Generalized Wintgen inequality for BI-SLANT submanifolds in conformal Sasakian space form with quarter-symmetric connection

Purpose – In 1979, P. Wintgen obtained a basic relationship between the extrinsic normal curvature the intrinsicGauss curvature,and squared meancurvature ofany surface in a Euclidean4-space with the equality holding if and only if the curvature ellipse is a circle. In 1999, P. J. De Smet, F. Dillen, L. Verstraelen and L. Vrancken gave a conjecture of Wintgen inequality, named as the DDVV-conjecture, for general Riemannian submanifolds in real space forms. Later on, this conjecture was proven to be true by Z. Lu and by Ge and Z. Tang independently. Since then, the study of Wintgen ’ s inequalities and Wintgen ideal submanifolds has attracted many researchers, and a lot of interesting results have been found during the last 15 years. The main purpose of this paper is to extend this conjecture of Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection. Design/methodology/approach – The authors used standard technique for obtaininggeneralized Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection. Findings – The authors establish the generalized Wintgen inequality for bi-slant submanifold in conformal Sasakian space form endowed with a quarter symmetric metric connection, and also find conditions under which the equality holds. Some particular cases are also stated. Originality/value – The research may be a challenge for new developments focused on new relationships in termsofvariousinvariants,fordifferenttypesofsubmanifoldsinthatambientspacewithseveralconnections.


Introduction
In 1980, I. Vaisman [1] introduced the concept of conformal changes (or deformation) of almost contact metric structures as follows: LetM be a (2n þ 1)-dimensional manifold endowed with an almost contact metric structure (w, ξ, η, g). A conformal change of the metric g leads to a metric which is no more compatible with the almost contact structure (w, ξ, η). This can be corrected by a convenient change of ξ and η which implies rather strong restrictions. Using this definition, a new type of almost contact metric structure (w, ξ, η, g) on a (2n þ 1)-dimensional manifoldM which is said to be a conformal Sasakian structure if the structure (w, ξ, η, g) is conformal related to a Sasakian structure ðw;ξ;η;gÞ.
The Wintgen inequality is a sharp geometric inequality for surfaces in four-dimensional Euclidean space involving Gauss curvature (intrinsic invariants), normal curvature and square mean curvature (extrinsic invariants). P. Wintgen [2] proved that the Gauss curvature K, the normal curvature K ⊥ and the squared mean curvature kHk 2 for any surfaceM 2 in E 4 satisfy the inequality [3] as follows: and the equality holds if and only if the ellipse of curvature ofM 2 in E 4 is a circle. Later, it was extended by I. V. Gaudalupe et al. [4] for arbitrary codimension m in real space forms M ðmþ2Þ ðcÞ as follows: In 1999, De Smet, Dillen, Verstraelen and Vrancken [5] conjectured the generalized Wintgen inequality for submanifolds in real space form. The conjecture is known as DDVV conjecture. It had been proved by Lu [6] and by Ge and Tang [7] independently. In 2014, Ion Mihai [8] established such inequality for Lagrangian submanifold in complex space form. They provided some applications and also stated such an inequality for slant submanifolds in complex space forms. However, the year 2014 is not the stopping point in investigating Wintgen inequality and some additional steps have been taken in the development of the theory. In fact, many remarkable articles were published in the recent years and several inequalities of this type have been obtained for other classes of submanifolds in several ambient spaces for example, for statistical submanifolds in statistical manifolds of constant curvature [9]; for Legendrian submanifolds in Sasakian space forms [10]; for submanifolds in statistical warped product manifolds [11]; for quaternionic CR-submanifolds in quaternionic space forms [12]; for submanifolds in generalized (κ, μ)-space forms [13]; for totally real submanifolds in LCS-manifolds [14] and so on. For more details, see [15].
In the present article, we obtain the generalized Wintgen inequalities for conformal Sasakian space forms. The equality case of the main inequality is investigated. Lastly, we discuss such inequality for various slant cases as an application of the obtained inequality.

Sasakian manifold
An odd-dimensional Riemannian manifold ðM; gÞ is said to be an almost contact metric manifold [16] if there exist a tensor w of type (1, 1), a vector field ξ (structure vector field) and a 1-form η onM satisfying w 2 X ¼ −X þ ηðXÞξ; ηðξÞ ¼ 1; (2.1) wξ ¼ 0; η8ξ ¼ 0; gðX ; ξÞ ¼ ηðXÞ; (2.2) AJMS and gðwX ; wY Þ ¼ gðX ; Y Þ À ηðXÞηðY Þ; for any X ; Y ∈ ΓðTMÞ. The two-form Φ is called the fundamental two-form inM and the manifold is said to be a contact metric manifold if Φ ¼ dη: A Sasakian manifold is a normal contact metric manifold. In fact, an almost contact metric manifold is Sasakian manifold if and only if we have for any X ; Y ∈ ΓðTMÞ, where ∇ denotes the Riemannian connection.
A plane section π in T pM is called a w-section if it is spanned by X and wX, where X is a unit tangent vector orthogonal to ξ. The sectional curvature of a w-section is called a w-sectional curvature. A Sasakian manifold with constant w-sectional curvature c is said to be a Sasakian space form and denoted byMðcÞ. The curvature tensor of a Sasakian space form MðcÞ is given by [16].
for any X ; Y ; Z ∈ ΓðTMÞ.

Conformal Sasakian manifold
A (2n þ 1)-dimensional Riemannian manifoldM endowed with the almost contact metric structure (w, η, ξ, g) called a conformal Sasakian manifold if for a C ∞ function such that ðM;w;ξ;gÞ is a Sasakian manifold. Let∇ and ∇ denote connections ofM related to metricsg and g, respectively. Using Koszul formula, we derive the following relation between the connections∇ and ∇: for any X ; Y ∈ ΓðMÞ so that ω(X) 5 X(f) and ω # is vector field of metrically equivalent to one form of ω, i.e. g(ω # , X) 5 ω(X). The vector field ω # 5 grad f is called the Lee vector field of conformal Sasakian manifoldM. The (2n þ 1)-dimensional conformal Sasakian manifold with constant sectional curvature c, denoted byMðcÞ, is called a conformal Sasakian space form and its curvature tensor is given by [3].

Quarter-symmetric metric connection
LetM be an (2n þ 1)-dimensional Riemannian manifold with Riemannian metric g and ∇ be the Levi-Civita connection onM. Let∇ be a linear connection defined by [17].
for any X ; Y ∈ ΓðMÞ, Λ 1 and Λ 2 are real constants and V is the vector field onM such that λðX Þ ¼ gðX ; VÞ, where λ is 1-form. If∇g ¼ 0, then∇ is known as quarter-symmetric metric connection and∇g ≠ 0, then∇ is known as quarter-symmetric non-metric connection.
Decomposing the vector field V onM uniquely into its tangent and normal components V T and V ⊥ , respectively. The special cases of (2.5) can be obtained as follows: (1) when Λ 1 5 Λ 2 5 1, then the above connection reduces to semi-symmetric metric connection and (2) when Λ 1 5 1 and Λ 2 5 0, then the above connection reduces to semi-symmetric nonmetric connection.
For any X ; Y ; Z ; W ∈ ΓðTMÞ, the curvature tensor with respect to∇ is given bỹ On using (2.5), the curvature tensor (2.6) takes the form [17] as follows: (2.7) where α and β are (0, 2)-tensors and defined as follows: The curvature tensor of conformal Saasakian space formMðcÞ with a quarter-symmetric connection∇ is given by For simplicity, we have put tr(α) 5 a and tr(β) 5 b.
Let M be an m-dimensional submanifold of a (2n þ 1)-dimensional conformal Saasakian space formMðcÞ. We consider the induced quarter-symmetric connection on M represented by∇ M and the induced Levi-Civita connection denoted by ∇ M . Let R and R M be the curvature tensors of∇ M and ∇ M . Then, the Gauss equation is given bỹ where h is the second fundamental form of M inM with respect to∇ and defined as follows: Here, h 0 is the second fundamental form of M inM with respect to ∇ and g denotes the Riemannian metric on M.
For any X ∈ ΓðTMÞ, we can write wX 5 PX þ SX, where the PX (respectively, SX) is the tangential component (respectively normal component) of wX. If P 5 0, then the submanifold is anti-invariant and if S 5 0, then the submanifold is invariant. The squared norm of P at p ∈ M is given as follows:

Generalized
Wintgen inequality where {e 1 , . . ., e m } is any orthonormal basis of T p M and p ∈ M. The structure vector field ξ can be decomposed as ξ 5 ξ T þ ξ ⊥ , where ξ T and ξ ⊥ are tangential and normal components of ξ.
The notion of bi-slant submanifolds was introduced by A. Carriazo et al. as a natural generalization of CR, slant, semi-slant and hemi-slant submanifolds (see [18][19][20]). Recently, S. Uddin and B.-Y. Chen studied bi-slant and pointwise bi-slant submanifolds for their warped products in [21,22]. A submanifold M of an almost contact-metric manifoldM is called bi-slant submanifolds, whenever we have where D θ 1 and D θ 2 are two orthogonal distributions of M with slant angle θ 1 and θ 2 , respectively.
Let M be a bi-slant submanifold of a conformal Sasakian space formM. We assume that from which we have [23] as follows: In fact, semi-slant, pseudo-slant, CR and slant submanifolds can be obtained from bi-slant submanifolds in particular. We can see the cases in the following Table 1: The special case of slant submanifold are invariant and anti-invariant if θ 5 0 and θ ¼ π 2 , respectively. The slant submanifold is said to be proper slant and proper bi-slant submanifold, if 0 < θ < π 2 and θ i lies between 0 and π 2 .  The mean curvature H of submanifold is given by Conveniently, let us put h r ij ¼ gðhðe i ; e i Þ; e r Þ; for any i, j 5 {1, . . ., m} and r 5 {m þ 1, . . ., 2n þ 1}. We denote by K and R ⊥ , the sectional curvature function and the normal curvature tensor on M, respectively. Then the normalized scalar curvature ρ is given by [8].
In term of the components of the second fundamental form, we can express the scalar normal curvature κ M of M by the formula [8].
and the normalized scalar normal curvature is given by [8].
Theorem 3.1. Let M be an m-dimensional bi-slant submanifold in conformal Sasakian space formMðcÞ of dimension (2n þ 1) endowed with a quarter-symmetric connection, then we have (3.5)

Generalized Wintgen inequality
Finally, by investigating the equality case of (3.5), the equality sign holds in (3.5) at a point p ∈ M if and only if the shape operators take the forms (3.6)-(3.8) with respect to some suitable tangent and normal orthonormal bases.
, An immediate consequence of Theorem 3.1 yields the following: Corollary 3.2. Let M be a minimal m-dimensional bi-slant submanifold in conformal Sasakian space formMðcÞ of dimension (2n þ 1) endowed with a quarter-symmetric connection, then we have (3.15) Corollary 3.3. Let M be an m-dimensional submanifold in conformal Sasakian space formMðcÞ of dimension (2n þ 1) endowed with a quarter-symmetric connection, then we have Table 2: For the semi-symmetric metric connection Λ 1 5 Λ 2 5 1, we have (3.16) Corollary 3.5. Let M be an m-dimensional submanifold in conformal Sasakian space form MðcÞ of dimension (2n þ 1) endowed with a semi-symmetric metric connection, then we have Table 3: For the semi-symmetric non-metric connection Λ 1 5 1 and Λ 2 5 0, we have Theorem 3.6. Let M be an m-dimensional bi-slant submanifold in conformal Sasakian space formMðcÞ of dimension (2n þ 1) endowed with a semi-symmetric non-metric connection, then we have Corollary 3.7. Let M be an m-dimensional submanifold in conformal Sasakian space form MðcÞ of dimension (2n þ 1) endowed with a semi-symmetric non-metric connection, then we have Table 4 as follows: where (x 1 , x 2 , x 3 ) are standard coordinates in R 3 . We choose the vector fields which are linearly independent at each point ofM. Let g be the Riemannian metric defined by Let η be the an 1-form defined by We define the (1, 1) tensor field w as The linear property of g and w yield that for any U ; V ∈ ΓðMÞ. Thus, ðM; w; ξ; η; gÞ defines an almost contact metric manifold with ξ 5 v 3 [24]. Then we have Similarly, The Riemannian connection ∇ of the metric g is given by By Koszul's formula, we obtain the following ðM;w;ξ;η; gÞ is Sasakian manifold. SoM is a conformal Sasakian manifold but not Sasakian. Since by the definition, we have for any v 1 ; v 2 ∈ ΓðMÞ (for instance ∇ v 2 v 3 ≠ 0). By using the above results, we can find the nonvanishing components of Riemannian curvature, Ricci curvature tensor and scalar curvature as follows: In view of above expressions, we turn up the following: Note that the sectional curvature of manifoldM with almost contact-metric structure ðM;w;ξ;η; gÞ is Moreover, the non-vanishing components of Ricci curvature tensor, and scalar curvature are given by Example 4.2 Let us consider a three-dimensional manifold where (x 1 , x 2 , x 3 ) are standard coordinates in R 3 . We choose the vector fields which are linearly independent at each point ofM. Let g be the Riemannian metric defined by Let η be the an 1-form defined by We define the (1, 1) tensor field w by