Pinching Results for Doubly Warped Products’ Pointwise Bi-Slant Submanifolds in Locally Conformal Almost Cosymplectic Manifolds with a Quarter-Symmetric Connection

: In this research paper, we establish geometric inequalities that characterize the relationship between the squared mean curvature and the warping functions of a doubly warped product pointwise bi-slant submanifold. Our investigation takes place in the context of locally conformal almost cosymplectic manifolds, which are equipped with a quarter-symmetric metric connection. We also consider the cases of equality in these inequalities. Additionally, we derive some geometric applications of our obtained results.


Introduction
In 2000, B. Unal [1] introduced the concept of doubly warped products as an extension of warped products [2].According to Unal, given two Riemannian manifolds N 1 and N 2 with Riemannian metrics g 1 and g 2 , respectively, and positive differentiable functions f 1 on N 1 and f 2 on N 2 , the doubly warped product N = f 2 N 1 × f 1 N 2 of dimension n is defined on the product manifold N 1 × N 2 equipped with the warped metric g = f 2  2 g 1 + f 2 1 g 2 .The metric g is given by [ where t 1 : N 1 × N 2 → N 1 and t 2 : N 1 × N 2 → N 2 are the natural projections, * denotes the tangent maps, and f 1 and f 2 are the warping functions on N 1 and N 2 , respectively.It is worth noting that if either f 1 or f 2 is constant on N (but not both), then N reduces to a single warped product.Similarly, if both f 1 and f 2 are constant functions on N, then N becomes locally a Riemannian product.A doubly warped product manifold is considered proper if both f 1 and f 2 are non-constant functions on N.
Shifting the focus, the question of whether a Riemannian manifold can be immersed in a space form is a crucial matter in submanifold theory, tracing its roots back to Nash's renowned embedding theorem [3].However, Nash's original objective could not be realized due to the constraints imposed by intrinsic invariants in governing extrinsic properties of submanifolds.In order to surmount these obstacles, Chen introduced novel Riemannian invariants and established optimal connections between intrinsic and extrinsic invariants on submanifolds.
In this paper, we embark on an investigation concerning the isometric immersion of doubly warped products into locally conformal almost cosymplectic manifolds endowed with a quarter symmetric metric connection.We obtained inequalities possessing a remarkable character, as they establish upper bounds for the warping functions in relation to mean curvature, scalar curvature, and pointwise constant φ-sectional curvature c.These results not only generalize but also encompass other inequalities as specific cases, which we obtain as a geometric application of the results.

Preliminaries
Consider Ñ, a Riemannian manifold equipped with the Riemannian metric g.Let ∇ denote the Levi-Civita connection on Ñ.We also introduce ∇, a linear connection defined by [28], given as follows: Here, χ 1 and χ 2 are arbitrary elements of Ñ, µ 1 and µ 2 are real constants, and Q is a vector field on Ñ such that π(χ 1 ) = g(χ 1 , Q), where π represents a one-form.If ∇g = 0, the connection ∇ is referred to as a quarter-symmetric metric connection.Conversely, if ∇g ̸ = 0, it is known as a quarter-symmetric non-metric connection.A quarter-symmetric connection (generalization of semi-symmetric metric connection and semi-symmetric non-metric connection) plays a crucial role in understanding the curvature properties of Riemannian manifolds.It possesses certain symmetry properties, and studying this connection helps in understanding the underlying symmetries of the manifold.

Remark 1.
We can obtain special cases of (2) as follows: (i) In the case where µ 1 = µ 2 = 1, the above connection reduces to a semi-symmetric metric connection.(ii) When µ 1 = 1 and µ 2 = 0, the above connection reduces to a semi-symmetric non-metric connection.
We can describe the curvature tensor with respect to ∇ as Analogously, the curvature tensor can be defined in relation to ∇. Utilizing (2), we find that the curvature tensor can be described as follows according to [28]: where are (0, 2) tensors, for any vector fields χ 1 , χ 2 , χ 3 , and χ 4 of Ñ.
Let N denote an n-dimensional submanifold that resides within a (2m + 1)-dimensional cosymplectic space form Ñ. We examine the induced quarter-symmetric connection denoted by ∇ and the induced Levi-Civita connection denoted by ∇ on N. By uniquely decomposing the vector field Q on N into its tangent component Q T and normal component Q ⊥ , we express Q as Q = Q T + Q ⊥ .The Gauss formula, with respect to ∇ and ∇, can be represented as follows: for each χ 1 , χ 2 ∈ Γ(TN), where σ is the second fundamental form of N in Ñ and σ(χ If a smooth manifold Ñ has a dimension of (2m + 1) and possesses an endomorphism φ of its tangent bundle T Ñ, along with a structure vector field ξ and a 1-form η, then it is termed as a locally conformal almost cosymplectic manifold.The conditions specified below establish the necessary requirements for this characterization: where χ 1 , χ 2 tangent to N and u is the conformal function such that ω = uη (see [22]).
Consider the cases where the function u takes on the values u = 0 and u = 1.In the former case, Ñ is identified as a cosymplectic manifold, while in the latter case, it is recognized as a Kenmotsu manifold (refer to [29,30] for more details).
For an almost contact metric manifold Ñ, a plane section σ in T p Ñ is referred to as a φ-section if σ is orthogonal to the structural vector field ξ and φ(σ) = σ.If the sectional curvature K(σ) remains constant regardless of the choice of φ-section σ at each point p ∈ Ñ, then Ñ is said to have a pointwise constant φ-sectional curvature.
Let us assume that N is a submanifold within an almost contact metric manifold Ñ, equipped with an induced metric g.If ∇ and ∇ ⊥ represent the induced connections on the tangent bundle TN and the normal bundle T ⊥ N of N, respectively, then the Weingarten map is defined by for every χ 1 , χ 2 ∈ TN and N ∈ T ⊥ N. Here, h and A N denote the second fundamental form and the shape operator (associated with the normal vector field N), respectively, characterizing the embedding of N into Ñ.They are related as follows: where g represents the Riemannian metric on Ñ as well as the metric induced on N.
In the context of Ñ2m+1 , we make the choice of {υ 1 , • • • , υ n } as an orthonormal tangent frame and {υ n+1 , • • • , υ 2m+1 } as an orthonormal normal frame on N. For any p ∈ Ñ and for a φ -section σ of T p Ñ, the function c defined by c(p) = K(p) is termed as the φ−sectional curvature of Ñ.In other words, in the case of a locally conformal almost cosymplectic manifold Ñ with a dimension of at least 5 and possessing pointwise φsectional curvature c, the curvature tensor R with respect to the Levi-Civita connection ∇ on Ñ can be represented as follows: Then, from (2) and (9), we have Similarly, we have Consider a vector field χ 1 tangent to the submanifold N. We can express Jχ 1 as the sum of its tangential component Tχ 1 and its normal component Fχ 1 .In the case where T = 0, the submanifold is classified as totally real, while a submanifold is considered holomorphic when F = 0.
To calculate the squared norm of T at a point p ∈ N, we can utilize the equation where {υ 1 , • • • , υ n } denotes any orthonormal basis of the tangent space TN of N.
In a research conducted by Chen [31], it was demonstrated that a submanifold N of an almost Hermitian manifold ( Ñ, J, g) is classified as pointwise slant if and only if the equation can be expressed as follows: where θ(p) represents a real-valued function on N. A pointwise slant submanifold is considered proper if it does not contain any totally real points or complex points.We can easily verify the following relationships: for any χ 1 , χ 2 ∈ Γ(TN).
Let us now introduce the concept of a pointwise bi-slant submanifold, as defined by Chen and Uddin in their work [9]: A submanifold N n of an almost Hermitian manifold Ñ4m is referred to as a pointwise bi-slant submanifold if it possesses a pair of orthogonal distributions U 1 and U 2 that satisfy the following conditions: Pointwise bi-slant submanifolds are a more general class of submanifolds, encompassing bi-slant, pointwise semi-slant, semi-slant, and CR-submanifolds as special cases.
Since N n is a pointwise bi-slant submanifold, we can define an adapted orthonormal frame as n = 2d 1 + 2d 2 , given by Consequently, we define it in such a way that Based on Equation ( 14), we can observe that g(υ 1 , Jυ 2 ) = cos θ 1 g(υ 1 , υ 2 ).As a result, we readily obtain the subsequent relation Hence, we have where When dealing with an almost contact metric manifold Ñ, the totally umbilicity and total geodesicity of a submanifold N are established by the conditions h(χ 1 , χ 2 ) = g(χ 1 , χ 2 )H and h(χ 1 , χ 2 ) = 0, respectively, where χ 1 and χ 2 belong to Γ(TN).Here, H represents the mean curvature vector pertaining to N. Furthermore, if H is found to be zero, it signifies that N is a minimal submanifold in Ñ.
We consider the isometric immersion ϕ a doubly warped product, into a Riemannian manifold Ñ characterized by a constant sectional curvature c.Let n 1 , n 2 , and n represent the dimensions of N 1 , N 2 , and N 1 × f N 2 , respectively.In this context, for unit vector fields χ 1 and χ 3 that are tangent to N 1 and N 2 , respectively, we have Let us define the sectional curvature of a general doubly warped product in terms of a local orthonormal frame {υ 1 , The sectional curvature can then be expressed as follows: Within this framework, we introduce another significant Riemannian intrinsic invariant known as the scalar curvature of Ñ2m+1 , denoted as τ(T p Ñ2m+1 ).At a certain point p in Ñ2m+1 , the scalar curvature can be expressed as follows: where κij = κ(υ i ∧ υ j ).It is clear that the first equality ( 18) is congruent to the following equation, which is frequently used in subsequent proofs: Similarly, scalar curvature τ(L p ) of L-plan is given by An orthonormal basis of the tangent space where ∥H∥ 2 is the squared norm of the mean curvature vector H of N. We define κ ij and κij as the sectional curvatures of the plane section spanned by e i and υ j at p in the submanifold N n and the Riemannian manifold Ñ2m+1 , respectively.Therefore, κ ij and κij represent the intrinsic and extrinsic sectional curvatures of the span {υ i , υ j } at p. Hence, from the Gauss equation, we obtain The subsequent implications arise from the Gauss equation and ( 21):

Main Inequalities
At the outset, we remind ourselves of an important result by B.-Y. Chen, which will come in handy at a later stage.Lemma 1. [32] For k ≥ 2 and real numbers w 1 , w 1 , . . ., At this juncture, we demonstrate the principal outcome of this section by means of a formal proof.Theorem 1.Let Ñ(c) be a (2m+1)-dimensional locally conformal almost cosymplectic manifold, and let ϕ : f 2 N 1 × f 1 N 2 → Ñ(c) denote an isometric immersion of an n-dimensional pointwise bi-slant doubly warped product submanifold into Ñ(c) with a quarter-symmetric connection.Then, the following statement holds true: (i) The squared norm of the mean curvature can be related to warping functions through the following expression: Here, ∇ and ∆ represent the gradient and Laplacian operators, respectively.H denotes the mean curvature vector of N n , while a and b correspond to the traces of α and β, respectively.(ii) The equality case in (24) is satisfied if and only if φ is a mixed totally geodesic isometric immersion, and n 1 H 1 = n 2 H 2 , where H 1 and H 2 are the partial mean curvature vectors of H along N Proof.By selecting {υ 1 , • • • , υ n } and {υ n+1 , • • • , υ 2m+1 } as an orthonormal tangent and normal frames on N, respectively, and substituting χ 1 = χ 4 = υ i and χ 2 = χ 3 = υ j into (10) while employing (11), we obtain Through the summation 1 ≤ i, j ≤ n of ( 25) and the utilization of ( 15), we arrive at {g(σ(υ i , υ i ), σ(υ j , υ j )) − g(σ(υ i , υ j ), σ(υ i , υ j ))}.
The aforementioned expression can be written as follows: Let us make the assumption that Thus, with reference to ( 26) and ( 27), we have Accordingly, (28) can be written in the form for the orthonormal frame {υ 1 , • • • , υ n }.Through the application of algebraic Lemma 1 and relation (29), we determine If we substitute ii and w 3 = ∑ n t=n 1 +1 σ n+1 tt in the above Equation ( 29), we find Thus, it can be inferred that w 1 , w 2 , w 3 satisfy Chen's Lemma (for k = 3), implying that 3 Therefore, the inequality 2w 1 w 2 ≥ b holds, and equality is attained if and only if w 1 + w 2 = w 3 .
In the specific case being examined, this implies that The equality sign holds in the above inequality if and only if Again, using Gauss equation, we derive Subsequently, by considering (21), we obtain the scalar curvature for the locally conformal almost cosymplectic space form with a quarter-symmetric connection as Now, making use of ( 32) and ( 35), we obtain By utilizing (27) in the preceding equation, we obtain In Equation ( 24), the equality holds if and only if the expression in Equations ( 32) and ( 33) and Moreover, from (33), we obtain This demonstrates that ϕ is an immersion that is mixed and totally geodesic.On the other hand, the converse part of (39) is true when considering the immersion of a pointwise bi-slant warped product into a locally almost cosymplectic space form.As a result, we can assert that the proof is fully established.
The above theorem readily implies the following corollary.
Corollary 1.Let Ñ(c) be a (2m+1)-dimensional locally conformal almost cosymplectic manifold, and let ϕ : with quarter-symmetric connection with semi-symmetric connection with semi-symmetric non-metric connection Pointwise semi-slant Remark 2. The above result is obtained by using Remark 1 and the definition of semi-slant, hemi-slant, and CR in Theorem 1.
Next, we have the following theorem.Theorem 2. Let Ñ(c) be a (2m+1)-dimensional locally conformal almost cosymplectic manifold, and let ϕ : denote an isometric immersion of an n-dimensional pointwise bi-slant doubly warped product submanifold into Ñ(c) with a quarter-symmetric connection.Then, the following statement holds true: where n i = dimN i , i=1,2, and ∆ i is the Laplacian operator on N i , i=1,2.The equality sign holds in (40) and is guaranteed to hold when the equality sign is present in (53).Furthermore, if n = 2, then the equality sign in (40) holds identically.
Proof.Suppose f 2 N 1 × f 1 N 2 is an isometric immersion of an n-dimensional pointwise bislant doubly warped product submanifold into Ñ(c), a manifold with pointwise φ-sectional curvature c and endowed with a quarter symmetric connection.Then, by applying the equation of Gauss, we obtain Now, we consider that (n 1 cos 2 θ 1 + n 2 cos 2 θ 2 ) Then from ( 41) and ( 42), it follows that Given an orthonormal frame {υ 1 , • • • , υ n }, the equation can be represented in the following form: which implies that ).
The above theorem readily implies the following corollary.
Corollary 2. Let Ñ(c) be a (2m + 1)-dimensional locally conformal almost cosymplectic manifold, and let ϕ : with quarter-symmetric connection with semi-symmetric connection with semi-symmetric non-metric connection Pointwise hemislant n 2 cos 2 θ + (n − 1)a Pointwise CR Remark 3. The above result is obtained by using Remark 1 and the definition of semi-slant, hemi-slant, CR in Theorem 2.

Conclusions
In this paper, we have established geometric inequalities that provide valuable insights into the relationship between the squared mean curvature and the warping functions of a doubly warped product pointwise bi-slant submanifold.These findings have been achieved within the framework of locally conformal almost cosymplectic manifolds, which are equipped with a quarter-symmetric metric connection.Furthermore, the paper also investigates the cases of equality in these inequalities, shedding light on the specific conditions under which these geometric relationships hold true.This analysis enhances our understanding of the intricate interplay between curvature and warping functions in the context of bi-slant submanifolds.The findings presented here significantly contribute to the existing body of knowledge in the field of locally conformal almost cosymplectic manifolds.Further exploration and utilization of these results are encouraged to advance our understanding of this area of mathematics.
denote an isometric immersion of n-dimensional different submanifolds into Ñ(c) equipped with different connections.Then, the following statement holds true: denote an isometric immersion of n-dimensional different submanifolds N into Ñ(c) with different connections.Then, the following statement holds true: