Geometry of Bi-Warped Product Submanifolds of Nearly Trans-Sasakian Manifolds

: In the present work, we consider two types of bi-warped product submanifolds, M = M T × f 1 M ⊥ × f 2 M φ and M = M φ × f 1 M T × f 2 M ⊥ , in nearly trans-Sasakian manifolds and construct inequalities for the squared norm of the second fundamental form. The main results here are a generalization of several previous results. We also design some applications, in view of mathematical physics, and obtain relations between the second fundamental form and the Dirichlet energy. The relationship between the eigenvalues and the second fundamental form is also established.


Introduction
Let M = M 1 × M 2 × M 3 × · · · × M k be the Cartesian product of Riemannian manifolds M 1 , M 2 , · · · , M k and let π i : M −→ M k denote the canonical projection maps for i = 1, · · · , k. If the positive-valued functions f 1 , · · · , f k are defined such that f 1 , · · · , f k : M 1 −→ (0, ∞), then the Riemannian metric g is defined as g(X, Y) = g(π 1 * X, π 1 * Y) + k ∑ i=1 f i • π g(π i * X, π i * Y), where * is the symbol for tangent map, for any X, Y tangent to M, then M is called a multiple warped product manifold [1,2]. If we choose two fibers of a multiple warped product M 1 × f 1 M 2 × · · · × f k M k , such that M = M 1 × f 1 M 2 × f 2 M 3 , then M is defined as a bi-warped product submanifold, which satisfies the following: where U 1 ∈ Γ(TM 1 ), Z ∈ Γ(T(M 2 × M 3 )) and Z i tangent to M i , for each i = 2, 3. Moreover, ∇ is the Levi-Civita connection and for more details, see [3][4][5]. An odd-dimensional analog of the nearly Kähler metric is the nearly Sasakian metric. The nearly Kähler cone over a nearly Sasakian Einstein manifold has applications in physics. The Sasakian geometry has been extensively studied, due to its recently perceived relevance in string theory. Sasakian Einstein metrics have received a lot of attention in physics-for example, related to p-brane solutions in superstring theory and the Maldacena conjecture [6]. On the other hand, the vanishing of Dirichlet energies is equivalent to the Dirichlet condition with the unique solution of Poisson's equation ∇ 2 f = −4 f ρ. This implies Neumann or Dirichlet boundary conditions, as classified by the electrostatic problem (see, e.g., [7]). In the present paper, we consider the bi-warped product submanifolds in a nearly trans-Sasakian manifold, inspired by the publication of Taştan's seminal work [5], and obtained some inequalities for the Dirichlet energy and the second fundamental form. The study of bi-warped product submanifolds with two distinct fibers has been a topic of great interest; see, e.g., Naghi et al. [4], Ali et al. [8], Siraj et al. [9][10][11], and Awatif et al. [12]. It has been noted that the class of bi-warped product submanifolds is a generalization of several classes, such as CR-warped products, warped product semi-slant submanifolds, and warped product pseudo-slant submanifolds. On the other hand, as a generalization of nearly cosymplectic, nearly Sasakian [13], nearly Kenmotsu [8,14], nearly α-Sasakian, and nearly β-Kenmotsu manifolds, nearly trans-Sasakian manifolds have been studied on a large scale; see [15][16][17][18][19]. Therefore, our objective was to remove the gap in the nearly trans-Sasakian manifold literature, as they are an interesting structure of the almost contact manifolds that have generalized many others structures. The main goal of this paper was to discuss the geometry of bi-warped product submanifolds of the types M T × f 1 M ⊥ × f 2 M φ and M = M φ × f 1 M T × f 2 M ⊥ in a nearly trans-Sasakian manifold. Some inequalities for the length of the second fundamental form are obtained, including the length of warping functions and slant immersions. Various inequalities, which have been derived in [13,14,[18][19][20][21][22][23][24][25][26][27], can be recovered from our inequalities under some conditions. Therefore, our results may find applications in mathematical physics.

Preliminaries
An odd-dimensional C ∞ -manifold ( M, g) associated with an almost contact structure (ψ, ζ, η) is referred to as an almost contact metric manifold if there exist a (1, 1) tensor field ψ, a vector field ζ (called a characteristic or Reeb vector field), and a 1-form η satisfying the following conditions: ∀ U 1 , W 2 ∈ Γ(T M). The above structure (ψ, η, ζ) can be reduced to a nearly trans-Sasakian manifold (cf. [18,19]) if the following holds: for any U 1 , V 1 ∈ Γ(T M), where is the Riemannian connection associated with the metric g on M. According to the structure, we have the following classifications: (i) A nearly trans-Sasakian M is a nearly cosymplectic if α = 0 and, β = 0 in (4).
The Gauss and Weingarten formulas, which specify the relation between Levi-Civita connections ∇ on a submanifold M and ∇ on an ambient manifold M, are given by (for more detail, see [28]): for every U 1 , W 2 ∈ Γ(TM) and ξ ∈ Γ(T ⊥ M). In addition, B and A ξ are the second fundamental form and shape operator, respectively, having the relation g(B(U 1 , W 2 ), ξ) = g(A ξ U 1 , W 2 ). If we replace U 1 = ξ, V 1 = ξ in (4), we find that ( ∇ ξ ψ)ξ = 0, which implies that ψ ∇ ξ ξ = 0. Applying ψ and using (4), we get ∇ ξ ξ = 0. From the Gauss formula, we get ∇ ξ ξ = 0 and h(ξ, ξ) = 0. For more classification, see [18,19]. We also have: in which FU 1 and TU 1 are normal and tangential elements of ψU 1 , respectively. If M is invariant and/or anti-invariant, then FU 1 and/or TU 1 are zero, respectively. Similarly, we have where tξ (respectively f ξ) are tangential (respectively normal) components of ψξ. The covariant derivative of the endomorphism ψ is defined by There is a motivating class of submanifolds, presented as slant submanifolds. For any non-zero vector U 1 tangential to M at point p, such that U 1 is not proportional to ζ p , 0 ≤ φ(U 1 ) ≤ π/2 refers to the angle between ψU 1 and T p M, which is named the Wirtinger angle. If φ(U 1 ) is constant for any U 1 ∈ T p M− < ζ > at point p ∈ M, then M is referred to as the slant submanifold [29] and φ is the slant angle of M. We consider the following necessary and sufficient for a submanifold M to be a slant submanifold [29,30]: for 0 ≤ φ ≤ π 2 and T is an endomorphism defined in (8). The following result is from Equation (11): ∀ U 1 , W 2 ∈ Γ(TM). The following are the definitions of semi-slant and psuedo-slant submanifolds. The mean curvature vector, H, for an orthonormal frame {e 1 , e 2 , · · · , e n } of the tangent space TM on M n is defined by where n = dim M. In addition, we set g(B(e α , e β ), B(e α , e β )), (15) where ||B|| 2 is the length of the second fundamental form.
The next practical consequence will be used later in this paper. 3 be a bi-warped product submanifold of a nearly trans-Sasakian manifold M. Then, M is only a single warped product if the structure vector field ζ is tangent to the fiber; that is, either ζ ∈ Γ(D 2 ) or ζ ∈ Γ(D 3 ).
By taking the inner product with ξ and using the fact that ξ ∈ Γ(D 2 ), we have where f 1 is a constant. Similarly, we can easily obtain that U(ln f 2 ) = 0, which implies that f 2 is constant. 3 be a bi-warped product submanifold of a nearly trans-Sasakian manifold M. Then, for each structure vector field ζ is tangent to M 1 .

Proof.
For any X 0 ∈ Γ(D 1 ), we expand the following: Using (1) and (8), we obtain Taking the inner product with X 0 , we obtain Similarly, we can find that Therefore, the statement is proved.
Remark 1. It can be noticed that, if the structure vector field ζ is tangent to any fiber, then the warped product will be trivial due to Proposition 1. On the other hand, Proposition 2 assures that the warped product is always non-trivial if the structure vector field ζ is tangent to any base manifold of a nearly trans-Sasakian manifold.
Proof. For all X 0 ∈ Γ(D), X 2 ∈ Γ(D ⊥ ) and X 4 ∈ Γ(D φ ), from which the following can be derived: By the use of (4), (10), and the normality of vector fields, we have Therefore, using (8)-(10) results in and we reached (i) , as desired. Additionally, we have By the use of (4) and (8), we arrive at Therefore, it follows from (6), (7), and (1) that Once more, by the use of (10)-(11), we construct From (36) and (37), we obtain the second part of (i) . For the third part, we calculate the following: By the use of (6)- (7) and (8), we have Replacing X 4 with X 5 in (38), we have Equations (38) and (39) give the second part. The rest of the terms are derived by interchanging X 0 with ψX 0 , as well as X 4 and X 5 with TX 4 and TX 5 , respectively. Thus, the proof is completed.
in a nearly trans-Sasakian manifoldM refers to a D ⊕ D ⊥ -mixed totally geodesic (respectively, D ⊕ D φ −mixed totally geodesic).

Remark 3. It easily proved that a proper bi-warped product submanifold
of a nearly trans-Sasakian manifoldM is trivial, by using the D ⊕ D φ -mixed totally geodesic and D ⊕ D ⊥ -mixed totally geodesic in (19) and (26). Now, we are in the position to give the proof of our main result. More precisely, we give the following inequality theorem for bi-warped product submanifolds of the type be a bi-warped product submanifold of a nearly trans-Sasakian manifoldM. Then, the following inequality is satisfied for the second fundamental form B: where is a n-dimensional proper bi-warped product submanifold of the nearly trans-Sasakian manifoldM 2m+1 . In addition, let the local orthonormal vector fields v 1 , . . . , v n of TM be as follows: Therefore, dim M T = n 1 = 2t + 1, dim M ⊥ = n 2 = l, and dim M φ = n 3 = 2k. Furthermore, the orthonormal frame fields v 1 , . . . , v 2m+1−n−l−2k of the normal sub-bundle T ⊥ M are as follows: According the definition B, we have The above equation can be broken into the components of submanifolds of M T , M φ , and M ⊥ as follows: where clearly dim µ = 2m + 1 − n − l − 2k. Expanding the above equation, according to the orthonormal bases of D φ , D, and D ⊥ (except for the last term), we arrive at Utilizing Equations (17), (18), and (26) in the preceding equation, we obtain Leaving out all terms except for the last two relations in the above equation, we find From the frame fields of tangent and normal sub-bundles of M, after ignoring the µ-components part in (41), we derive Utilizing Equations (19), (20), and (21) in the first terms, we substitute (26)-(35) to obtain From the orthonormal frame η(v a ) = 0 for 1 ≤ i ≤ 2t, and with some rearrangement in the remaining terms, we arrive at (v a ln f 1 ) 2 + 4kα 2 .
Using trigonometric identities and adding and subtracting some terms, we obtain For a nearly trans-Sasakian manifold, two conditions are satisfied: ζ ln f 1 = β and ζ ln f 2 = β for the structure vector field ζ tangent to the base M T . The following inequality is obtained by substituting into the proceeding equation: 10 csc 2 φ − 1 β 2 + 4kα 2 , from which we reach the final result, (40), which we wanted to prove.
If the inequality (40) becomes an equality, the missing third part in (41) gives: Using terms (i) and (ii), which were not considered in (17), and (42), we can derive B(D, D)⊥ ψD ⊥ and B(D, D)⊥FD φ . It is obtained that B(D, D) = 0. The missing first and fourth terms in (42) lead to the following: It can easily be seen that B(D ⊥ , D ⊥ ) = 0, by using the missing terms in (43) and (44). In addition, the second and third terms that were left in (42) give B(D φ , D φ )⊥ψD ⊥ and B(D φ , D φ )⊥FD φ . This leads to B(D φ , D φ ) = 0. In addition, term numbers five and six that were missed in (42)

Some Geometric Consequences
If we consider dim M φ = 2k = 0, then Theorem 1 gives the following: Theorem 2. (Theorem 4.1 of [18]) Let M = M T × f M ⊥ be a CR-warped product submanifold in a nearly trans-Sasakian manifold. Then, we have: If M ⊥ vanishes, then Theorem 1 implies the following: Theorem 3. (Theorem 4.1 of [19]) Let M = M T × f M φ be a warped product semi-slant submanifold of a nearly trans-SasakianM, then inequality (40) implies the following inequality: Therefore, Theorem 1 is an extension of Theorem 4.1 of [19]. Similarly, for α = 0, we have: be a bi-warped product submanifold of a nearly β-Kenmotsu manifoldM. Then, the second fundamental form B satisfies the following inequality: Inserting α = 0 and β = 1 into Theorem 1, we have Theorem 5. (Theorem 1 of [8]) Let M = M T × f 1 M ⊥ × f 2 M φ be a bi-warped product submanifold of a nearly Kenmotsu manifoldM. Then, the second fundamental form B satisfies the following inequality: If we put α = 0, β = 0 into Theorem 1, the following is obtained for the bi-warped product submanifold of a nearly cosymplectic manifold: be a bi-warped product submanifold of a nearly cosymplectic manifoldM. Then, the second fundamental form B satisfies the following inequality: Again, we get the following by using α = 0, β = 0, and dim M φ = 0. Theorem 7. (Theorem 3.2 of [31]) Let M = M T × f M ⊥ be a CR-warped product submanifold of a nearly cosymplectic manifold. Then, we have: Inserting α = 0, β = 0 and dim M ⊥ = 0 into Theorem 1, we obtain the following: By inserting β = 0 and α = 1, β = 0 into Theorem 1, the following results for the bi-warped product submanifolds of a nearly α-Sasakian manifold and a nearly Sasakian manifold, respectively, can be obtained: be a bi-warped product submanifold of a nearly α-Sasakian manifoldM. Then, the second fundamental form B satisfies the following inequality: where n 2 = dim M ⊥ , n 3 = 1 2 dim M φ . If the above inequality (40) becomes an equality, then M ⊥ , M φ are totally umbilical and M T is geodesic totally inM. The D ⊕ D ⊥ -mixed totally geodesic and D ⊕ D φ -mixed totally geodesic do not exist in M ofM.
be a bi-warped product submanifold of a nearly Sasakian manifoldM. Then, the second fundamental form B satisfies the following inequality: If inequality (40) becomes an equality, then M ⊥ , M φ are totally umbilical and M T is geodesic totally inM. The D ⊕ D ⊥ -mixed totally geodesic and D ⊕ D φ -mixed totally geodesic do not exist in M ofM.

Inequality for Bi-Warped Product Submanifold of the Type M
In this section, we consider the bi-warped product submanifold of type M = M φ × f 1 M T × f 2 M ⊥ in a nearly trans-Sasakian manifoldM, with respect to the tangent spaces of M T , M ⊥ and M φ , which are integral manifolds of D, D ⊥ , and D φ , respectively.
Thus, the required results are obtained.
In this direction, we provide a relationship between the squared norm of the second fundamental form and the warping function for the bi-warped product. Before giving the next relationship, we define an orthonormal frame. Taking ζ tangent to the base manifold M φ of an n-dimensional bi-warped product submanifold M = M φ × f 1 M T × f 2 M ⊥ in a (2n + 1)-dimensional nearly trans-Sasakian manifold M, we consider the dimensions dim(M T ) = n 1 , dim(M ⊥ ) = n 2 , and dim(M φ ) = n 3 . We provide proof of the main theorem as follows. For the second type of bi-warped product submanifold, M φ × f 1 M T × f 2 M ⊥ , we prove the following result: is a bi-warped product submanifold of a nearly trans-Sasakian manifold M. If D ⊥ − D φ is mixed totally geodesic and ζ is tangent to M φ , then the length of the second fundamental form B is defined as where n 1 = dim(M T ) and n 2 = dim(M ⊥ ). The gradients ∇(ln f 1 ) and ∇(ln f 2 ) of ln f 1 and ln f 2 are along M T and M ⊥ , respectively. If the inequality (56) becomes an equality, then M T and M ⊥ are totally umbilical submanifolds, and M φ is a totally geodesic submanifold in M. Furthermore, M is a D φ -totally geodesic submanifold of M.
Using (11) and the fact that ξ(ln f a ) = β, i = 1, 2 from Proposition 2, we finally obtain (60)-(67) show that M is a D φ -totally geodesic submanifold inM. This completes the proof of the theorem.

Remark 5.
It can be noted that only on a nearly Kenmotsu manifold does the warped product semi-slant submanifold of the type M = M φ × f M T exist; in other structures, it becomes a trivial case (see [22]).

Some Applications Related to Mathematical Physics
In this section, we investigate the Dirichlet energy, which satisfies the following for a compact submanifold M and differentiable function θ : M −→ R: where dV is a volume element. Considering this, we give the following Theorem by combining (40) and (68), where M T is compact without boundary.