**1. Introduction**

Let *M* be a compact minimal hypersurface of the unit sphere *Sn*+<sup>1</sup> with shape operator *A*. In his pioneering work, Simons [1] has shown that on a compact minimal hypersurface *M* of the unit sphere *Sn*+<sup>1</sup> either *A* = 0 (totally geodesic), or -*A*-2 = *n*, or -*A*-2 (*p*) > *n* for some point *p* ∈ *M*, where -*A*- is the length of the shape operator. This work was further extended in [2] and for compact constant mean curvature hypersurfaces in [3]. If for every point *p* in *M*, the square of the length of the second fundamental form of *M* is *n*, then it is known that *M* must be a subset of a Clifford minimal hypersurface

$$S^l\left(\sqrt{\frac{l}{n}}\right) \times S^m\left(\sqrt{\frac{m}{n}}\right) \dots$$

where *l*, *m* are positive integers, *l* + *m* = *n* (cf. Theorem 3 in [4]). Note that this result was independently proven by Lawson [2] and Chern, do Carmo, and Kobayashi [5]. One of the interesting questions in differential geometry of minimal hypersurfaces of the unit sphere *Sn*+<sup>1</sup> is to characterize minimal Clifford hypersurfaces. Minimal hypersurfaces have also been studied in (cf. [6–8]). In [2], bounds on Ricci curvature are used to find a characterization of the minimal Clifford hypersurfaces in the unit sphere *S*4. Similarly in [3,9–11], the authors have characterized minimal Clifford hypersurfaces in the odd-dimensional unit spheres *S*3 and *S*5 using constant contact angle. Wang [12] studied compact minimal hypersurfaces in the unit sphere *Sn*+<sup>1</sup> with two distinct principal curvatures, one of them being simple and obtained the following integral inequality,

$$\int\_{\mathcal{M}} \left\| A \right\|^2 \leq n \operatorname{Vol}(\mathcal{M})\_\* $$

where *Vol*(*M*) is the volume of *M*. Moreover, he proved that equality in the above inequality holds if and only if *M* is the Clifford hypersurface,

$$S^1\left(\sqrt{\frac{1}{n}}\right)\times S^m\left(\sqrt{\frac{n-1}{n}}\right).$$

.

In this paper, we are interested in studying compact minimal hypersurfaces of the unit sphere *S*2*n*+<sup>1</sup> using the Sasakian structure (*ϕ*, *ξ*, *η*, *g*) (cf. [13]) and finding characterizations of minimal Clifford hypersurface of *S*2*n*<sup>+</sup>1. On a compact minimal hypersurface *M* of the unit sphere *S*2*n*<sup>+</sup>1, we denote by *N* the unit normal vector field and define a smooth function *f* = *g*(*ξ*, *<sup>N</sup>*), which we call the *Reeb function* of the minimal hypersurface *M*. Also, on the hypersurface *M*, we have a smooth vector field *v* = *ϕ*(*N*), which we call the *contact vector field* of the hypersurface (*v* being orthogonal to *ξ* belongs to contact distribution). Instead of demanding two distinct principal curvatures one being simple, we ask the contact vector field *v* of the minimal hypersurface in *S*2*n*+<sup>1</sup> to be conformal vector field and obtain an inequality similar to Wang's inequality and show that the equality holds if and only if *M* is isometric to a Clifford hypersurface. Indeed we prove

**Theorem 1.** *Let M be a compact minimal hypersurface of the unit sphere S*2*n*+<sup>1</sup> *with Reeb function f and contact vector field v a conformal vector field on M. Then,*

$$\int\_{M} (1 - f^2) \left\| A \right\|^2 \le 2n \int\_{M} \left( 1 - f^2 \right)^2$$

*and the equality holds if and only if M is isometric to the Clifford hypersurface Sl* 0*l* 2*n* × *Sm* 0 *m* 2*n , where l* + *m* = 2*n.*

Also in [12], Wang studied embedded compact minimal non-totally geodesic hypersurfaces in *Sn*+<sup>1</sup> those are symmetric with respect to *n* + 2 pair-wise orthogonal hyperplanes of *Rn*<sup>+</sup>2. If *M* is such a hypersurface, then it is proved that

$$\int\_{\mathcal{M}} \left\| A \right\|^2 \geq nVol(\mathcal{M})\_\star$$

and the equality holds precisely if *M* is a Clifford hypersurface. Note that compact embedded hypersurface has huge advantage over the compact immersed hypersurface, as it divides the ambient unit sphere *Sn* into two connected components.

In our next result, we consider compact immersed minimal hypersurface *M* of the unit sphere *S*2*n*+<sup>1</sup> such that the Reeb function *f* is a constant along the integral curves of the contact vector field *v* and show that above inequality of Wang holds, and we ge<sup>t</sup> another characterization of minimal Clifford hypersurface in the unit sphere *S*2*n*<sup>+</sup>1. Precisely, we prove the following.

**Theorem 2.** *Let M be a compact minimal hypersurface of the unit sphere S*2*n*+<sup>1</sup> *with Reeb function f a constant along the integral curves of the contact vector field v. Then,*

$$\int\_{M} \left\| A \right\|^{2} \geq 2nVol(M).$$

*and the equality holds if and only if M is isometric to the Clifford hypersurface Sl* 0*l* 2*n* × *Sm* 0 *m* 2*n , where l* + *m* = 2*n.*

## **2. Preliminaries**

Recall that conformal vector fields play an important role in the geometry of a Riemannian manifolds. A conformal vector field *v* on a Riemannian manifold (*<sup>M</sup>*, *g*) has local flow consisting of conformal transformations, which is equivalent to

$$
\mathfrak{E}\_{\overline{\upsilon}} \mathfrak{g} = 2\rho \mathfrak{g}.\tag{1}
$$

The smooth function *ρ* appearing in Equation (1) defined on *M* is called the potential function of the conformal vector field *v*. We denote by (*ϕ*, *ξ*, *η*, *g*) the Sasakian structure on the unit sphere *S*2*n*+<sup>1</sup> as a totally umbilical real hypersurface of the complex space form (*Cn*<sup>+</sup>1, *J*,,), where *J* is the complex structure and , is the Euclidean Hermitian metric. The Sasakian structure (*ϕ*, *ξ*, *η*, *g*) on *S*2*n*+<sup>1</sup> consists of a (1, 1) skew symmetric tensor field *ϕ*, a smooth unit vector field *ξ*, a smooth 1-form *η* dual to *ξ*, and the induced metric *g* on *S*2*n*+<sup>1</sup> as real hypersurface of *Cn*+<sup>1</sup> and they satisfy (cf. [13])

$$
\varrho^2 = -I + \eta \otimes \mathfrak{J}, \ \eta \circ \varrho = 0, \ \eta(\mathfrak{J}) = 1, \ \lg(\varrho X, \varrho \mathcal{Y}) = \lg(X, \mathcal{Y}) - \eta(X)\eta(\mathcal{Y}), \tag{2}
$$

and

$$\left(\overline{\nabla}\varphi\right)\left(X,Y\right) = \mathfrak{g}\left(X,Y\right)\mathfrak{k} - \eta\left(Y\right)X, \quad \overline{\nabla}\_X\mathfrak{k} = -\varrho X,\tag{3}$$

where *X*,*Y* are smooth vector fields, ∇ is Riemannian connection on *S*2*n*+<sup>1</sup> and the covariant derivative

$$(\nabla \varphi) \left( X, \mathcal{Y} \right) = \nabla\_X \varphi \mathcal{Y} - \varphi \left( \nabla\_X \mathcal{Y} \right) \dots$$

We dente by *N* and *A* the unit normal and the shape operator of the hypersurface *M* of the unit sphere *S*2*n*<sup>+</sup>1. We denote the induced metric on the hypersurface *M* by the same letter *g* and denote by ∇ the Riemannian connection on the hypersurface *M* with respect to the induced metric *g*. Then, the fundamental equations of hypersurface are given by (cf. [14])

$$
\nabla\_X \mathbf{Y} = \nabla\_X \mathbf{Y} + \mathbf{g}(AX, \mathbf{Y}), \ \nabla\_X \mathbf{N} = -AX, \ X, \mathbf{Y} \in \mathfrak{X}(M), \tag{4}
$$

$$R(X,Y)Z = \operatorname{g}(Y,Z)X - \operatorname{g}(X,Z)Y + \operatorname{g}(AY,Z)AX - \operatorname{g}(AX,Z)AY,\tag{5}$$

$$\left(\nabla A\right)\left(X, Y\right) = \left(\nabla A\right)\left(Y, X\right), \quad X, Y \in \mathfrak{X}(M), \tag{6}$$

where X(*M*) is the Lie algebra of smooth vector fields and *<sup>R</sup>*(*<sup>X</sup>*,*<sup>Y</sup>*)*<sup>Z</sup>* is the curvature tensor field of the hypersurface *M*. The Ricci tensor of the minimal hypersurface *M* of the unit sphere *S*2*n*+<sup>1</sup> is given by

$$\operatorname{Ric}(X,Y) = (2n-1)\operatorname{g}(X,Y) - \operatorname{g}(AX,AY), \quad X,Y \in \mathfrak{X}(M) \tag{7}$$

and

$$\sum\_{i=1}^{2n} \left(\nabla A\right) \left(e\_i, e\_i\right) = 0 \tag{8}$$

holds for a local orthonormal frame {*<sup>e</sup>*1,...,*e*2*n*} on the minimal hypersurface *M*.

Using the Sasakian structure (*ϕ*, *ξ*, *η*, *g*) on the unit sphere *S*2*n*<sup>+</sup>1, we analyze the induced structure on a hypersurface *M* of *S*2*n*<sup>+</sup>1. First, we have a smooth function *f* on the hypersurface *M* defined by *f* = *g*(*ξ*, *<sup>N</sup>*), which we call the *Reeb function* of the hypersurface *M*, where *N* is the unit normal vector field. As the operator *ϕ* is skew symmetric, we ge<sup>t</sup> a vector field *v* = *ϕN* defined on *M*, which we call the *contact vector field* of the hypersurface *M*. Note that the vector field *v* is orthogonal to *ξ*, and therefore lies in the contact distribution of the Sasakian manifold *S*2*n*<sup>+</sup>1. We denote by *u* = *ξT* the tangential component of *ξ* to the hypersurface *M* and, consequently, we have *ξ* = *u* + *f N*. Let *α* and *β* be smooth 1-forms on *M* dual to the vector fields *u* and *v*, respectively, that is, *α*(*X*) = *g*(*<sup>X</sup>*, *u*) and *β*(*X*) = *g*(*<sup>X</sup>*, *<sup>v</sup>*), *X* ∈ <sup>X</sup>(*M*). For *X* ∈ <sup>X</sup>(*M*), we set *JX* = (*ϕ<sup>X</sup>*)*<sup>T</sup>* the tangential component of *ϕX* to the hypersurface, which gives a skew symmetric (1, 1) tensor field *J* on the hypersurface *M*. It follows

that *ϕX* = *JX* − *β*(*X*)*<sup>N</sup>*. Thus, we ge<sup>t</sup> a structure (*J*, *u*, *v*, *α*, *β*, *f* , *g*) on the hypersurface *M* and using properties in Equations (2) and (3) of the Sasakian structure (*ϕ*, *ξ*, *η*, *g*) on the unit sphere *S*2*n*+<sup>1</sup> and Equation (4), it is straightforward to see that the structure (*J*, *u*, *v*, *α*, *β*, *f* , *g*) on the hypersurface *M* has the properties described in the following Lemma.

**Lemma 1.** *Let M be a hypersurface of the unit sphere S*2*n*+1*. Then, M admits the structure* (*J*, *u*, *v*, *α*, *β*, *f* , *g*) *satisfying*


 -∇ *f* -2*.*

*(vi)* ∇ *f* = −*Au* + *v,*

*(vii) u*-2 = *v*-2 = (1 − *f* 2)*, g*(*<sup>u</sup>*, *v*) = 0*,*

*where* ∇ *f is the gradient of the Reeb function f .*

Let Δ*f* be the Laplacian of the Reeb function *f* of the minimal hypersurface *M* of the unit sphere *S*2*n*+<sup>1</sup> defined by Δ*f* = div∇ *f* . Then using Lemma 1 and 12Δ*f* 2 = *f*Δ*f* + -∇ *f* -2 and Equations (6) and (8), we ge<sup>t</sup> the following:

**Lemma 2.** *Let M be a minimal hypersurface of the unit sphere S*2*n*+1*. Then, the Reeb function f satisfies*

$$\begin{aligned} (i) \qquad &\Delta f = -\left(2n + \|A\|^2\right)f, \\ (ii) \qquad &\frac{1}{2}\Delta f^2 = -\left(2n + \|A\|^2\right)f^2 + \cdots \end{aligned}$$

On the hypersurface *M* of the unit sphere *S*2*n*<sup>+</sup>1, we define a (1, 1) tensor field Ψ = *JA* − *A J*, then it follows that *g*(Ψ*X*,*<sup>Y</sup>*) = *g*(*<sup>X</sup>*, <sup>Ψ</sup>*<sup>Y</sup>*), *X*,*Y* ∈ <sup>X</sup>(*M*), that is, Ψ is symmetric and that *tr*Ψ = 0. Next, we prove the following:

**Lemma 3.** *Let M be a compact minimal hypersurface of the unit sphere S*2*n*+1*. Then,*

$$\int\_M \left(1 - f^2\right) \|A\|^2 = \int\_M \left(2n - 2n(2n+1)f^2 + \frac{1}{2} \left\|\Psi\right\|^2\right).$$

**Proof.** Using Equation (7), we have *Ric*(*<sup>v</sup>*, *v*)=(<sup>2</sup>*n* − 1) *v*-2 − -*Av*-2. Now, using Lemma 1, we ge<sup>t</sup>

$$(\mathcal{E}\_{\mathbb{S}}\mathbb{g})\left(X,Y\right) = -2f\mathbb{g}(X,Y) - \mathbb{g}(\Psi X,Y)\_{\prime\prime}$$

which on using the fact that *tr*Ψ = 0, gives

$$\left|\mathcal{E}\_{\upsilon}\mathbf{g}\right|^{2} = 8nf^{2} + \left\|\mathbf{Y}\right\|^{2}\dots$$

Also, using (iii) of Lemma 1, we have

$$\left\| \|JA\| \right\|^2 = \left\| A\| \right\|^2 - \left\| Au\| \right\|^2 - \left\| Av\|^2 \right\|^2$$

which together with second equation in (iv) of Lemma 1 and the fact that *trJA* = 0, implies

$$\left\| \|\nabla v\|\|^2 = 2nf^2 + \left\|A\right\|^2 - \left\|Au\right\|^2 - \left\|Av\right\|^2.$$

Note that second equation in (iv) of Lemma 1 also gives

$$\text{div}\mathbf{v}v = -2nf.$$

Now, inserting above values in the following Yano's integral formula (cf. [15])

$$\int\_{M} \left( \operatorname{Ric}(\upsilon, \upsilon) + \frac{1}{2} \left| \mathcal{E}\_{\upsilon} \underline{g} \right|^{2} - \left\| \nabla \upsilon \right\|^{2} - \left( \operatorname{div} \upsilon \right)^{2} \right) = 0,$$

we ge<sup>t</sup>

$$\int\_{M} \left( (2n - 1) \left\| v \right\|^2 + 2nf^2 + \frac{1}{2} \left\| \Psi \right\|^2 - \left\| A \right\|^2 + \left\| Au \right\|^2 - 4n^2 f^2 \right) = 0. \tag{9}$$

Also, (vi) of Lemma 1, gives *Au* = *v* − ∇ *f* , that is, -*Au*-2 = *v*-2 + -∇ *f* -2 − <sup>2</sup>*v*(*f*), which on using div(*f v*) = *v*(*f*) + *f*div*v* = *v*(*f*) − 2*n f* 2, gives

$$\left\|Au\right\|^2 = \left\|v\right\|^2 + \left\|\nabla f\right\|^2 - 2\text{div}(fv) - 4nf^2.$$

Inserting above value of -*Au*-2 in Equation (9), yields

$$\int\_{M} \left( 2n \left\| v \right\|^2 - 2nf^2 + \frac{1}{2} \left\| \Psi \right\|^2 - \left\| A \right\|^2 + \left\| \nabla f \right\|^2 - 4n^2 f^2 \right) = 0. \tag{10}$$

Integrating (ii) of Lemma 2, we ge<sup>t</sup>

$$\int\_{\mathcal{M}} \left\| \nabla f \right\|^2 = \int\_{\mathcal{M}} \left( 2n + \left\| A \right\|^2 \right) f^2 \, d\mu$$

which together with *v*-2 = 1 − *f* 2 and Equation (10) proves the integral formula.

**Lemma 4.** *Let M be a minimal hypersurface of the unit sphere S*2*n*+1*. Then, the contact vector field v is a conformal vector field if and only if JA* = *AJ.*

**Proof.** Suppose that *A J* = *JA*. Then, using Lemma 1 and symmetry of shape operator *A* and skew symmetry of the operator *J*, we have

$$(\mathfrak{E}\_{\overline{\upsilon}}\mathbb{g})\left(X,Y\right) = \mathbb{g}(\nabla\_X\upsilon,Y) + \mathbb{g}(\nabla\_Y\upsilon,X) = -2f\mathfrak{g}(X,Y), \qquad X \in \mathfrak{X}(M),$$

which proves that *v* is a conformal vector field with potential function −*f* . Conversely, suppose *v* is conformal vector field with potential function *ρ*. Then, using Equation (1), we have

$$(\mathfrak{E}\_v \mathfrak{g})(X, Y) = \mathfrak{g}(\nabla\_X \upsilon, Y) + \mathfrak{g}(\nabla\_Y \upsilon, X) = 2\mathfrak{g}(X, Y)\mathfrak{g}$$

which on using Lemma 1, gives

$$
\log(-fAX - fX, \mathcal{Y}) + \lg(-fAY - f\mathcal{Y}, X) = 2\rho \mathcal{g}(X, \mathcal{Y}),
$$

that is,

$$
\lg(A|X-JAX,Y) = 2(\rho+f)\lg(X,Y).
$$

Choosing a local orthonormal frame {*<sup>e</sup>*1,...,*e*2*n*} on the minimal hypersurface *M* and taking *X* = *Y* = *ei* in above equation and summing, we ge<sup>t</sup> *ρ* = −*f* . This gives *g*(*AJX* − *JAX*,*<sup>Y</sup>*) = 0, *X*,*Y* ∈ <sup>X</sup>(*M*), that is, *A J* = *JA*.

**Lemma 5.** *Let M be a minimal hypersurface of the unit sphere S*2*n*+1*. If the contact vector field v is a conformal vector field on M, then*

$$Au = \frac{\left\|A\right\|^2}{2n}v.$$

**Proof.** Suppose *v* is a conformal vector field. Then, by Lemma 4, we have *JA* = *A J* . Note that for the Hessian operator *Af* of the Reeb function *f* using Lemma 1, we have

$$A\_f(X) = \nabla\_X \nabla f = \nabla\_X(\upsilon - Au) = -fAX - fX - \nabla\_X Au, \qquad X \in \mathfrak{X}(M),$$

which on using (vi) of Lemma 1, gives

$$A\_f(X) = -f(X + A^2X) - (\nabla A)(X, \mu).$$

Taking covariant derivative in above equation gives

$$\begin{aligned} \left(\nabla A\_f\right)(\mathbf{X},\mathbf{Y}) &= -\mathbf{X}(f)(\left(\mathbf{Y} + A^2\mathbf{Y}\right) - f\left(\nabla A^2\right)(\mathbf{X},\mathbf{Y}) - \left(\nabla^2 A\right)(\mathbf{X},\mathbf{Y},\mathbf{u})) \\ &+ \left(\nabla A\right)(\mathbf{Y},f\mathbf{X}) - f\left(\nabla A\right)(\mathbf{Y},A\mathbf{X}), \end{aligned}$$

where we used (iv) of Lemma 1. Now, on taking a local orthonormal frame {*<sup>e</sup>*1,...,*e*2*n*} on the minimal hypersurface *M* and taking *X* = *Y* = *ei* in above equation and summing, we ge<sup>t</sup>

$$\begin{split} \sum\_{i=1}^{2n} \left( \nabla A\_f \right) \left( e\_i, e\_i \right) &= \quad -\nabla f - A^2 \nabla f - f \sum\_{i=1}^{2n} \left( \nabla A^2 \right) \left( e\_i, e\_i \right) - \sum\_{i=1}^{2n} \left( \nabla^2 A \right) \left( e\_i, e\_i, u \right) \\ &\quad + \sum\_{i=1}^{2n} \left( \nabla A \right) \left( e\_i, Je\_i \right) - f \sum\_{i=1}^{2n} \left( \nabla A \right) \left( e\_i, Ae\_i \right). \end{split}$$

Note that for the minimal hypersurface, we have

$$\begin{aligned} \sum\_{i=1}^{2n} \left( \nabla A \right) \left( \mathbf{e}\_i, A \mathbf{e}\_i \right) &= \sum\_{i=1}^{2n} \left( \nabla\_{\mathbf{e}\_i} A^2 \mathbf{e}\_i - A \left( \left( \nabla A \right) \right) \left( \mathbf{e}\_i, \mathbf{e}\_i \right) + A \left( \nabla\_{\mathbf{e}\_i} \mathbf{e}\_i \right) \right) \\ &= \sum\_{i=1}^{2n} \left( \nabla A^2 \right) \left( \mathbf{e}\_i, \mathbf{e}\_i \right). \end{aligned}$$

Thus, the previous equation takes the form

$$\sum\_{i=1}^{2t} \left(\nabla A\_f\right)(e\_i, e\_i) = -\nabla f - A^2 \nabla f - 2f \sum\_{i=1}^{2t} \left(\nabla A^2\right)(e\_i, e\_i) - \sum\_{i=1}^{2t} \left(\nabla^2 A\right)(e\_i, e\_i, u) + \sum\_{i=1}^{2t} \left(\nabla A\right)(e\_i, \left[e\_i\right]). \tag{11}$$

Now, using the definition of Hessian operator, we have

$$\mathcal{R}(X,Y)\nabla f = \left(\nabla A\_f\right)(X,Y) - \left(\nabla A\_f\right)(Y,X).$$

which gives

$$Ric(Y, \nabla f) = g\left(Y, \sum\_{i=1}^{2n} \left(\nabla A\_f\right)(e\_{i\nu} e\_i)\right) - Y\left(\Delta f\right)$$

and we conclude

$$Q(\nabla f) = -\nabla(\Delta f) + \sum\_{i=1}^{2n} \left(\nabla A\_f\right)(\varepsilon\_{i\prime}\varepsilon\_i),\tag{12}$$

where *Q* is the Ricci operator defined by *Ric*(*<sup>X</sup>*,*<sup>Y</sup>*) = *g*(*QX*,*<sup>Y</sup>*), *X*,*Y* ∈ <sup>X</sup>(*M*). Using (i) of Lemma 2, we have

$$\nabla \left( \Delta f \right) = -2\mathbf{n} \nabla f - \left\| A \right\|^2 \nabla f - f \nabla \left\| A \right\|^2$$

and, consequently, using *Q*(*X*)=(<sup>2</sup>*n* − 1)*X* − *A*2*X* (outcome of Equation (7)), the Equation (12) takes the form

$$\sum\_{i=1}^{2n} \left(\nabla A\_f \right) (e\_i, e\_i) = (2n - 1)\nabla f - A^2 \left(\nabla f\right) - 2n\nabla f - \left\|A\right\|^2 \nabla f - f \nabla \left\|A\right\|^2 \,,$$

that is,

$$\sum\_{i=1}^{2n} \left(\nabla A\_f\right) \left(\mathbf{e}\_i, \mathbf{e}\_i\right) = -\nabla f - A^2 \left(\nabla f\right) - \left\|A\right\|^2 \nabla f - f \nabla \left\|A\right\|^2. \tag{13}$$

Also, note that

$$\begin{aligned} X\left(\left\|\left|A\right\|\right\|^2\right) &=& X\left(\sum\_{i=1}^{2n} \mathcal{g}\left(A\varepsilon\_i, A\varepsilon\_i\right)\right) = 2\sum\_{i=1}^{2n} \mathcal{g}\left(\left(\nabla A\right)\left(X, \varepsilon\_i\right), A\varepsilon\_i\right) \\ &=& 2\sum\_{i=1}^{2n} \mathcal{g}\left(X, \left(\nabla A\right)\left(\varepsilon\_i, A\varepsilon\_i\right)\right), \end{aligned}$$

where we have used Equation (6) and symmetry of the shape operator *A*. Therefore, the gradient of the function -*A*-2 is

$$\nabla \parallel A \parallel^2 = 2 \sum\_{i=1}^{2n} \left( \nabla A \right) \left( e\_{i\nu} A e\_i \right)\_{\nu}$$

and, consequently, Equation (13), takes the form

$$\sum\_{i=1}^{2n} \left(\nabla A\_f\right) \left(\varepsilon\_i, \varepsilon\_i\right) = -\nabla f - A^2 \left(\nabla f\right) - \left\|A\right\|^2 \nabla f - 2f \sum\_{i=1}^{2n} \left(\nabla A\right) \left(\varepsilon\_i, A\varepsilon\_i\right). \tag{14}$$

Using Equations (11) and (14), we conclude

$$-\left\|A\right\|^{2}\nabla f = -\sum\_{i=1}^{2n} \left(\nabla^{2}A\right) \left(e\_{i}, e\_{i}, u\right) + \sum\_{i=1}^{2n} \left(\nabla A\right) \left(e\_{i}, l\varepsilon\_{i}\right). \tag{15}$$

Now, using Equations (6) and (8) and the Ricci identity, we have

$$\sum\_{i=1}^{2n} \left(\nabla^2 A\right)(\varepsilon\_i, \varepsilon\_i, \mu) = \sum\_{i=1}^{2n} \left(\nabla^2 A\right)(\varepsilon\_i, \mu, \varepsilon\_i) = \sum\_{i=1}^{2n} \left(R(\varepsilon\_i, \mu) A \varepsilon\_i - A R(\varepsilon\_i, \mu) \varepsilon\_i\right).$$

which on using Equation (5) and *trA* = 0 gives

$$\sum\_{i=1}^{2n} \left(\nabla^2 A\right) \left(\varepsilon\_{i\prime} \varepsilon\_{i\prime} u\right) = -\left\|A\right\|^2 A u + 2nAu. \tag{16}$$

Also, using *JA* = *A J*, we have

$$\begin{aligned} \sum\_{i=1}^{2n} \left( \nabla A \right) \left( \mathbf{e}\_{i\prime} \, \mathrm{J} \mathbf{e}\_{i} \right) &= \sum\_{i=1}^{2n} \left( \nabla\_{\mathbf{e}\_{i}} \mathrm{J} \mathbf{e}\_{i} - A \left( \left( \nabla f \right) \left( \mathbf{e}\_{i\prime} \mathbf{e}\_{i} \right) + \int \left( \nabla\_{\mathbf{e}\_{s}} \mathbf{e}\_{i} \right) \right) \right) \\ &= \sum\_{i=1}^{2n} \left( \left( \nabla f \right) \left( \mathbf{e}\_{i\prime} \, \mathrm{A} \mathbf{e}\_{i} \right) - A \left( \left( \nabla f \right) \left( \mathbf{e}\_{i\prime} \mathbf{e}\_{i} \right) \right) \right) \end{aligned}$$

which on using (v) of Lemma 1, yields

$$\sum\_{i=1}^{2n} \left(\nabla A\right) \left(e\_i, Je\_i\right) = \left\|A\right\|^2 v - 2nAu. \tag{17}$$

Finally, using (vi) of Lemma 1 and Equations (16) and (17) in Equation (15), we ge<sup>t</sup>

$$-\left\|A\right\|^2 \left(-Au+v\right) = \left\|A\right\|^2 Au - 2nAu + \left\|A\right\|^2 v - 2nAu$$

and this proves the Lemma.

## **3. Proof of Theorem 1**

As the contact vector field *v* is a conformal vector field by Lemma 4, we have *JA* = *A J*, that is, Ψ = 0. Then Lemma 3 implies

$$\int\_{M} \left(1 - f^2\right) \left\| A \right\|^2 = \int\_{M} \left(2n - 2n(2n + 1)f^2\right) \,\mu$$

that is,

*M* 1 − *f* 2 -*A*-2 = *M* <sup>2</sup>*n*(<sup>1</sup> − *f* 2) − 4*n f* 2 . (18)

Therefore, we ge<sup>t</sup> the inequality

$$\int\_{\mathcal{M}} \left(1 - f^2\right) \left\|A\right\|^2 \le \int\_{\mathcal{M}} 2n(1 - f^2).$$

Moreover, if the equality holds, then by Equation (18), we ge<sup>t</sup> *f* = 0, which in view of (vi), (vii) of Lemma 1, we conclude that *Au* = *v* and that the contact vector field *v* is a unit vector field. As *v* is a conformal vector field, combining *Au* = *v* with Lemma 5, we ge<sup>t</sup> -*A*-2 *v* = 2*nv*, that is, -*A*-2 = 2*n*. Therefore, *M* is a Clifford hypersurface (cf. [5]).

The converse is trivial.

## **4. Proof of Theorem 2**

As the Reeb function *f* is a constant along the integral curves of the contact vector field *v*, we have *v*(*f*) = 0. Note that div(*f v*) = *v*(*f*) + *f*div*v* = −2*n f* 2, which on integration gives *f* = 0, and consequently, the contact vector field *v* is a unit vector field. Then Lemma 3, implies

$$\int\_{M} \left\| A \right\|^{2} = \int\_{M} \left( 2n + \frac{1}{2} \left\| \Psi \right\|^{2} \right) \,\tag{19}$$

which proves the inequality

$$\int\_{\mathcal{M}} \left\| \mathcal{A} \right\|^2 \geq 2n \operatorname{Vol}(\mathcal{M}).$$

If the equality holds, then by Equation (4.1), we ge<sup>t</sup> that Ψ = 0, that is, *JA* = *A J*. Thus, by Lemma 4, the contact vector field *v* is a conformal vector field. Using Lemma 5, we ge<sup>t</sup> -*A*-2 = 2*n*. Therefore, *M* is a Clifford hypersurface (cf. [5]).

The converse is trivial.

**Author Contributions:** Conceptualization, S.D. and I.A.-D.; methodology, S.D.; software, I.A.-D.; validation, S.D. and I.A.-D.; formal analysis, S.D.; investigation, I.A.-D.; resources, S.D.; data curation, I.A.-D.; writing—original draft preparation, S.D. and I.A.-D.; writing—review and editing, S.D. and I.A.-D.; visualization, I.A.-D.; supervision, S.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding. **Acknowledgments:** This work is supported by King Saud University, Deanship of Scientific Research, College of Science Research Center.

**Conflicts of Interest:** The authors declare no conflict of interest.
