Hypersurfaces of a Sasakian Manifold

: We extend the study of orientable hypersurfaces in a Sasakian manifold initiated by Watanabe. The Reeb vector ﬁeld ξ of the Sasakian manifold induces a vector ﬁeld ξ T on the hypersurface, namely the tangential component of ξ to hypersurface, and it also gives a smooth function ρ on the hypersurface, which is the projection of the Reeb vector ﬁeld on the unit normal. First, we ﬁnd volume estimates for a compact orientable hypersurface and then we use them to ﬁnd an upper bound of the ﬁrst nonzero eigenvalue of the Laplace operator on the hypersurface, showing that if the equality holds then the hypersurface is isometric to a certain sphere. Also, we use a bound on the energy of the vector ﬁeld ∇ ρ on a compact orientable hypersurface in a Sasakian manifold in order to ﬁnd another geometric condition (in terms of mean curvature and integral curves of ξ T ) under which the hypersurface is isometric to a sphere. Finally, we study compact orientable hypersurfaces with constant mean curvature in a Sasakian manifold and ﬁnd a sharp upper bound on the ﬁrst nonzero eigenvalue of the Laplace operator on the hypersurface. In particular, we show that this upper bound is attained if and only if the hypersurface is isometric to a sphere, provided that the Ricci curvature of the hypersurface along ∇ ρ has a certain lower bound.


Introduction
It is well known that Sasakian manifolds are considered the odd dimensional analogue of Kählerian manifolds and therefore Sasakian space forms can be regarded as the counterpart of complex space forms [1]. One of the important branches of differential geometry is the submanifold theory and here some very challenging topics come from the geometry of submanifolds in real, complex and Sasakian space forms. In this setting, in many studies, a key role is played by the Gauss, Codazzi and Ricci Equations for submanifolds, as these take a manageable form. The differential geometry of hypersurfaces in a complex space form has been widely studied over the years (see, e.g., [2][3][4][5][6][7][8][9][10][11]), but though Sasakian manifolds are very important due to their elegant geometry (see the excellent monograph [12]) as well as their important applications in theoretical physics (see [13] and the references therein), not as many studies have been realized for hypersurfaces in a Sasakian ambient space. In this context, a well known result is that of Watanabe (cf. [14]), who used the Obata's differential Equation (cf. [15,16]) in order to prove that a complete and connected totally umbilical hypersurface of a (2n + 1)-dimensional Sasakian manifold of constant mean curvature H is isometric with a sphere of radius carried out in [17], where the author found two much stronger theorems that give sufficient conditions for a hypersurface in a Sasakian manifold to be isometric to a sphere.
Given an orientable hypersurface M of a (2n + 1)-dimensional Sasakian manifold M (ϕ, ξ, η, g) with unit normal vector field N and shape operator A, then there are two vector fields ξ T and t naturally defined on the hypersurface M. More exactly, ξ T is the tangential component of the Reeb vector field ξ to M, while t is defined by ϕ (N) = −t. Recall that the study of real hypersurfaces of complex and Sasakian space forms become more convenient, owing to a simpler form of Gauss Equation for expression of curvature tensor field of hypersurface and also due to a handy form of Codazzi Equation, which is lacking in the study of hypersurfaces of a general Sasakian manifold.
In this paper, we show that focussing on the investigation of hypersurfaces in general Sasakian manifolds, this deficiency can be compensated by the power of the Reeb field. In Section 2, we derive basic formulae for an orientable hypersurface M of a (2n + 1)-dimensional Sasakian manifold M (ϕ, ξ, η, g). In Section 3, we find an estimate of volume for a compact orientable hypersurface M and use it together with an additional condition, namely the vector field ξ T is a principal direction, to find an upper bound for the first nonzero eigenvalue λ 1 of the Laplace operator on M. Also, we show that if the eigenvalue λ 1 attains this upper bound, then M is isometric to a sphere (cf. Theorems 1 and 2). Moreover, in the same section, we find other special conditions on a compact orientable hypersurface M that assure both constancy of the mean curvature H and the isometry of M with a certain sphere (cf. Theorem 3).
In Section 4, we use an upper bound for the energy of the gradient vector field ∇ρ and the condition that the mean curvature H of the compact orientable hypersurface M in a (2n + 1)-dimensional Sasakian manifold is constant along the integral curves of ξ T in order to show that in this case, H is also a constant and M is isometric with a sphere S 2n (r) with radius r = 1 √ 1+H 2 (cf. Theorem 4). Finally, in the last section of the paper, we study hypersurfaces of constant mean curvature H, also known as CMC-hypersurfaces, in a (2n + 1)-dimensional Sasakian manifold M (ϕ, ξ, η, g). We prove that on a compact orientable CMC-hypersurface M, if ξ T is a principal direction with constant principal curvature µ, then necessarily H = µ. We show that for a compact orientable CMC-hypersurface M with Aξ T = µξ T , the first nonzero eigenvalue λ 1 of the Laplace operator satisfies λ 1 ≤ 2n 1 + H 2 and the equality case holds for a certain hypersurface if and only if M is isometric to the sphere S 2n 1 √ 1+H 2 (cf. Theorem 5).

Preliminaries
Let M (ϕ, ξ, η, g) be a (2n + 1)-dimensional Sasakian manifold and ∇ be the Riemannian connection on M. Then we have (cf. [1,12]): for all X, Y ∈ X (M), where X (M) is the Lie algebra of smooth vector fields on M, while the covariant derivative ∇ϕ of ϕ is defined by We denote by R, Ric, Q the curvature tensor field, the Ricci tensor field and the Ricci operator of the Sasakian manifold M (ϕ, ξ, η, g). Then, for all X, Y ∈ X (M), we have (cf. [1]): where Ricci operator Q is a symmetric operator related to Ricci tensor Ric by [18] Ric (X, Y) = g Q (X) , Y .
Recall that for all vector fields X, Y orthogonal to ξ, we have Let M be an orientable hypersurface of a (2n + 1)-dimensional Sasakian manifold M (ϕ, ξ, η, g) with unit normal vector field N and shape operator A. Then we have the Gauss and Weingarten formulae (cf. [19]) where we denote by the same letter g the induced metric on M and by ∇ the Riemannian connection on M. Clearly, X (M) is the Lie algebra of smooth vector fields on M. Please note that owing to skew-symmetry of the operator ϕ, ϕ (N) is orthogonal to N and we get a smooth vector field t ∈ X (M), defined by ϕ (N) = −t. Let α be smooth 1-form on hypersurface M dual to t, that is, α(U) = g (t, U), U ∈ X (M). Also, we define an operator F : and it is easy to see that F is a skew-symmetric operator. Now, define a smooth function ρ on the hypersurface M by ρ = g (ξ, N). Then we have where ξ T ∈ X (M) is the tangential component of the Reeb vector field ξ. We denote by β the smooth 1-form on M dual to ξ T , i.e, β(U) = g(ξ T , U). Then using ϕ (N) = −t and Equations (1), (2), (7) and (8), it follows that and Also, using Equations (3), (6), (8) and ϕ (N) = −t, we conclude where ∇ρ is the gradient of the function ρ and the covariant derivative ∇F of F is given by for U, V ∈ X (M).
Please note that F is skew-symmetric and A symmetric, we have tr (F • A) = 0 and using first two equations in Equation (13), we conclude that where H is the mean curvature of the hypersurface M given by 2nH = trA. Thus, if M is a compact hypersurface of a (2n If λ 1 is the first nonzero eigenvalue of the Laplace operator ∆ acting on smooth functions on M, then using first Equation in (16) Using Equation (6), we have Moreover, the curvature tensor R of the hypersurface M is given by Choosing a local orthonormal frame {e 1 , ..., e 2n } on the hypersurface and using Equation (19), we get the following expression for the Ricci tensor Ric of the hypersurface M and consequently, we conclude Also, note that on an orientable hypersurface M of a (2n + 1)-dimensional Sasakian manifold M (ϕ, ξ, η, g) there are two globally defined orthogonal vector fields ξ T , t and they span a plane section of the tangent bundle of M. Thus, we have the sectional curvature K ξ T , t given by (21) Proof. Observe that Equation ϕ(N) = −t and (1) imply that g t, ξ T = 0. Thus, using Equation (13), we have Also, using Equation (15), we have Integrating this Equation and using Equation (22), we get the desired result.

Volume and First Eigenvalue Estimates
In this section, first we find the volume estimate for a compact hypersurface M of a (2n + 1)-dimensional Sasakian manifold M (ϕ, ξ, η, g) and use it to find an upper bound for the first nonzero eigenvalue λ 1 of the Laplace operator ∆ acting on smooth functions of hypersurface M under the assumption that ξ T is a principal direction.
where λ 1 is the first nonzero eigenvalue of the Laplace operator on M.
Proof. Using Equation (9) and (13), we have Integrating above Equation and using Lemma 2, we get Using inequality (17), we derive and this gives the required estimate. Now, suppose that the vector field ξ T is a principal direction of the hypersurface M with constant principal curvature µ, that is, Aξ T = µξ T . Then it turns out that t is also a principal direction as seen in the following.
Proof. Suppose Aξ T = µξ T holds for a constant µ. Then the last Equation in (13), gives ∇ρ = −µξ T − t, and using first two Equations in (13), we derive the following expression for the Hessian operator A ρ of the function ρ Note the Hessian operator A ρ is symmetric and thus, using above equation, we conclude that Taking U = ξ T in above equation and using Equation (10), we get If ρ = 0, then Equation (9) implies that both ξ T , t are unit vector fields and as ϕ (N) = −t, we get g ξ T , t = −g (ξ, ϕ (N)) = 0. Also, as ρ = 0, Equation (13) implies µξ T = −t and taking the inner product with t gives t 2 = 0 a contradiction to the fact that t is a unit vector field (under the assumption ρ = 0). Hence ρ = 0 on M. Whereas M is connected and ρ = 0, Equation (25) implies At = µt. Now, we shall prove the main results of this section.

Theorem 2.
Let M be a compact and connected orientable hypersurface of a (2n + 1)-dimensional Sasakian manifold M (ϕ, ξ, η, g). If Aξ T = µξ T for a constant µ and the squared length of shape operator is bounded above by 2nc for a constant c, 0 < c ≤ 1, then the first nonzero eigenvalue λ 1 of Laplace operator on M satisfies Moreover, if equality holds, then M is isometric to the sphere S 2n Proof. Suppose M is a compact and connected orientable hypersurface satisfying Aξ T = µξ T for a constant µ and A 2 ≤ 2nc, for a constant c, 0 < c ≤ 1. Then by Lemma 3, we have At = µt. Now, define operators Γ and Ψ by Then it follows that Γ is skew-symmetric and Ψ is symmetric with trΨ = 0 and we have where we have used tr (Γ • Ψ) = 0. Using Equation (24) (which holds for Aξ T = µξ T ), we conclude Please note that for a local orthonormal frame {e 1 , ..., e 2n } on M, using Equation (12) we have where we have used Equation (9). Thus, the above Equation and (27) give Also, using Equation (12) we have Combining Equations (26), (28) and (29), we conclude that is, On the other hand, Theorem 1 in our setting implies Thus, using Equation (30) in above inequality, we conclude Using next A 2 ≤ 2nc, we get However, the above inequality implies and as 1 − c ≥ 0, we conclude (4n If the equality λ 1 = 4n + 1 − c holds, then inequality (31) implies c = 1 and 1 − ρ 2 Ψ 2 = 0. If 1 − ρ 2 = 0, then ρ = ±1 together with first equation in (16) gives a contradiction. Hence on connected M, we get Ψ = 0, i.e., F • A = A • F, and consequently Γ = 2F • A. Then Equation (27), gives µF (U) = F (AU), i.e., F (AU − µU) = 0. Operating F in this last equation and using (11) , we conclude On the other hand, it is easy to see that At = µt implies Similarly, we obtain β (AU − µU) = 0 and consequently Equation (32), gives AU = µU, U ∈ X (M). Hence, M is totally umbilical hypersurface of the Sasakian manifold M (ϕ, ξ, η, g) with constant mean curvature µ and therefore isometric to the sphere S 2n 1 √ 1+µ 2 (cf. [14]).
In the next result, we use a bound on the squared length of the operator Ψ to find conditions on a complete and connected hypersurface of a Sasakian manifold to be isometric to a certain sphere. Theorem 3. Let M be a complete and connected orientable hypersurface of a (2n + 1)-dimensional Sasakian manifold M (ϕ, ξ, η, g) with mean curvature H. If Aξ T = µξ T for a constant µ and the squared length of operator Ψ satisfies then H is a constant and M is isometric to the sphere S 2n Proof. Since, Aξ T = µξ T , by Lemma 3, we have At = µt. Now, using Equation (30), we have that is, Now, using the bound on Ψ 2 , we conclude A 2 − 2nH 2 ≤ 0. However, Schwartz's inequality implies A 2 − 2nH 2 ≥ 0 and thus we have the equality A 2 = 2nH 2 , which holds if and only if where I stands for the identity operator. Please note that in view of Equation (33), we have Aξ T = Hξ T , and the hypothesis Aξ T = µξ T , we conclude If ξ T = 0, then by Equation (9), we get ρ 2 = 1 and consequently, t = 0. In this case, Equation (11) implies F 2 = −I. Now, using the second Equations in (13) and (33), we have F(U) = ρHU.
Combining the above relation with F 2 = −I we get H 2 = −1, hence a contradiction. Therefore, as M is connected, Equation (34) implies H = µ, that is, H is a constant. Thus, by Equation (33), we see that M is a totally umbilical hypersurface of constant mean curvature H and consequently it is isometric to the sphere S 2n 1 √ 1+H 2 .

A Bound on Energy of a Vector Field
Recall that on a compact Riemannian manifold (M, g), the energy of a smooth vector field u on M is defined by In this section, we use a bound on the energy of the vector field ∇ρ on a compact orientable hypersurface M to find another condition under which M is isometric to a sphere. Proof. As the mean curvature H is constant along the integral curves of ξ T , we have where {e 1 , ..., e 2n } is a local orthonormal frame on M. Now, using both Equations (18) and (35), we get As {e 1 , ..., e 2n , N} is local orthonormal frame on M and A is a symmetric operator, above equation takes the form However, taking into account (4), we derive (N, N).
Integrating above equation and using t 2 = 1 − ρ 2 and Lemma 2, we get Inserting next the above Equation in (39), we get If the energy E (∇ρ) satisfies the given condition in hypothesis, then Equation (40) reads Using Schwartz's inequality A 2 ≥ 2nH 2 in inequality (41), we conclude If ρ = 0, then ξ T , t is an orthonormal set globally defined on M and equations in (13) take the form Also, the two equations of (10) imply F (t) = 0 and F ξ T = 0 and thus using Equation (43), we get Thus, we compute R(ξ T , t)t = ∇ ξ T F (At) , and consequently, which is contrary to the hypothesis. Hence, ρ = 0 and due to the fact that M is connected, it follows that Equation (42) implies A 2 = 2nH 2 and this inequality holds if and only if Now, we proceed to show that H is a constant. In view of Equation (44), the equations in (13) change to ∇ U t = ρU + HF (U) , ∇ U ξ T = −F (U) + ρHU, ∇ρ = −Hξ T − t.
Using the above Equations, we compute the Hessian operator A ρ (U) = ∇ U ∇ρ and get the following Please note that A ρ is symmetric, and as such the above equation gives Choosing V = ξ T in the above equation and using hypothesis that H is constant along the integral curves of ξ T , we get If ξ T 2 = 0, we derive ρ = ±1, which gives a contradiction to the integral formula (16). Hence, as M is connected, Equation (45) implies U (H) = 0, U ∈ X (M). Therefore, H is a constant and by Equation (44) we get that M is a totally umbilical hypersurface of constant mean curvature H. Consequently, we deduce that M is isometric to S 2n Recall that the odd dimensional unit sphere S 2n+1 viewed as a hypersurface of the complex space C n+1 admits a standard Sasakian structure (ϕ, ξ, η, g) (see [1] for details). As a particular case of the above theorem, we have the following result.
Hence, all the conditions in the statement are met.

CMC-Hypersurfaces
In this section, we study compact and connected oriented hypersurfaces of constant mean curvature (briefly CMC-hypersurfaces) of a Sasakian manifold. It is interesting to note that on compact orientable CMC hypersurfaces, if ξ T is a principal direction, then Aξ T = Hξ t holds, where H is the constant mean curvature. We also find a sharp upper bound for the first nonzero eigenvalue of the Laplace operator on compact and orientable CMC-hypersurfaces with ξ T a principal direction. These two equations imply If ρ = 0, then Equation (9) implies that the set ξ T , t is an orthonormal set. However, the last equation in (13) and Aξ t = µξ T , gives µξ T = −t. Taking the inner product in this last equation, gives t 2 = 0 a contradiction. Hence, ρ = 0 and consequently, Equation (46) confirms that H = µ.
Also, as Aξ t = µξ T , we have g Aξ t , t = 0 and Lemma 2 implies Inserting this equation in inequality (48), we conclude Now, using the argument given in the proof of Lemma 4, we see that ρ = 0. Hence, from above inequality, we conclude λ 1 ≤ 2n 1 + H 2 .