The First Eigenvalue Estimates of Warped Product Pseudo-Slant Submanifolds

The aim of this paper is to construct a sharp general inequality for warped product pseudo-slant submanifold of the type M = M⊥ × f Mθ , in a nearly cosymplectic manifold, in terms of the warping function and the symmetric bilinear form h which is known as the second fundamental form. The equality cases are also discussed. As its application, we establish a bound for the first non-zero eigenvalue of the warping function whose base manifold is compact.


Introduction
The clue of warped product manifolds is related to the generalization of Riemannian products. The role of warped product submanifolds in studying Riemannian geometry was studied actively from the pioneering work of Chen [1]. The work of Chen is about the characterization of CR-warped products in Kaehler manifolds, and derives the inequality for the second fundamental form. In fact, distinct classes of warped product submanifolds of the different kinds of structures were studied by several geometers (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14]). Recently, Ali et al. [15], established general inequalities for warped product pseudo-slants isometrically immersed in nearly Kenmotsu manifolds for mixed, totally geodesic submanifolds. Moreover, some results on the existence of the warped product pseudo-slant submanifolds in a nearly cosymplectic manifold in terms of endomorphisms were proved by Uddin in [14]. We have noticed that the warped product pseudo-slant submanifolds of the form M ⊥ × f M θ , and M θ × f M ⊥ , is a CR-warped product submanifold with slant angle θ = 0. For contradict that warped product pseudo-slant submanifolds always not generalize CR-warped product submanifold which was show in [13]. However, some interesting inequalities have been obtained by many geometers (see [4,10,12,[16][17][18][19][20]) for distinct warped product submanifolds in the different types of ambient manifolds. In [5], Al-Solamy derived the inequality for mixed, totally geodesic warped product pseudo-slant submanifolds of type M = M θ × f M ⊥ , in a nearly cosymplectic manifold. On other hand, the warped product pseudo-slant submanifold in a nearly cosymplectic manifold of the type M = M ⊥ × f M θ was studied by Uddin et al. [14]. We consider the non-trivial warped product pseudo-slant submanifold M = M ⊥ × f M θ , such that M θ and M ⊥ are proper-slant and anti-invariant submanifolds, respectively. In this case, considering that M is not mixed and totally geodesic, we announce our first result as follows.
Further, ∇ ln f is the gradient of ln f over M ⊥ ; (ii) If the equality sign holds in (1), then M ⊥ is a totally geodesic submanifold with satisfying conditions: As an application of Theorem 1, the next result comes from the idea of the eigenvalue comparison theorem of Cheng [21], which has proved that M is complete and isometric to the standard unit sphere, provided that Ric(M) ≥ 1 and d(M) = π by using the first non-zero eigenvalue of the Laplacian operator. After that, Mihai [12] obtained the first eigenvalue for CR-warped products into the Sasakian space form. Therefore, we used the method of the maximum principle for the first non-zero eigenvalue λ 1 which is defined in [22], and made use of Theorem 1 to deduce the following.

Theorem 2.
Assuming M is a nearly cosymplectic manifold, let M = M ⊥ × f M θ be a warped product pseudo-slant submanifold of M, such that M ⊥ is a compact submanifold of M. Then, we have: where λ 1 is a first non-zero eigenvalue of the warping function ln f . The dimensions are defined in Theorem 1 (ii).

Preliminaries
An odd (2n + 1)-dimensional Riemannian manifold ( M, g) is called a nearly cosymplectic manifold, if it consists of an endomorphism ϕ of its tangent bundle T M, a structure vector field ξ, and a 1-form η, which satisfies the following: for any vector field U, V on M such that ∇ denotes the Riemannian connection with respect to the Riemannian metric g (see [14]). Furthermore, the fundamental 2-form denoted by Φ, i.e., Let the Lie algebras of vector fields tangent and normal to a submanifold M be denoted as Γ(TM) and Γ(T ⊥ M). Moreover, the induced connection on T ⊥ M is denoted ∇ ⊥ , and ∇ is the Levi-Civita connection of M. Thus, the Gauss and Weingarten formulas are defined as: for each U, V ∈ Γ(TM) and N ∈ Γ(T ⊥ M), where h and A N are the second fundamental form and the shape operator, respectively, for the submanifold of M into M, which are related as: For any U ∈ Γ(TM), we have: where PU and FU are the tangential and normal components of ϕU, respectively. Similarly, for any N ∈ Γ(T ⊥ M), we have: where tN (resp. f N) are tangential (resp. normal) components of ϕN.
The submanifold M is defined as a slant submanifold if, for each non-zero vector X tangent to M, the angle θ(X) between ϕX and T p M− < ξ > is constant. We shall use Theorem 2.1 [5] in our subsequent proof as a characterization of the slant submanifold.
In this sequel, by using the slant distribution given in [23,24], we shall give the definition of pseudo-slant submanifolds of almost-contact manifolds. For deeper classifications of pseudo-slant submanifolds, we refer to [7]. First, we give the definition of a pseudo-slant submanifold: Definition 1. If the tangent bundle TM of submanifold M in an almost-contact manifold is decomposed as TM = D θ ⊕ D ⊥ ⊕ < ξ >, where D θ and D ⊥ are slant and anti-invariant distributions such that PD θ ⊆ D θ and ϕD ⊥ ⊆ T ⊥ M, respectively-in this case, where < ξ > is a one-dimensional distribution spanned by the structure field ξ, then M is called pseudo-slant submanifold.
If µ is an invariant subspace of T ⊥ M, then for the pseudo-slant case, the normal bundle T ⊥ M can be decomposed as:

Warped Product Submanifolds of the Form M ⊥ × f M θ
Let (M 1 , g 1 ) and (M 2 , g 2 ) be two Riemannian manifolds. Then, the warped product of M 1 and M 2 is the Riemannian product M 1 × f M 2 = (M 1 × f M 2 , g) with the metric g = g 1 + f 2 g 2 and f being a positive differential function defined on M 1 . Then, from Lemma 7.3 [25], we have: for any vector fields U and W tangent to M 1 and M 2 , respectively, where ∇ denote the Levi-Civitas connection on M (see [25]). If the warping function f is constant, then a warped product manifold M = M 1 × f M 2 is called a simply Riemannian product or a trivial warped product manifold. Now, we consider the warped product pseudo-slant submanifold of the form M = M ⊥ × f M θ , such that ξ ∈ Γ(TM ⊥ ) and other cases is leaved because they were studied by Al-Solamy [5], and then we obtain some lemmas for use in our main result. Thus, from (11) and from Theorem 2.1 in [5], we obtain: Then, using (13), we can derive: Using (7) and the (11), the above equation then becomes: By using (13), (9), and from (Equation (2.12) in [5]), we derive the following: From (9), (11), and Theorem 2.1 [5] for the slant submanifold, it is not hard to come by the following: for any U ∈ Γ(TM θ ) and W ∈ Γ(TM ⊥ ). (8) and (11), we have:
We also derived from the covariant derivative of ϕ and (8), that is: Then, from (7) and (13), we obtained: which implies: That is, the first part of the proof of lemma ends up and part (ii) can be constructed by rearranging X by PX in (16). for any U ∈ Γ(TM θ ) and W ∈ Γ(TM ⊥ ) Proof. Assuming U ∈ Γ(TM θ ) and W ∈ Γ(TM ⊥ ), we have: From the covariant derivative of the tensor field ϕ, it follows that: Taking into account (7) and (8), we have: Thus, from (13) and (9), it can easily be found that: g(h(U, U), ϕW) = (W ln f )g(U, PU) + g(A FU U, W).
Since U and PU are orthogonal and (10), we arrive at: which is the first part of lemma. Then, by switching X by PX in (17), we were able to find the proof of the second part of the lemma.

Main Proof of Inequality for Warped Product of the form M ⊥ × f M θ
In this section, we establish a geometric inequality for warped product pseudo-slant submanifolds in terms of the symmetric bilinear form and warping functions with included immersion.
Considering only the first, second, and fourth terms, they then imply that: Then, using the components of the adapted frame for FD θ , we derive: