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

In 1980, I. Vaisman [1] introduced the concept of conformal changes (or deformation) of almost contact metric structures as follows: Let $M̃$ be a (2n + 1)-dimensional manifold endowed with an almost contact metric structure (φ, ξ, η, g). A conformal change of the metric g leads to a metric which is no more compatible with the almost contact structure (φ, 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 (φ, ξ, η, g) on a (2n + 1)-dimensional manifold $M̃$ which is said to be a conformal Sasakian structure if the structure (φ, ξ, η, g) is conformal related to a Sasakian structure $(φ̃,ξ̃,η̃,g̃)$⁠.

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 $‖H‖2$ for any surface $M̃2$ in $E4$ satisfy the inequality [3] as follows:

and the equality holds if and only if the ellipse of curvature of $M̃2$ in $E4$ is a circle. Later, it was extended by I. V. Gaudalupe et al. [4] for arbitrary codimension m in real space forms $M̃(m+2)(c)$ 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.

An odd-dimensional Riemannian manifold $(M̃,g)$ is said to be an almost contact metric manifold [16] if there exist a tensor φ of type (1, 1), a vector field ξ (structure vector field) and a 1-form η on $M̃$ satisfying

and

for any $X,Y∈Γ(TM̃)$⁠. The two-form Φ is called the fundamental two-form in $M̃$ and the manifold is said to be a contact metric manifold if

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 $X,Y∈Γ(TM̃)$⁠, where $∇¯$ denotes the Riemannian connection.

A plane section π in $TpM̃$ is called a φ-section if it is spanned by X and φX, where X is a unit tangent vector orthogonal to ξ. The sectional curvature of a φ-section is called a φ-sectional curvature. A Sasakian manifold with constant φ-sectional curvature c is said to be a Sasakian space form and denoted by $M̃(c)$⁠. The curvature tensor of a Sasakian space form $M̃(c)$ is given by [16].

for any $X,Y,Z∈Γ(TM̃)$⁠.

A (2n + 1)-dimensional Riemannian manifold $M̃$ endowed with the almost contact metric structure (φ, η, ξ, g) called a conformal Sasakian manifold if for a C^∞ function

there exists [3].

such that $(M̃,φ̃,ξ̃,g̃)$ is a Sasakian manifold.

Let $∇̃$ and $∇¯$ denote connections of $M̃$ related to metrics $g̃$ and g, respectively. Using Koszul formula, we derive the following relation between the connections $∇̃$ and $∇¯$⁠:

for any $X,Y∈Γ(M̃)$ so that ω(X) = X(f) and ω^# is vector field of metrically equivalent to one form of ω, i.e. g(ω^#, X) = ω(X). The vector field ω^# = grad  f is called the Lee vector field of conformal Sasakian manifold $M̃$⁠.

The (2n + 1)-dimensional conformal Sasakian manifold with constant sectional curvature c, denoted by $M̃(c)$⁠, is called a conformal Sasakian space form and its curvature tensor is given by [3].

for any $X,Y,Z,W,ω#,ξ∈Γ(TM̃)$⁠, where $B≔∇¯ω−12ω⊗ω$⁠.

Let $M̃$ be an (2n + 1)-dimensional Riemannian manifold with Riemannian metric g and ∇ be the Levi-Civita connection on $M̃$⁠. Let $∇¯̃$ be a linear connection defined by [17].

for any $X,Y∈Γ(M̃)$⁠, Λ₁ and Λ₂ are real constants and $V$ is the vector field on $M̃$ such that $λ(X)=g(X,V)$⁠, where λ is 1-form. If $∇¯̃g=0$⁠, then $∇¯̃$ is known as quarter-symmetric metric connection and $∇¯̃g≠0$⁠, then $∇¯̃$ is known as quarter-symmetric non-metric connection. Decomposing the vector field $V$ on $M̃$ uniquely into its tangent and normal components $VT$ and $V⊥$⁠, respectively.

The special cases of (2.5) can be obtained as follows:

1. when Λ₁ = Λ₂ = 1, then the above connection reduces to semi-symmetric metric connection and

2. when Λ₁ = 1 and Λ₂ = 0, then the above connection reduces to semi-symmetric non-metric connection.

For any $X,Y,Z,W∈Γ(TM̃)$⁠, the curvature tensor with respect to $∇¯̃$ is given by

On using (2.5), the curvature tensor (2.6) takes the form [17] as follows:

where α and β are (0, 2)-tensors and defined as follows:

and

The curvature tensor of conformal Saasakian space form $M̃(c)$ with a quarter-symmetric connection $∇¯̃$ is given by

For simplicity, we have put tr(α) = a and tr(β) = b.

Let $M$ be an m-dimensional submanifold of a (2n + 1)-dimensional conformal Saasakian space form $M̃(c)$⁠. We consider the induced quarter-symmetric connection on $M$ represented by $∇̃M$ and the induced Levi-Civita connection denoted by $∇M$⁠. Let $R$ and $RM$ be the curvature tensors of $∇̃M$ and $∇M$⁠. Then, the Gauss equation is given by

where h is the second fundamental form of $M$ in $M̃$ with respect to $∇¯̃$ and defined as follows:

Here, h′ is the second fundamental form of $M$ in $M̃$ with respect to $∇¯$ and g denotes the Riemannian metric on $M$⁠.

For any $X∈Γ(TM)$⁠, we can write φX = PX + SX, where the PX (respectively, SX) is the tangential component (respectively normal component) of φX. If P = 0, then the submanifold is anti-invariant and if S = 0, then the submanifold is invariant. The squared norm of P at $p∈M$ is given as follows:

where {e₁, …, e_m} is any orthonormal basis of $TpM$ and $p∈M$⁠. The structure vector field ξ can be decomposed as ξ = ξ^T + 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 [18–20]). 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 manifold $M̃$ is called bi-slant submanifolds, whenever we have

1. $TMm=Dθ1⊕Dθ1⊕〈ξ〉$ and

2. $φDθ1⊥Dθ2$ and $φDθ2⊥Dθ1$

where $Dθ1$ and $Dθ2$ are two orthogonal distributions of $M$ with slant angle θ₁ and θ₂, respectively.

Let $M$ be a bi-slant submanifold of a conformal Sasakian space form $M̃$⁠. We assume that dim$(M)=m=2m1+2m2+1$⁠, where dim$(Dθ1)=m1$ and dim$(Dθ2)=m2$⁠. Let {e₁, …, e_m = ξ} be an orthonormal basis of $TpM$ at p in $M$ with

$e1,e2=secθ1Pe1,…,e2m1−1,e2m1=secθ1Pe2m1−1,e2m1+1,e2m1+2=secθ2Pe2m1+1,…,e2m1+2m2−1,e2m1+2m2=secθ2Pe2m1+2m2−1,e2m1+2m2+1=ξ$⁠,

from which we have [23] as follows:

$g2(φei+1,ei)=cos2θ1,fori=1,2,…,2m1−1cos2θ2,fori=2m1+1,…,2m1+2m2−1.$

Thus, we have

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 = 0 and $θ=π2$⁠, respectively. The slant submanifold is said to be proper slant and proper bi-slant submanifold, if $0<θ<π2$ and θ_i lies between 0 and $π2$⁠.

In [10], Mihai discussed the generalized Wintgen inequality for Legendrian submanifolds in Sasakian space forms. He also stated such an inequality for contact slant submanifolds in Sasakian space forms. Thus, in this section, we obtain such an inequality in terms of the invariant $ρM$ (called normalized scalar-normal curvature) for bi-slant submanifolds of dimension m in a (2n + 1)-dimensional conformal Sasakian space form $M̃$⁠. Consider the local orthonormal tangent frame {e₁, …, e_m} of the tangent bundle $TM$ of $M$ and a local orthonormal normal frame {e_m+1, …, e_2n+1} of the normal bundle $T⊥M$ of $M$ in $M̃$⁠. At any $p∈M$⁠, the scalar curvature τ at that point is given by

The mean curvature $H$ of submanifold is given by

Conveniently, let us put

for any i, j = {1, …, m} and r = {m + 1, …, 2n + 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 $κM$ of $M$ by the formula [8].

and the normalized scalar normal curvature is given by [8].

Moreover, the equality case holds in the above inequality at a point $p∈M$ if and only if, with respect to some suitable orthonormal basis $e1,…,em$ of $TpM$ and $ξ1,…,ξ2n−m+1$ of $Tp⊥M$⁠, the shape operators S_γ, γ = 1, …, 2n − m + 1, take the forms as follows:

whereμ₁, μ₂, μ₃, andζare real numbers.

By taking summation 1 ≤ i < j ≤ m of (3.9) and using (2.7), we have

where

Using (2.10) and (3.1), we obtain

On the other hand, we have

Further, from [6], we have

Now, combining (3.3), (3.12) and (3.14), we have

Taking into account (3.2), (3.11) and (3.14), we obtain the required 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: