallel \mathcal{C}\parallel }^2\}\hfill \\ \hfill & +2f_3(1-r-n)\hfill \\ \hfill & -r^2{\parallel H\parallel }^2+\parallel {\mathcal{T}}^{\mathcal{H}}{\parallel }^2+2\delta \left(N\right)+2\parallel {\mathcal{A}}^{\mathcal{H}}{\parallel }^2-2{\parallel {\mathcal{T}}^{\mathcal{V}}\parallel }^2,\hfill \end{array}$$

(55)

and

$$\begin{array}{cc}\hfill 2\tilde{\tau }+2{\tau }^{*}& \ge f_1(n+r)(n+r-1)\hfill \\ \hfill & +3f_2\{(r-1){cos}^2\theta +{2\parallel \mathcal{B}\parallel }^2+{\parallel \mathcal{C}\parallel }^2\}\hfill \\ \hfill & +2f_3(1-r-n)\hfill \\ \hfill & -r^2{\parallel H\parallel }^2+\parallel {\mathcal{T}}^{\mathcal{H}}{\parallel }^2-3\parallel {\mathcal{A}}^{\mathcal{V}}{\parallel }^2+2\delta \left(N\right)-2{\parallel {\mathcal{T}}^{\mathcal{V}}\parallel }^2.\hfill \end{array}$$

(56)

The equality in (55) and (56) is achieved at every point $p\in \tilde{M}$ if and only if the horizontal distribution $\mathcal{H}$ admits an integrable structure.

Proof. 

From (43), (46), (48), (49), and (50), we obtain our result. □

Theorem 11 naturally leads to the next result.

Corollary 1. 

Let $\mathsf{\Psi }$ be an SRS mapping the GSSF $\left(\overline{M}(f_1,f_2,f_3),\varrho \right)$ onto a Riemannian manifold $\left(M,{\varrho }_2\right)$, with the assumption that the vector field ξ is entirely contained in the vertical distribution and each fiber is totally geodesic. Then,

$$\begin{array}{cc}\hfill 2\tilde{\tau }+2{\tau }^{*}& \le f_1(n+r)(n+r-1)\hfill \\ \hfill & +3f_2\{(r-1){cos}^2\theta +{2\parallel \mathcal{B}\parallel }^2+{\parallel \mathcal{C}\parallel }^2\}\hfill \\ \hfill & +2f_3(1-r-n)+2{\parallel {\mathcal{A}}^{\mathcal{H}}\parallel }^2,\hfill \end{array}$$

(57)

and

$$\begin{array}{cc}\hfill 2\tilde{\tau }+2{\tau }^{*}& \ge f_1(n+r)(n+r-1)\hfill \\ \hfill & +3f_2\{(r-1){cos}^2\theta +{2\parallel \mathcal{B}\parallel }^2+{\parallel \mathcal{C}\parallel }^2\}\hfill \\ \hfill & +2f_3(1-r-n)-3{\parallel {\mathcal{A}}^{\mathcal{V}}\parallel }^2.\hfill \end{array}$$

(58)

The equality in (57) and (58) is achieved at every point $p\in \tilde{M}$ if and only if the horizontal distribution $\mathcal{H}$ admits an integrable structure.

5. Explicit Constructions of Slant Submersions from GSSF

In this section, we present two explicit constructions of proper slant submersions from generalized Sasakian space forms. These examples are designed to illustrate the geometric principles discussed earlier, demonstrating how one can systematically construct slant distributions that satisfy the constant angle condition with respect to the structure tensor $\varphi$.

The first example is based on a local coordinate model on ${\mathbb{R}}^5$, equipped with the standard contact metric structure, and shows how a slant submersion can be achieved through careful linear combinations of coordinate directions. The second example employs a warped product manifold structure, where the generalized Sasakian geometry arises naturally from a warping function over a Kähler base, and a slant submersion is constructed by projecting onto a direction inclined at a constant angle. These constructions not only exemplify the theory but also provide practical templates for generating further examples in more general settings.

Example 1. 

Consider the 5-dimensional manifold $\overline{M}={\mathbb{R}}^5$ with coordinates $(x_1,x_2,y_1,y_2,z)$. Define a contact metric structure $(\varphi ,\xi ,\eta ,\varrho )$ on $\overline{M}$ as follows:

The contact 1-form is given by

$$\eta =dz-y_{1}d{x}_1-y_{2}d{x}_2.$$

The Reeb vector field is

$$\xi ={\displaystyle \frac{\partial }{\partial z}}.$$

The almost contact structure ϕ acts on the coordinate vector fields as

$$\varphi \left({\displaystyle \frac{\partial }{\partial x_i}}\right)={\displaystyle \frac{\partial }{\partial y_i}},\varphi \left({\displaystyle \frac{\partial }{\partial y_i}}\right)=-{\displaystyle \frac{\partial }{\partial x_i}},\varphi \left({\displaystyle \frac{\partial }{\partial z}}\right)=0,i=1,2.$$

The Riemannian metric is defined as

$$\varrho =\sum _{i=1}^2\left(d{x}_i^2+d{y}_i^2\right)+\eta \otimes \eta .$$

This structure makes $(M,\varphi ,\xi ,\eta ,\varrho )$ a Sasakian manifold, and hence, a generalized Sasakian space form with constant curvature functions.

Now, define a smooth map $F:\overline{M}\to {\mathbb{R}}^2$ by

$$F(x_1,x_2,y_1,y_2,z)=(x_1+y_2,x_2-y_1).$$

Compute the differential $F_{*}$ and determine the kernel:

$$ker{F}_{*}=span\left\{V_1={\displaystyle \frac{\partial }{\partial y_1}}+{\displaystyle \frac{\partial }{\partial x_2}},V_2={\displaystyle \frac{\partial }{\partial y_2}}-{\displaystyle \frac{\partial }{\partial x_1}},V_3={\displaystyle \frac{\partial }{\partial z}}=\xi \right\}.$$

We compute the image of the vertical frame under ϕ:

$$\begin{array}{cc}\hfill \varphi \left(V_1\right)& =-{\displaystyle \frac{\partial }{\partial x_1}}+{\displaystyle \frac{\partial }{\partial y_2}},\hfill \\ \hfill \varphi \left(V_2\right)& =-{\displaystyle \frac{\partial }{\partial x_2}}-{\displaystyle \frac{\partial }{\partial y_1}},\hfill \\ \hfill \varphi \left(V_3\right)& =0.\hfill \end{array}$$

It is easily verified that each $\varphi \left(V_i\right)$ has nonzero components both along and orthogonal to $ker{F}_{*}$. Moreover, the angle between each $\varphi \left(V_i\right)$ and $ker{F}_{*}$ is a constant $\theta =\pi /4$. Therefore, F is a proper slant submersion from the generalized Sasakian space form M with slant angle $\theta ={\displaystyle \frac{\pi }{4}}$.

Example 2. 

Let $\overline{M}=\mathbb{R}{\times }_{e^t}{\mathbb{R}}^2$ be the warped product of $\mathbb{R}$ and ${\mathbb{R}}^2$ with warping function $f\left(t\right)=e^t$. Let $(t,x,y)$ be coordinates on $\overline{M}$, and define a Riemannian metric on $\overline{M}$ as

$$\varrho =d{t}^2+e^{2t}(d{x}^2+d{y}^2).$$

We define a contact metric structure $(\varphi ,\xi ,\eta ,\varrho )$ on $\overline{M}$ by

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & \eta =dt,\xi ={\displaystyle \frac{\partial }{\partial t}},\hfill \\ \hfill & \varphi \left({\displaystyle \frac{\partial }{\partial x}}\right)={\displaystyle \frac{\partial }{\partial y}},\hfill \\ \hfill & \varphi \left({\displaystyle \frac{\partial }{\partial y}}\right)=-{\displaystyle \frac{\partial }{\partial x}},\hfill \\ \hfill & \varphi \left({\displaystyle \frac{\partial }{\partial t}}\right)=0.\hfill \end{array}\right.\end{array}$$

Then, $(\overline{M},\varphi ,\xi ,\eta ,\varrho )$ becomes a generalized Sasakian space form with variable curvature functions depending on t.

Now, define a smooth map $F:\overline{M}\to \mathbb{R}$ by

$$F(t,x,y)=xcos\alpha +ysin\alpha$$

for a fixed angle $\alpha \in \left(0,{\displaystyle \frac{\pi }{2}}\right)$.

Then,

$$ker{F}_{*}=span\left\{\xi ={\displaystyle \frac{\partial }{\partial t}},V=-sin\alpha {\displaystyle \frac{\partial }{\partial x}}+cos\alpha {\displaystyle \frac{\partial }{\partial y}},\right\},$$

and

$$\varphi \left(V\right)=-sin\alpha {\displaystyle \frac{\partial }{\partial y}}-cos\alpha {\displaystyle \frac{\partial }{\partial x}}.$$

We observe that $\varphi \left(V\right)$ has a constant angle $\theta =\alpha$ with the vertical distribution $ker{F}_{*}$. Hence, F is a proper slant submersion from the generalized Sasakian space form $\overline{M}$ with the slant angle $\theta =\alpha$.

6. Conclusions and Future Directions

In this study, we have examined key inequalities involving the scalar and Ricci curvatures of slant submersions in GSSF. Through the establishment of geometric constraints, we have extended the existing framework of curvature inequalities and identified necessary conditions for achieving equality in these bounds. Our findings provide deeper insights into the curvature behavior of slant submersions within contact structures, further enriching the understanding of their geometric properties.

For future research, one possible direction is to investigate how these inequalities extend to other classes of almost contact and contact metric manifolds, such as trans-Sasakian structures. Additionally, exploring the impact of higher-order curvature invariants on slant submersions could provide a broader perspective on their geometric significance. Another promising avenue is the application of these results to physical theories, particularly in the study of submersions in generalized relativistic settings. Further analysis of stability conditions and harmonicity properties of such submersions could also yield valuable insights into their structural behavior.