A Note on Minimal Hypersurfaces of an Odd Dimensional Sphere

Abstract

We obtain the Wang-type integral inequalities for compact minimal hypersurfaces in the unit sphere $S^{2n+1}$ with Sasakian structure and use these inequalities to find two characterizations of minimal Clifford hypersurfaces in the unit sphere $S^{2n+1}$.

1. Introduction

Let M be a compact minimal hypersurface of the unit sphere $S^{n+1}$ with shape operator A. In his pioneering work, Simons [1] has shown that on a compact minimal hypersurface M of the unit sphere $S^{n+1}$ either $A=0$ (totally geodesic), or ${∥A∥}^2=n$, or ${∥A∥}^2\left(p\right)>n$ for some point $p\in M$, 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

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

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 $S^{n+1}$ is to characterize minimal Clifford hypersurfaces. Minimal hypersurfaces have also been studied in (cf. [6,7,8]). 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 [3,9,10,11], 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 $S^{n+1}$ with two distinct principal curvatures, one of them being simple and obtained the following integral inequality,

$${\int }_M{∥A∥}^2\le nVol\left(M\right),$$

where $Vol\left(M\right)$ 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 $S^{2n+1}$ using the Sasakian structure $\left(\phi ,\xi ,\eta ,g\right)$ (cf. [13]) and finding characterizations of minimal Clifford hypersurface of $S^{2n+1}$. On a compact minimal hypersurface M of the unit sphere $S^{2n+1}$, we denote by N the unit normal vector field and define a smooth function $f=g(\xi ,N)$, which we call the Reeb function of the minimal hypersurface M. Also, on the hypersurface M, we have a smooth vector field $v=\phi \left(N\right)$, which we call the contact vector field of the hypersurface (v being orthogonal to $\xi$ 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 $S^{2n+1}$ 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 $S^{2n+1}$ with Reeb function f and contact vector field v a conformal vector field on M. Then,

$${\int }_M(1-f^2){∥A∥}^2\le 2n{\int }_M\left(1-f^2\right)$$

and the equality holds if and only if M is isometric to the Clifford hypersurface $S^l\left(\sqrt{\frac{l}{2n}}\right)\times S^m\left(\sqrt{\frac{m}{2n}}\right)$, where $l+m=2n$.

Also in [12], Wang studied embedded compact minimal non-totally geodesic hypersurfaces in $S^{n+1}$ those are symmetric with respect to $n+2$ pair-wise orthogonal hyperplanes of $R^{n+2}$. If M is such a hypersurface, then it is proved that

$${\int }_M{∥A∥}^2\ge nVol\left(M\right),$$

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 $S^n$ into two connected components.

In our next result, we consider compact immersed minimal hypersurface M of the unit sphere $S^{2n+1}$ 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 get another characterization of minimal Clifford hypersurface in the unit sphere $S^{2n+1}$. Precisely, we prove the following.

Theorem 2.

Let M be a compact minimal hypersurface of the unit sphere $S^{2n+1}$ with Reeb function f a constant along the integral curves of the contact vector field v. Then,

$${\int }_M{∥A∥}^2\ge 2nVol\left(M\right)$$

and the equality holds if and only if M is isometric to the Clifford hypersurface $S^l\left(\sqrt{\frac{l}{2n}}\right)\times S^m\left(\sqrt{\frac{m}{2n}}\right)$, where $l+m=2n$.

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 $(M,g)$ has local flow consisting of conformal transformations, which is equivalent to

$${\pounds }_{v}g=2\rho g.$$

(1)

The smooth function $\rho$ appearing in Equation (1) defined on M is called the potential function of the conformal vector field v. We denote by $(\phi ,\xi ,\eta ,g)$ the Sasakian structure on the unit sphere $S^{2n+1}$ as a totally umbilical real hypersurface of the complex space form $(C^{n+1},\overline{J},⟨,⟩)$, where $\overline{J}$ is the complex structure and $⟨,⟩$ is the Euclidean Hermitian metric. The Sasakian structure $(\phi ,\xi ,\eta ,g)$ on $S^{2n+1}$ consists of a $(1,1)$ skew symmetric tensor field $\phi$, a smooth unit vector field $\xi$, a smooth 1-form $\eta$ dual to $\xi$, and the induced metric g on $S^{2n+1}$ as real hypersurface of $C^{n+1}$ and they satisfy (cf. [13])

$${\phi }^2=-I+\eta \otimes \xi ,\eta \circ \phi =0,\eta \left(\xi \right)=1,g(\phi X,\phi Y)=g(X,Y)-\eta \left(X\right)\eta \left(Y\right),$$

(2)

and

$$\left(\overline{\nabla }\phi \right)(X,Y)=g(X,Y)\xi -\eta \left(Y\right)X,{\overline{\nabla }}_X\xi =-\phi X,$$

(3)

where $X,Y$ are smooth vector fields, $\overline{\nabla }$ is Riemannian connection on $S^{2n+1}$ and the covariant derivative

$$\left(\overline{\nabla }\phi \right)(X,Y)={\overline{\nabla }}_X\phi Y-\phi \left({\overline{\nabla }}_{X}Y\right).$$

We dente by N and A the unit normal and the shape operator of the hypersurface M of the unit sphere $S^{2n+1}$. 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])

$${\overline{\nabla }}_{X}Y={\nabla }_{X}Y+g(AX,Y),{\overline{\nabla }}_{X}N=-AX,X,Y\in \mathfrak{X}\left(M\right),$$

(4)

$$R(X,Y)Z=g(Y,Z)X-g(X,Z)Y+g(AY,Z)AX-g(AX,Z)AY,$$

(5)

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

(6)

where $\mathfrak{X}\left(M\right)$ is the Lie algebra of smooth vector fields and $R(X,Y)Z$ is the curvature tensor field of the hypersurface M. The Ricci tensor of the minimal hypersurface M of the unit sphere $S^{2n+1}$ is given by

$$Ric(X,Y)=(2n-1)g(X,Y)-g(AX,AY),X,Y\in \mathfrak{X}\left(M\right)$$

(7)

and

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

(8)

holds for a local orthonormal frame $\left\{e_1,\dots ,e_{2n}\right\}$ on the minimal hypersurface M.

Using the Sasakian structure $(\phi ,\xi ,\eta ,g)$ on the unit sphere $S^{2n+1}$, we analyze the induced structure on a hypersurface M of $S^{2n+1}$. First, we have a smooth function f on the hypersurface M defined by $f=g(\xi ,N)$, which we call the Reeb function of the hypersurface M, where N is the unit normal vector field. As the operator $\phi$ is skew symmetric, we get a vector field $v=\phi N$ defined on M, which we call the contact vector field of the hypersurface M. Note that the vector field v is orthogonal to $\xi$, and therefore lies in the contact distribution of the Sasakian manifold $S^{2n+1}$. We denote by $u={\xi }^T$ the tangential component of $\xi$ to the hypersurface M and, consequently, we have $\xi =u+fN$. Let $\alpha$ and $\beta$ be smooth 1-forms on M dual to the vector fields u and v, respectively, that is, $\alpha \left(X\right)=g(X,u)$ and $\beta \left(X\right)=g(X,v)$, $X\in \mathfrak{X}\left(M\right)$. For $X\in \mathfrak{X}\left(M\right)$, we set $JX={\left(\phi X\right)}^T$ the tangential component of $\phi X$ to the hypersurface, which gives a skew symmetric $(1,1)$ tensor field J on the hypersurface M. It follows that $\phi X=JX-\beta \left(X\right)N$. Thus, we get a structure $\left(J,u,v,\alpha ,\beta ,f,g\right)$ on the hypersurface M and using properties in Equations (2) and (3) of the Sasakian structure $(\phi ,\xi ,\eta ,g)$ on the unit sphere $S^{2n+1}$ and Equation (4), it is straightforward to see that the structure $\left(J,u,v,\alpha ,\beta ,f,g\right)$ on the hypersurface M has the properties described in the following Lemma.

Lemma 1.

Let M be a hypersurface of the unit sphere $S^{2n+1}$. Then, M admits the structure $\left(J,u,v,\alpha ,\beta ,f,g\right)$ satisfying

(i) 

$J^2=-I+\alpha \otimes u+\beta \otimes v$,

(ii) 

$Ju=-fv$, $Jv=fu$,

(iii) 

$g(JX,JY)=g(X,Y)-\alpha \left(X\right)\alpha \left(Y\right)-\beta \left(X\right)\beta \left(Y\right)$,

(iv) 

${\nabla }_{X}u=-JX+fAX$, ${\nabla }_{X}v=-fX-JAX$,

(v) 

$\left(\nabla J\right)(X,Y)=g(X,Y)u-\alpha \left(Y\right)X+g(AX,Y)v-\beta \left(Y\right)AX$,

(vi) 

$\nabla f=-Au+v$,

(vii) 

${∥u∥}^2={∥v∥}^2=(1-f^2)$, $g(u,v)=0$,

where $\nabla f$ is the gradient of the Reeb function f.

Let $\Delta f$ be the Laplacian of the Reeb function f of the minimal hypersurface M of the unit sphere $S^{2n+1}$ defined by $\Delta f=\mathrm{div}\nabla f$. Then using Lemma 1 and $\frac{1}{2}\Delta f^2=f\Delta f+{∥\nabla f∥}^2$ and Equations (6) and (8), we get the following:

Lemma 2.

Let M be a minimal hypersurface of the unit sphere $S^{2n+1}$. Then, the Reeb function f satisfies

(i) 

$\Delta f=-\left(2n+{∥A∥}^2\right)f$,

(ii) 

$\frac{1}{2}\Delta f^2=-\left(2n+{∥A∥}^2\right)f^2+{∥\nabla f∥}^2$.

On the hypersurface M of the unit sphere $S^{2n+1}$, we define a $(1,1)$ tensor field $\mathrm{\Psi }=JA-AJ$, then it follows that $g(\mathrm{\Psi }X,Y)=g(X,\mathrm{\Psi }Y)$, $X,Y\in \mathfrak{X}\left(M\right)$, that is, $\mathrm{\Psi }$ is symmetric and that $tr\mathrm{\Psi }=0$. Next, we prove the following:

Lemma 3.

Let M be a compact minimal hypersurface of the unit sphere $S^{2n+1}$. Then,

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

Proof. 

Using Equation (7), we have $Ric(v,v)=(2n-1){∥v∥}^2-{∥Av∥}^2$. Now, using Lemma 1, we get

$$\left({\pounds }_{v}g\right)(X,Y)=-2fg(X,Y)-g(\mathrm{\Psi }X,Y),$$

which on using the fact that $tr\mathrm{\Psi }=0$, gives

$${\left|{\pounds }_{v}g\right|}^2=8n{f}^2+{∥\mathrm{\Psi }∥}^2.$$

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

$${∥JA∥}^2={∥A∥}^2-{∥Au∥}^2-{∥Av∥}^2,$$

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

$${∥\nabla v∥}^2=2n{f}^2+{∥A∥}^2-{∥Au∥}^2-{∥Av∥}^2.$$

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

$$\mathrm{div}v=-2nf.$$

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

$${\int }_M\left(Ric(v,v)+\frac{1}{2}{\left|{\pounds }_{v}g\right|}^2-{∥\nabla v∥}^2-{\left(\mathrm{div}v\right)}^2\right)=0,$$

we get

$${\int }_M\left((2n-1){∥v∥}^2+2n{f}^2+\frac{1}{2}{∥\mathrm{\Psi }∥}^2-{∥A∥}^2+{∥Au∥}^2-4n^2{f}^2\right)=0.$$

(9)

Also, (vi) of Lemma 1, gives $Au=v-\nabla f$, that is, ${∥Au∥}^2={∥v∥}^2+{∥\nabla f∥}^2-2v\left(f\right)$, which on using $\mathrm{div}\left(fv\right)=v\left(f\right)+f\mathrm{div}v=v\left(f\right)-2n{f}^2$, gives

$${∥Au∥}^2={∥v∥}^2+{∥\nabla f∥}^2-2\mathrm{div}\left(fv\right)-4n{f}^2.$$

Inserting above value of ${∥Au∥}^2$ in Equation (9), yields

$${\int }_M\left(2n{∥v∥}^2-2n{f}^2+\frac{1}{2}{∥\mathrm{\Psi }∥}^2-{∥A∥}^2+{∥\nabla f∥}^2-4n^2{f}^2\right)=0.$$

(10)

Integrating (ii) of Lemma 2, we get

$${\int }_M{∥\nabla f∥}^2={\int }_M\left(2n+{∥A∥}^2\right)f^2,$$

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 $S^{2n+1}$. Then, the contact vector field v is a conformal vector field if and only if $JA=AJ$.

Proof. 

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

$$\left({\pounds }_{v}g\right)(X,Y)=g({\nabla }_{X}v,Y)+g({\nabla }_{Y}v,X)=-2fg(X,Y),X\in \mathfrak{X}\left(M\right),$$

which proves that v is a conformal vector field with potential function $-f$. Conversely, suppose v is conformal vector field with potential function $\rho$. Then, using Equation (1), we have

$$\left({\pounds }_{v}g\right)(X,Y)=g({\nabla }_{X}v,Y)+g({\nabla }_{Y}v,X)=2\rho g(X,Y),$$

which on using Lemma 1, gives

$$g(-JAX-fX,Y)+g(-JAY-fY,X)=2\rho g(X,Y),$$

that is,

$$g(AJX-JAX,Y)=2(\rho +f)g(X,Y).$$

Choosing a local orthonormal frame $\left\{e_1,\dots ,e_{2n}\right\}$ on the minimal hypersurface M and taking $X=Y=e_i$ in above equation and summing, we get $\rho =-f$. This gives $g(AJX-JAX,Y)=0$, $X,Y\in \mathfrak{X}\left(M\right)$, that is, $AJ=JA$. ☐

Lemma 5.

Let M be a minimal hypersurface of the unit sphere $S^{2n+1}$. If the contact vector field v is a conformal vector field on M, then

$$Au=\frac{{∥A∥}^2}{2n}v.$$

Proof. 

Suppose v is a conformal vector field. Then, by Lemma 4, we have $JA=AJ$ . Note that for the Hessian operator $A_f$ of the Reeb function f using Lemma 1, we have

$$A_f\left(X\right)={\nabla }_X\nabla f={\nabla }_X(v-Au)=-JAX-fX-{\nabla }_{X}Au,X\in \mathfrak{X}\left(M\right),$$

which on using (vi) of Lemma 1, gives

$$A_f\left(X\right)=-f(X+A^{2}X)-(\nabla A)(X,u).$$

Taking covariant derivative in above equation gives

$$\begin{array}{ccc}\hfill \left(\nabla A_f\right)(X,Y)& =& -X\left(f\right)((Y+A^{2}Y)-f\left(\nabla A^2\right)(X,Y)-\left({\nabla }^{2}A\right)(X,Y,u)\hfill \\ & & +\left(\nabla A\right)(Y,JX)-f\left(\nabla A\right)(Y,AX),\hfill \end{array}$$

where we used (iv) of Lemma 1. Now, on taking a local orthonormal frame $\left\{e_1,\dots ,e_{2n}\right\}$ on the minimal hypersurface M and taking $X=Y=e_i$ in above equation and summing, we get

$$\begin{array}{ccc}\hfill \sum _{i=1}^{2n}\left(\nabla A_f\right)(e_i,e_i)& =& -\nabla f-A^2\nabla f-f\sum _{i=1}^{2n}\left(\nabla A^2\right)(e_i,e_i)-\sum _{i=1}^{2n}\left({\nabla }^{2}A\right)(e_i,e_i,u)\hfill \\ & & +\sum _{i=1}^{2n}\left(\nabla A\right)(e_i,J{e}_i)-f\sum _{i=1}^{2n}\left(\nabla A\right)(e_i,A{e}_i).\hfill \end{array}$$

Note that for the minimal hypersurface, we have

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

Thus, the previous equation takes the form

$${\sum }_{i=1}^{2n}\left(\nabla A_f\right)\left(e_i,e_i\right)=-\nabla f-A^2\nabla f-2f{\sum }_{i=1}^{2n}\left(\nabla A^2\right)\left(e_i,e_i\right)-{\sum }_{i=1}^{2n}\left({\nabla }^{2}A\right)\left(e_i,e_i,u\right)+{\sum }_{i=1}^{2n}\left(\nabla A\right)\left(e_i,J{e}_i\right).$$

(11)

Now, using the definition of Hessian operator, we have

$$R(X,Y)\nabla f=\left(\nabla A_f\right)(X,Y)-\left(\nabla A_f\right)(Y,X),$$

which gives

$$Ric(Y,\nabla f)=g\left(Y,\sum _{i=1}^{2n}\left(\nabla A_f\right)(e_i,e_i)\right)-Y\left(\Delta f\right)$$

and we conclude

$$Q(\nabla f)=-\nabla (\Delta f)+\sum _{i=1}^{2n}\left(\nabla A_f\right)(e_i,e_i),$$

(12)

where Q is the Ricci operator defined by $Ric(X,Y)=g(QX,Y)$, $X,Y\in \mathfrak{X}\left(M\right)$. Using (i) of Lemma 2, we have

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

and, consequently, using $Q\left(X\right)=(2n-1)X-A^{2}X$ (outcome of Equation (7)), the Equation (12) takes the form

$$\sum _{i=1}^{2n}\left(\nabla A_f\right)(e_i,e_i)=(2n-1)\nabla f-A^2\left(\nabla f\right)-2n\nabla f-{∥A∥}^2\nabla f-f\nabla {∥A∥}^2,$$

that is,

$$\sum _{i=1}^{2n}\left(\nabla A_f\right)(e_i,e_i)=-\nabla f-A^2\left(\nabla f\right)-{∥A∥}^2\nabla f-f\nabla {∥A∥}^2.$$

(13)

Also, note that

$$\begin{array}{ccc}\hfill X\left({∥A∥}^2\right)& =& X\left(\sum _{i=1}^{2n}g\left(A{e}_i,A{e}_i\right)\right)=2\sum _{i=1}^{2n}g\left(\left(\nabla A\right)(X,e_i),A{e}_i\right)\hfill \\ & =& 2\sum _{i=1}^{2n}g\left(X,\left(\nabla A\right)(e_i,A{e}_i)\right),\hfill \end{array}$$

where we have used Equation (6) and symmetry of the shape operator A. Therefore, the gradient of the function ${∥A∥}^2$ is

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

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

$$\sum _{i=1}^{2n}\left(\nabla A_f\right)(e_i,e_i)=-\nabla f-A^2\left(\nabla f\right)-{∥A∥}^2\nabla f-2f\sum _{i=1}^{2n}\left(\nabla A\right)(e_i,A{e}_i).$$

(14)

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

$$-{∥A∥}^2\nabla f=-\sum _{i=1}^{2n}\left({\nabla }^{2}A\right)(e_i,e_i,u)+\sum _{i=1}^{2n}\left(\nabla A\right)(e_i,J{e}_i).$$

(15)

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

$$\sum _{i=1}^{2n}\left({\nabla }^{2}A\right)(e_i,e_i,u)=\sum _{i=1}^{2n}\left({\nabla }^{2}A\right)(e_i,u,e_i)=\sum _{i=1}^{2n}\left(R(e_i,u)A{e}_i-AR(e_i,u)e_i\right),$$

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

$$\sum _{i=1}^{2n}\left({\nabla }^{2}A\right)(e_i,e_i,u)=-{∥A∥}^{2}Au+2nAu.$$

(16)

Also, using $JA=AJ$, we have

$$\begin{array}{ccc}\hfill \sum _{i=1}^{2n}\left(\nabla A\right)(e_i,J{e}_i)& =& \sum _{i=1}^{2n}\left({\nabla }_{e_i}JA{e}_i-A\left(\left(\nabla J\right)(e_i,e_i\right)+J\left({\nabla }_{e_o}{e}_i\right)\right)\hfill \\ & =& \sum _{i=1}^{2n}\left(\left(\nabla J\right)(e_i,A{e}_i)-A\left(\left(\nabla J\right)(e_i,e_i\right)\right),\hfill \end{array}$$

which on using (v) of Lemma 1, yields

$$\sum _{i=1}^{2n}\left(\nabla A\right)(e_i,J{e}_i)={∥A∥}^{2}v-2nAu.$$

(17)

Finally, using (vi) of Lemma 1 and Equations (16) and (17) in Equation (15), we get

$$-{∥A∥}^2\left(-Au+v\right)={∥A∥}^{2}Au-2nAu+{∥A∥}^{2}v-2nAu$$

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 $JA=AJ$, that is, $\mathrm{\Psi }=0$. Then Lemma 3 implies

$${\int }_M\left(1-f^2\right){∥A∥}^2={\int }_M\left(2n-2n(2n+1)f^2\right),$$

that is,

$${\int }_M\left(1-f^2\right){∥A∥}^2={\int }_M\left(2n(1-f^2)-4n{f}^2\right).$$

(18)

Therefore, we get the inequality

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

Moreover, if the equality holds, then by Equation (18), we get $f=0$, 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 get ${∥A∥}^{2}v=2nv$, that is, ${∥A∥}^2=2n$. 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 $v\left(f\right)=0$. Note that $\mathrm{div}\left(fv\right)=v\left(f\right)+f\mathrm{div}v=-2n{f}^2$, which on integration gives $f=0$, and consequently, the contact vector field v is a unit vector field. Then Lemma 3, implies

$${\int }_M{∥A∥}^2={\int }_M\left(2n+\frac{1}{2}{∥\mathrm{\Psi }∥}^2\right),$$

(19)

which proves the inequality

$${\int }_M{∥A∥}^2\ge 2nVol\left(M\right).$$

If the equality holds, then by Equation (4.1), we get that $\mathrm{\Psi }=0$, that is, $JA=AJ$. Thus, by Lemma 4, the contact vector field v is a conformal vector field. Using Lemma 5, we get ${∥A∥}^2=2n$. Therefore, M is a Clifford hypersurface (cf. [5]).

The converse is trivial.