Improved Chen’s Inequalities for Submanifolds of Generalized Sasakian-Space-Forms

: In this article, we derive Chen’s inequalities involving Chen’s δ -invariant δ M , Riemannian invariant δ ( m 1 , . . . , m k ) , Ricci curvature, Riemannian invariant Θ k ( 2 ≤ k ≤ m ) , the scalar curvature and the squared of the mean curvature for submanifolds of generalized Sasakian-space-forms endowed with a quarter-symmetric connection. As an application of the obtain inequality, we ﬁrst derived the Chen inequality for the bi-slant submanifold of generalized Sasakian-space-forms


Introduction
In submanifold theory, obtaining the relationship between an intrinsic invariant and an extrinsic invariant has been the primary goal of many geometers in recent decades.Chen invariants were introduced by B.Y. Chen [1] to tackle the question raised by Chen concerning the existence of minimal immersions into a Euclidean space of arbitrary dimension [2].Chen's δ-invariant δ M of a Riemannian manifold M introduced by Chen is where τ is the scalar curvature of M.
In [1], Chen obtained an inequality for a Riemannian submanifold M m of a real space form M with constant sectional curvature c as where H is the mean curvature of the submanifold M m .Equation ( 2) is known as the first Chen inequality.
Qu and Wang [15] introduced the notion of a special type of quarter-symmetric connection as a generalization of a semi-symmetric metric connection [16] and a semi-symmetric non-metric connection [17].They studied the Einstein warped product and multiple warped products with a quarter-symmetric connection [15].In [18], the authors obtained Chen's inequalities for submanifolds of real space forms endowed with a quarter-symmetric connection.Mihai and Özg ür [19] obtained the Chen inequalities for submanifolds of complex space forms and Sasakian-space-forms with a semi-symmetric metric connection.Wang [20] obtained Chen inequalities for submanifolds of complex space forms and Sasakian-space-forms with quarter-symmetric connections which improved the results of Mihai and Özg ür [19].Sular [21] obtained Chen inequalities for submanifolds of generalized space forms with a semi-symmetric metric connection.Al-Khaldi et al. [22] obtained the Chen-Ricci inequalities Lagrangian submanifold in generalized complex space form and a Legendrian submanifold in a generalized Sasakian-space-form endowed with the quarter-symmetric connection.
As a continuation of their studies, we obtained Chen inequalities for submanifolds of generalized Sasakian-space-form admitting a quarter-symmetric connection.The significance of this study is that it generalizes a large number of previously obtained results, some of which are [20,21].The paper is organized as follows.In Section 2, we recall the properties of the quarter-symmetric connection.In Section 3, we establish the B.Y. Chen inequalities for submanifolds of a generalized Sasakian-space-form endowed with a quarter-symmetric connection.First, we prove the following inequality and also look at its equality case.
Theorem 1.Let M m , m ≥ 3 be an m-dimensional submanifold of a (2n + 1)-dimensional generalized Sasakian-space-form M( f 1 , f 2 , f 3 ) endowed with a quarter-symmetric connection ∇, then where Π is a two-plane section T x M, x ∈ M.
Next, we obtain bounds for the Riemannian invariant δ(m 1 , . . ., m k ) and a Ricci curvature in terms of the scalar curvature of the r-plane section L, squared mean curvature and some special functions.Among others, we obtain the inequality involving the Riemannian invariant Θ k , 2 ≤ k ≤ m, as follows: Using Theorem 1 in Section 4, we derive Chen inequalities for the bi-slant submanifold of generalized Sasakian-space-forms.

Preliminaries
Suppose that M m+p is an (m + p)-dimensional Riemannian manifold with Riemannian metric g.A linear connection ∇ is known as a quarter-symmetric connection if its torsion tensor T is presented by where Λ is a 1-form, P is a vector field given by Λ(X 1 ) = g(X 1 , P), and ϕ is (1, 1)-tensor.
In [15], the authors introduced a special type of quarter-symmetric connection defined as: where ∇ denote the Levi-Civita connection.It is easy to see that the quarter-symmetric connection ∇ includes the semi-symmetric metric connection (ψ 1 = ψ 2 = 1) and the semi-symmetric non-metric connection ( Similarly, the curvature tensor R of ∇ can be defined as the same.Let M m be an m-dimensional submanifold of an (m + p)-dimensional Riemannian manifold M m+p endowed with the quarter-symmetric connection ∇ and the Levi-Civita connection ∇.Let ∇ and ∇ denote the induced quarter-symmetric connection and the induced Levi-Civita connection on the submanifold M. The Gauss formula with respect to ∇ and ∇ can be presented as where h and h are the second fundamental forms associated with the quarter-symmetric connection ∇ and the Levi-Civita connection ∇, respectively, and are related as follows: where P ⊥ is the normal component of the vector field P on M. If P T represents that tangent component of the vector field P on M, then P = P T + P ⊥ .The curvature tensor R with respect to the quarter-symmetric connection ∇ on M m+p can be expressed as [15]: where β 1 and β 2 are symmetric (0, 2)-tensor fields defined as and Moreover, we assume that tr(β 1 ) = λ and tr(β 2 ) = µ.
Suppose that R and R are the curvature tensors of ∇ and ∇, respectively.Then the Gauss equation with respect to the quarter-symmetric connection is as follows [15]: Let {e 1 , . . ., e m } and {e m+1 , . . ., e m+p } be an orthonormal frame of T x M and T ⊥ x M at the point x ∈ M. Then the mean curvature vector of M associated with ∇ is H = 1 m ∑ m i=1 h(e i , e j ).Similarly, the mean curvature vector of M associated to ∇ is H = 1 m ∑ m i=1 h(e i , e j ).In addition, the squared length of h is h 2 = ∑ m i,j=1 g(h(e i , e j ), h(e i , e j ).Now, we recall some of the Riemannian invariants introduced by Chen [23] in a Riemannian manifold.Let L be an r-dimensional subspace of T x M, x ∈ M, r ≥ 2 and {e 1 , . . ., e r } an orthonormal basis of L. The scalar curvature τ of the r-plane section L is given by where K ij is the sectional curvature of the plane section spanned by e i and e j at x ∈ M. Suppose that Π ⊂ T x M is a two-plane section and K(Π) is the sectional curvature of M for a plane section Π in T x M, x ∈ M. Then The scalar curvature τ(x) of M at the point x is presented by where {e 1 , . . ., e m } is an orthonormal basis for T x M.

B. Y. Chen Inequalities
First, we recall the well-known lemma obtained by Chen [1], which is as follows: Now, let M be a (2n + 1)-dimensional almost contact metric manifold with the structure (ϕ, η, g, ξ) where ϕ is a (1, 1)-tensor, η is a 1-form which is dual to the Reeb vector field ξ, and g is a Riemannian metric on M which satisfies the follows [24]: Because of these conditions, we have An almost contact metric manifold ( M, ϕ, η, ξ, g) whose curvature tensor satisfies for any vector field X 1 , X 2 , X 3 ∈ Γ(T M) and f 1 , f 2 , f 3 being differentiable functions on M is said to be a generalized Sasakian-space-form denoted by M( f 1 , f 2 , f 3 ).The notion of a generalized Sasakian-space-form M( f 1 , f 2 , f 3 ) was introduced by Alegre et al. [25], generalizing three important contact space forms, that is, the Sasakian-space-form 6) and ( 11), we obtain Let M m be a submanifold of a generalized Sasakian-space-form M( f 1 , f 2 , f 3 ) of dimension (2n + 1).For any tangent vector field X 1 on M, we can write ϕX 1 = T X 1 + F X 1 , where T X 1 is the tangential component, and F X 1 is the normal component of ϕX 1 .The squared norm of T at x ∈ M is defined as where {e 1 , . . ., e m } is any orthonormal basis of the tangent space T x M and decomposing the structural vector field ξ = ξ T + ξ ⊥ , where ξ T and ξ ⊥ denotes the tangential and normal components of ξ.Moreover, we set Θ 2 (Π) = g 2 (T e 1 , e 2 ) = g 2 (ϕe 1 , e 2 ), where {e 1 , e 2 } is the orthonormal basis of two-plane section Π.
where Π is a two-plane section T x M, x ∈ M. If in addition, P is a tangent vector field on M m , then H = H and the equality case holds at a point x ∈ M if and only if there exists an orthonormal basis {e 1 , . . ., e m } of T x M and an orthonormal basis {e m+1 , . . ., e 2n+1 } of T ⊥ x M such that the shape operators of M in M( f 1 , f 2 , f 3 ) at x have the following forms: Proof.Let x ∈ M and {e 1 , e 2 , . . ., e m }, {e m+1 , . . ., e 2n+1 } be an orthonormal basis of T x M and T ⊥ x M, respectively, then from (7), ( 10) and ( 12) we obtain We set, then ( 14) becomes For a chosen orthonormal basis, (16) can be written as: Now, let Π = span{e 1 , e 2 }, then from ( 7) and ( 12) we obtain R(e 1 , e 2 , e 2 , e 1 ) = ), P) and Making use of ( 18) and ( 19) in ( 9), we obtain Combining ( 14) and (20) gives Making use of Lemma 2.4 [26], we have 2n+1 In view of the last expression in (21), we obtain Now, if P is a tangent vector field on M, then (5) implies h = h and H = H.If the equality case (23) holds at a point x ∈ M, then the equality cases of ( 17) and ( 22) hold, which gives Therefore, choosing a suitable orthonormal basis, the shape operators take the desired forms.
Corollary 1.Under the same arguments as in Theorem 2, 1.
If the structure vector field ξ is tangent to M, we have

2.
If the structure vector field ξ is normal to M, we have Remark 1.It should be noted that Theorem 2 generalizes the Theorem 6 obtained in [20].Moreover, taking different values of f i , i = 1, 2, 3, we can obtain similar inequalities as Theorem 1 for the Kenmotsu space form and the Cosympletic space form endowed with certain types of connections by restricting the values of ψ i , i = 1, 2.
Corollary 2. Let M m , m ≥ 3 be an m-dimensional submanifold of a (2n + 1)-dimensional generalized Sasakian-space-form M( f 1 , f 2 , f 3 ) endowed with a semi-symmetric non-metric connection, then where Π is a two-plane section T x M, x ∈ M.
For an integer k ≥ 0, we denote by S(m, k) the set of k-tuples (m 1 , . . ., m k ) of integers ≥ 2 satisfying m 1 < m and m 1 , . . ., m k ≤ m.In addition, let S(m) be the set of unordered k-tuples with k ≥ 0 for a fixed m.Then, for each k-tuple (m 1 , . . ., m k ) ∈ S(m), Chen introduced a Riemannian invariant δ(m 1 , . . ., m k ) as follows [23] where L 1 , . . ., L k run over all k mutually orthogonal subspaces of T x M such that dimL j = m j , j ∈ {1, . . ., k}.For simplicity, we set As the generalization of Theorem 2, we state and prove the following results using the methods used in [26].
for any k-tuples (m 1 , . . ., m k ) ∈ S(m).If P is a tangent vector field on M, the equality case holds at x ∈ M m if and only if there exists an orthonormal basis {e 1 , . . ., e m } of T x M and an orthonormal basis {e m+1 , . . ., e 2n+1 } of T ⊥ x M such that the shape operators of M in M( f 1 , f 2 , f 3 ) at x have the following forms: where a 1 , . . ., a m satisfy Proof.Chooseanorthonormalbasis {e 1 , . . ., e m } of T x M andanorthonormalbasis {e m+1 , . . ., e 2n+1 } of T ⊥ x M such that mean curvature vector H is in the direction of the normal vector to e m+1 .We set and consider the following sets Let L 1 , . . ., L k be a mutually orthogonal subspace of T x M with dimL j = m j , defined by From (7), ( 8) and ( 12), we obtain We set where Then in view of this and (26), Equation ( becomes which can be written as where α j , β j ∈ D j for all j = 1, . . ., k. Now applying Lemma 2.3 [26] in (27), we obtain where 14), ( 25) and ( 28) gives where, a(m 1 , . . ., The equality case (29) at a point x ∈ M holds if and only if all the previous inequalities hold; thus, the shape operators take the desired forms.Remark 3. Restricting the values of f i , i = 1, 2, 3 and ψ i for i = 1, 2, we can obtain similar bounds as Theorem 3 for certain contact space forms endowed with certain connections.
Theorem 4. Let M m , m ≥ 3 be an m-dimensional submanifold of a (2n + 1)-dimensional generalized Sasakian-space-form M( f 1 , f 2 , f 3 ) endowed with a quarter-symmetric connection ∇, then (i) For each unit vector X 1 in T x M, we have (ii) If H(x) = 0, then a unit tangent vector X 1 at x satisfies the equality case of (30) if and only if (iii) The equality of (30)  Proof.Choosing the orthonormal basis {e 1 , . . ., e m } such that e 1 = X 1 , where X 1 ∈ T x M is a unit tangent vector at the point x on M. In view of ( 7) and ( 12), then proceeding similarly as the proof of Theorem 4 in [20], one can easily obtain the desire results.
By choosing an orthonormal frame {e 1 , . . ., e k } of L such that e 1 = X 1 , a unit tangent vector, Chen [23] defined the k-Ricci curvature of L at X 1 by For an integer k, 2 ≤ k ≤ m, the Riemannian invariant Θ k on M is defined by where L runs over all k-plane sections in T x M and X 1 runs over all unit vectors in L.
From [26], we have Let us choose {e 1 , . . ., e m } and {e m+1 , . . ., e 2n+1 } as an orthonormal basis of T x M and T ⊥ x M, x ∈ M, respectively, where e m+1 is parallel to the mean curvature vector H.In addition, let {e 1 , . . ., e m } diagonalize the shape operator A e m+1 .Then, In consequence of the above assumptions, Equation ( 14) can be written as follows: Using the Cauchy-Schwartz inequality, we have Combining ( 32) and ( 34), we can state the following: Theorem 5. Let M m , m ≥ 3 be an m-dimensional submanifold of a (2n + 1)-dimensional generalized Sasakian-space-form M( f 1 , f 2 , f 3 ) endowed with a quarter-symmetric connection ∇.Then for any integer k, 2 ≤ k ≤ m and any point x ∈ M, we have As a particular case of Theorem 5, we obtained Theorem 6.2 [21] which is as follows:
Then for any integer k, 2 ≤ k ≤ m and any point x ∈ M, we have Corollary 4. Let M m , m ≥ 3 be an m-dimensional submanifold of a (2n + 1)-dimensional generalized Sasakian-space-form M( f 1 , f 2 , f 3 ) endowed with a semi-symmetric non-metric connection.
Then for any integer k, 2 ≤ k ≤ m and any point x ∈ M, we have Remark 4. Restricting function f i , i = 1, 2, 3, we can easily obtain similar inequality in the case of the Sasakian, Kenmotsu and Cosympletic space forms.

Some Applications
The notion of slant submanifolds in almost contact geometry was introduced by Lotta [27].A submanifold M of an almost contact metric manifold ( M, ϕ, ξ, η, g) tangent to the structure vector field ξ is said to be a contact slant submanifold if, for any point x ∈ M and any vector X 1 ∈ T x M linearly independent on ξ x , the angle between the vector ϕX 1 and the tangent space T x M is constant.This angle is known as the slant angle of M. The concept of slant submanifold is further generalized as follows: Definition 1 ([28]).A submanifold M of an almost contact metric manifold M is called a bi-slant submanifold, whenever we have ϕD θ 1 ⊥ D θ 2 and ϕD θ 2 ⊥ D θ 1 .

3.
For i = 1, 2, the distribution D i is slant with slant angle θ i .
Now, as a consequence of Theorem 2, we can state the following: Theorem 6.Let M be a (m = 2d 1 + 2d 2 + 1)-dimensional bi-slant submanifold of a (2n + 1)dimensional generalized Sasakian-space-form M( f 1 , f 2 , f 3 ) endowed with a quarter-symmetric connection ∇, then we have H 2 +(m + Making use of the above facts in Theorem 2, the proof is straightforward.
In a similar manner, Theorems 3, 4 and 5 can be stated for a bi-slant submanifold of a generalized Sasakian-space-form.Moreover, restricting the values of θ i , i = 1, 2, similar results can be obtained for a large class of submanifolds such as slant, semi-slant, hemi-slant, semi-invariant submanifolds.Moreover, by taking different values of f i , i = 1, 2, 3, we can derive similar inequalities for the Sasakian, Kenmotsu and Cosympletic space forms.