Certain investigations of sequential warped product submanifolds on cosymplectic manifolds

In a special class of almost contact metric manifolds known as cosymplectic manifolds, the current study aims to establish the existence result and a few inequalities for sequential warped product submanifolds. These results and inequalities represent fruitful connections between the primary intrinsic and extrinsic invariants. Furthermore, findings related to Dirichlet energy have been addressed. Finally, some exceptional cases resulting in several inequities are examined.


Introduction
The Nash theorem was created to ensure that Riemannian manifolds are always perceived as Riemannian submanifolds [23].Gromov in [16], however, noted that this hope had not come true.The fundamental cause of this is that the extrinsic aspects of the submanifolds are not within the control of the known intrinsic facts.The establishment of meaningful connections between the primary intrinsic and extrinsic invariants of submanifolds is a fundamental concern in submanifold theory.There is growing interest in stabilizing this concern by developing several kinds of geometric equalities and inequalities that admit invariants.The research along this path gained momentum after Chen [7][8][9], employed the warped product technique as a method to develop inequalities involving invariants and deduced some significant information.After that, several differential geometers established geometric inequalities to analyze the relationships between extrinsic and intrinsic parameters not only for warped products but also for their generalizations, i.e., double and multiple warped products in different ambient spaces (see [10,12,13,24,32,33]). The sequential warped product (briefly, SWP), whose base remains a warped product manifold, is a new generalization of warped products (in short, WP) [5].Shenawy revealed SWP in [31].Thereafter, De, Shenawy, and Unal did a full investigation of SWP in [14].
With certain examples, Pahan and Pal [25], Karaca and Ozgür [18] extended the study of sequential warped space with (quasi-) Einstein's structure.Sahin [28] discovered the presence of SWP submanifolds in Kähler manifolds and presented geometric inequalities and equalities such that the associations between the intrinsic and extrinsic invariants are illustrated.In [19], the author recently generalized the Chen inequality involving invariants in nearly Kähler manifolds.Recently, Perktas-Blaga [27] continued the study on Sasakian manifolds and provided some nonexistence results.Motivated by these developments, in this article, the authors first present a numerical example in support of the existence of SWP of the form ( T × f ⊥ ) × h θ on cosymplectic manifolds: an odd-dimensional counterpart of a Kaehler manifold, and then generalized Chen's type inequality and equalities involved Dirichlet energy and curvature to analyze the geometric invariants using the SWP technique for such type of SWP on a cosymplectic manifold.Other types of SWP on a cosymplectic manifold may be thought of as open problems.A succinct summary of the article is provided below.We first recall some key concepts and definitions of an almost contact, cosymplectic, and their submanifolds in Sect. 2 and Sect.3.Then, in Sect. 4 we derive and review a few results related to SWP for future use.Finally, in Sect. 5 we establish some important results that extend inequalities for various warped products, inequalities involving Dirichlet energy and curvature in cosymplectic manifolds.We also look at the specific instances and corresponding equality.

Preliminaries
Let ¯ be a 2m + 1-dimensional smooth manifold and X( ¯ ) be the Lie algebra of smooth vector fields on ¯ 2m+1 .Then, (ϕ, ξ , η) is said to have an almost contact structure on ¯ 2m+1 [22], if there exist an endomorphism ϕ of type (1, 1), a smooth global vector field ξ and a 1-form η satisfying where I is the identity map.If an almost contact structure admits Riemann metric g such that then it is called an almost contact Riemann structure (ϕ, ξ , η, g) to ¯ 2m+1 .Furthermore, ¯ 2m+1 associated with the structure (ϕ, ξ , η, g) is called an almost contact metric manifold ¯ (ϕ, ξ , η, g).
From the above definition and (1), it is easy to obtain that Let ¯ (k) be a cosymplectic space form, then the Riemannian curvature on ¯ (k) for arbitrary U, V , W , X ∈ X( ¯ ) is given by

Geometry of submanifolds
Now, we review some important formulas and definitions for future usage concerning submanifolds of dimension d, immersed isometrically in ¯ 2m+1 .(For details refer to [6].)The formulas of Gauss-Weingarten are defined by the expressions: ∀U, V ∈ X( ): a space tangent to and N ∈ X( ⊥ ): a space normal to .The induced tangent and normal connections on X( ) and X( ⊥ ) are denoted by the symbols ∇ and ∇ ⊥ , respectively.Then, the shape operator and the second fundamental form (abbreviated SFF) are represented by the A N at N and σ , respectively, in such a way that If p is any point in and {x 1 , . . ., x d , x d+1 , . . ., x 2m+1 } is an orthonormal frame of the tangent space X p ( ¯ ) = X p ( ) ⊕ X p ( ⊥ ) such that {x 1 , . . ., x d } ∈ X p ( ) and {x d+1 , . . ., x 2m+1 } ∈ X p ( ⊥ ), then H(p) = 1 d traceσ , gives the mean curvature vector of and σ 2 is computed by By setting can be represented as If σ (U, V ) equals g(U, V )H (0), then is umbilical (geodesic).For H = 0, is minimal.Moreover, if for all U ∈ X( ), where tU (t N ) and nU (n N ) are tangential (normal) parts of ϕU (ϕN ), then employing (1), (2), and (11), we obtain In view of Definition 1 and ( 6) and ( 12), we have where (∇ U ϕ)V , (∇ U t)V and (∇ U n)V are defined by for all U, V ∈ X( ).Next, the Gauss-Codazzi equations are defined, respectively, by, for every U, V , W and X ∈ X( ), where R and R denote the Riemann curvature on and ¯ 2m+1 .Next, the gradient of the smooth function f on is given by for any U a ∈ X( ).
Here, we refresh certain key definitions and findings from [26] for further usage.
Definition 2 Let be a submanifold of ¯ 2m+1 .If at a point p in , any tangent vector field W /{0} not proportional to the characteristic vector field ξ ∈ X( ), the angle symbolized by θ , between ϕ(W ) and X p ( ) does not depend on p nor W , then is pointwise slant.Consequently, θ is identified as a function on , and thus known as the slant function.
Next, from the definition and [20,30], we write the iff condition for a submanifold of ¯ 2m+1 to be pointwise slant, given as follows for any vector field: Additionally, we have for W 1 , W 2 ∈ X( ) that g(tW 1 , tW 2 ) = cos 2 g(ϕW 1 , ϕW 2 ), ( 23) Definition 3 A submanifold is pointwise semislant (resp., pseudoslant), if there are distributions D T : an invariant and D θ : a pointwise slant on (resp., D ⊥ : an antiinvariant and D θ ) with slant function θ and characteristic vector field ξ such that i.The tangent space X( ) admits the orthogonal direct decomposition

Sequential warped product submanifolds
Shenawy in [31] and De et al. in [14] defined SWP as: consider three Riemannian manifolds B, F 1 , and Sahin recently analyzed the possibility of warped products of type ( T × f ⊥ ) × h θ for Kaehler manifolds and discovered some significant findings [28].Motivated by the work of Sahin, in this section we continue the study for cosymplectic manifolds: an odddimensional counterpart of a Kaehler manifold, and derive several geometric characterizations for such types of submanifolds in cosymplectic manifold.
For a vector field on a factor manifold and its lift to the sequential warped product manifold, we use the same notation.We start by keeping in mind the following statements about SWP manifolds for future use.
Proof Consider ξ ∈ X( ⊥ ), then by the use of Proposition 1, we attain In view of ( 4), (6), and ( 7), we have U(ln f ) = 0.This shows that f is constant on T .From the above discussion, we conclude that is a CR-slant warped product submanifold.On the other hand, ξ ∈ X( θ ), then by the use of Proposition 1, we achieve for every U ∈ X( T ) and V ∈ X( ⊥ ).In light of ( 4), ( 6), (7), and ( 25), we observe that U(ln h) = V (ln h) = 0.This shows that h is constant on both the factors T and ⊥ .This completes the proof.
Next, we first present a numerical example that shows the existence of SWP of the form ( T × f ⊥ ) × h θ and then continue the study by presenting several important results for such submanifolds.
Example 1 Consider a 13-dimensional Euclidean space E 13 with coordinates (x 1 , . . ., x 6 , y 1 , . . ., y 6 , t) and Euclidean metric g.The almost contact structure (ϕ, ξ , η) on E 13 is described as One can easily verify that the Euclidean space E 13 with almost contact structure is a cosymplectic manifold.Consider a subset ⊂ E 13 immersed as a submanifold by the following immersion The tangent subspace of at each point is spanned by the basis By straightforward computation, we observed that the distribution The induced metric on is given by This shows that is a SWP manifold of E 13 with warping functions Remark 2 From here onwards, we consider and study the SWP submanifolds of the type ( T × f ⊥ ) × h θ when the characteristic vector field ξ is tangent to T .
Proof Consider U ∈ X( T ) and V ∈ X( ⊥ ), then by consequence of the first part of Proposition 1, we have Hence, applying ( 4) and ( 6) in the above expression, we obtain the first part.By the use of the second part of Proposition 1, we derive . With the help of ( 4) and ( 6) in the last relation, we obtain the second part.
Here, we present some crucial findings for later use.
Moreover, employing (1) in Lemma 2, we obtain that Replacing W 1 by tW 1 in Lemma 2, then utilizing (22), we obtain Replacing W 1 by tW 1 in ( 28) and ( 29), and then employing (22) in the obtained expression, we arrive at Replacing W 2 by tW 2 in ( 29), (32), and (33) and using (23), yields Replacing U 1 by ϕU 1 in (34), we achieve that Proof As a result of ( 6) and (11), we obtain g(σ (U, V ), nW ) = g( ∇V U, ϕW )g(∇ V U, tW ).Now, employing the first part of Lemma 2 into the last expression, we achieve the first part.Likewise, we can prove the second part.By the consequence of (3), (6), and (11), we obtain With the help of Definition 1 and the third part of Proposition 2, we have Applying (11), (22), and (23), we achieve the third part.
Proof By use of ( 3) and ( 6), we obtain g(σ (W , tW ), ϕV ) = -g(ϕ ∇W tW , V ).Now, utilizing (11) and Definition 1 in the last expression, we observe that By virtue of ( 8) and ( 22), we concede that Using the second part of Proposition 1 and the property of the covariant derivative, we achieve (36).
Proof By the use of ( 16) and ( 6), we have ϕ∇ V U -∇ V ϕU = σ (ϕU, V )ϕσ (U, V ).Employing the fact that T is an invariant submanifold, then by the application of Proposition 1(i) the above expression reduces to the following form: Now, by taking an inner product with ϕσ (U, V ) into (39), we achieve the first part.Reusing ( 16) and ( 6), we have ϕ∇ W U -∇ W ϕU = σ (ϕU, W )ϕσ (W , V ).By the application of Proposition 1(i) in the above expression, we obtain Now, by taking the inner product with ϕσ (U, W ) into (39), we have In view of (35) and (41), we obtain the second part.

Then, we deduce dim(D
Here, we present our first main result that represents a relationship between the warping function and the second fundamental form and acts as a generalization for earlier existing literature in this regard. holds, where σ symbolizes the SFF, ∇ T (f ): gradient(f ) on T , ∇ T (h) and ∇ ⊥ (h): gradients(h) on T and ⊥ .
Proof By the use of ( 10), we have Since X( ⊥ ) = X(nD ⊥ ) ⊕ X(nD θ ) ⊕ ν, then the preceding expression is simplified into its subsequent form as The nD ⊥ and nD θ components are found in the first and second terms, while νcomponents are found in the final term in the above expression.Further, we only use the frame fields described above to calculate the components in the first and second terms as follows: By using the above-mentioned frame field, we concede that Now, employing the first and second parts of Lemma 1 and the first and second parts of Lemma 3 in the above expression With the help of ( 4) and ( 6), we have σ (ξ , ξ ) = 0.In order to reduce the complexity of the computation, we extract the final three terms from the equation above, and by using the orthonormal frame previously defined, we reach Next, utilizing (31)-( 35) and the second parts of Lemmas 1 and 2, we have Leaving the last term and by adding and subtracting the same quantity in (48), and by the application of Proposition 3, we have Using (21) in the above expression, we derive (42).Hence, the theorem is proved.

Particular cases
If in ineq.(42) of Theorem 2: i. r = 0 and the norm of ∇ T (h) h and ∇ ⊥ (h) h vanishes, then SWP becomes a CR-warped product [34], and ii. q = 0 and the norm of It is to be noted that the same result was obtained at [26] for Sasakian manifolds, but now for cosymplectic ones.iii.p = 0 and the norm of ∇ T (f ) f and ∇ ⊥ (h) h vanishes, then SWP becomes pointwise pseudoslant WP ⊥ × h θ [2, Th. 4.1] and iv. ∇ ⊥ (h) h vanishes, then SWP becomes a biwarped product T × f ⊥ × h θ [11] such that then, the claims that follow hold: The present theorem can be proved using the same steps as Theorem 4.2 of [19].
Proof Suppose that σ satisfies (50), then by (44), we have Hence, straightforwardly from (51), Lemma 1, Lemma 2, and Proposition 2, we obtain (1).Also from (46), we obtain that By the use of (46), we have From ( 51) and (53), we have By the consequence (54), we accomplished the proof of the second part.By virtue of (51) and (52), we have Since ⊥ is a totally umbilical submanifold in T × f ⊥ using this fact in (55), we conclude that ⊥ is totally umbilical in M. Similarly, we prove the fourth part.Let σ ⊥ be the second fundamental form on ⊥ , then by the utilization of Proposition 1, we have . Now, assume σ and σ θ are the second fundamental tensors of M and θ , respectively, then we have . By the consequence of (49) and Proposition 2, we obtain , where H θ is the mean curvature of θ .Using (1), we obtain . Now, by the definition of gradient, we achieve the last part.This completes the proof.
Our next result presents one of the ways to analyze Dirichlet problems.Several authors in different senses have analyzed the existence of solutions for Dirichlet problems [3,11].Here, we investigate the relation involving Dirichlet energy of functions and the second fundamental form for SWP.Jackson et al. in [17] stated this as: The Dirichlet energy of smooth function ψ and compact submanifold over its volume element dV is defined by As a result, inspired by the publications mentioned above, we derive an important finding: By using the property of integration in (58), we obtain By utilization of ∇ = ∇ T + ∇ ⊥ + ∇ θ and (56) in the above equation, we obtain (57).

Particular cases
In inequality (57) of Theorem 4 if: i. r = 0 and the norm of ∇ T (h) h , ∇ ⊥ (h) h vanishes, then SWP is a CR-warped product with 4qE(ln f ) ≤ σ 2 dV .
ii. q = 0 and the norm of ∇ T (f ) f vanishes, then SWP is WP T × h θ [4] with iii. p = 0 and the norm of Remark 3 The Dirichlet energy in the special cases (1), (2), and (4) can be easily reduced to Corollary 6.1, Corollary 6.2, and Theorem 6.1 of [11] after certain computations.
Motivated by Sahin [28,29], we give the Lawson-Simons-type inequality [21] (possesses important applications in the theory of integral currents [15] for SWP in a cosymplectic manifold with constant holomorphic sectional curvature k (briefly: Cosymplectic space form denoted by ( ¯ 2m+1 , k)).We also extract exceptional cases for the same.
where ⊥ (h) denotes the Laplacian of h on ⊥ .Moreover, if the equality holds then = By virtue of ( 19) and (61), we have Further, utilizing Proposition 1 in the above expression, we compute By the definition of a Laplacian, the above expression reduces to the given form From ( 64), (66), and (69), we have (60).If equality holds then we have The above equation shows that the functions f and h are constant on T .Therefore, reduces to T × ⊥ × h , θ is a single warped product.By the direct consequence of (66) and (69), we conclude that σ (D T , D ⊥ ) ⊥ ν and σ (D T , D θ ) ⊥ ν.This completes the proof.