A Note on Minimal Hypersurfaces of an Odd Dimensional Sphere

: We obtain the Wang-type integral inequalities for compact minimal hypersurfaces in the unit sphere S 2 n + 1 with Sasakian structure and use these inequalities to ﬁnd two characterizations of minimal Clifford hypersurfaces in the unit sphere S 2 n + 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 (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 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, 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, In this paper, we are interested in studying compact minimal hypersurfaces of the unit sphere S 2n+1 using the Sasakian structure (ϕ, ξ, η, g) (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(ξ, N), 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 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, 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 M A 2 ≥ nVol(M), 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.

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 (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 2n+1 as a totally umbilical real hypersurface of the complex space form (C n+1 , J, , ), where J is the complex structure and , is the Euclidean Hermitian metric. The Sasakian structure (ϕ, ξ, η, g) on S 2n+1 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 2n+1 as real hypersurface of C n+1 and they satisfy (cf. [13]) where X, Y are smooth vector fields, ∇ is Riemannian connection on S 2n+1 and the covariant derivative 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]) where X(M) 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 and holds for a local orthonormal frame {e 1 , . . . , e 2n } on the minimal hypersurface M.
Using the Sasakian structure (ϕ, ξ, η, 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(ξ, N), 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 get 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 2n+1 . We denote by u = ξ T 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(X, u) and β(X) = g(X, v), X ∈ X(M). For X ∈ X(M), we set JX = (ϕX) T 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)N. Thus, we get 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 2n+1 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 2n+1 . Then, M admits the structure (J, u, v, α, β, f , g) satisfying where ∇ f is the gradient of the Reeb function f .
Let ∆ f be the Laplacian of the Reeb function f of the minimal hypersurface M of the unit sphere S 2n+1 defined by ∆ f = div∇ f . Then using Lemma 1 and 1 2 ∆ f 2 = f ∆ f + ∇ 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 On the hypersurface M of the unit sphere S 2n+1 , we define a (1, 1) tensor field Ψ = J A − AJ, then it follows that g(ΨX, Y) = g(X, ΨY), X, Y ∈ X(M), that is, Ψ is symmetric and that trΨ = 0. Next, we prove the following: Lemma 3. Let M be a compact minimal hypersurface of the unit sphere S 2n+1 . Then, Proof. Using Equation (7), we have Ric(v, v) = (2n − 1) v 2 − Av 2 . Now, using Lemma 1, we get which on using the fact that trΨ = 0, gives Also, using (iii) of Lemma 1, we have which together with second equation in (iv) of Lemma 1 and the fact that trJ A = 0, implies Note that second equation in (iv) of Lemma 1 also gives divv = −2n f . Now, inserting above values in the following Yano's integral formula (cf. [15]) Also, (vi) of Lemma 1, gives gives Inserting above value of Au 2 in Equation (9), yields Integrating (ii) of Lemma 2, we get which together with v 2 = 1 − f 2 and Equation (10) proves the integral formula. Proof. Suppose that AJ = J A. Then, using Lemma 1 and symmetry of shape operator A and skew symmetry of the operator J, we have 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 which on using Lemma 1, gives Choosing a local orthonormal frame {e 1 , . . . , e 2n } on the minimal hypersurface M and taking X = Y = e i in above equation and summing, we get ρ = − f . This gives g(AJX − J AX, Y) = 0, X, Y ∈ X(M), that is, AJ = J A.

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 Proof. Suppose v is a conformal vector field. Then, by Lemma 4, we have J A = AJ . Note that for the Hessian operator A f of the Reeb function f using Lemma 1, we have which on using (vi) of Lemma 1, gives Taking covariant derivative in above equation gives where we used (iv) of Lemma 1. Now, on taking a local orthonormal frame {e 1 , . . . , e 2n } on the minimal hypersurface M and taking X = Y = e i in above equation and summing, we get Note that for the minimal hypersurface, we have Thus, the previous equation takes the form Now, using the definition of Hessian operator, we have and we conclude where Q is the Ricci operator defined by Ric(X, Y) = g(QX, Y), X, Y ∈ X(M). Using (i) of Lemma 2, we have and, consequently, using Q(X) = (2n − 1)X − A 2 X (outcome of Equation (7)), the Equation (12) takes the form Also, note that where we have used Equation (6) and symmetry of the shape operator A. Therefore, the gradient of the function A 2 is and, consequently, Equation (13), takes the form Using Equations (11) and (14), we conclude Now, using Equations (6) and (8) and the Ricci identity, we have which on using Equation (5) and trA = 0 gives Also, using J A = AJ, we have which on using (v) of Lemma 1, yields Finally, using (vi) of Lemma 1 and Equations (16) and (17) in Equation (15), we get and this proves the Lemma.

Proof of Theorem 1
As the contact vector field v is a conformal vector field by Lemma 4, we have J A = AJ, that is, Therefore, we get the inequality 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.

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 divv = −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 which proves the inequality M A 2 ≥ 2nVol(M).
If the equality holds, then by Equation (4.1), we get that Ψ = 0, that is, J A = 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.