On characterizations of hypersurfaces in a Sasakian space forms with commuting operators

Abstract Let be a real hypersurface in a Sasakian space form . In this paper, we prove that if holds on , then is a Hopf hypersurface, where and denote the Jacobi operator structure and the induced operator from the Lie derivative with respect to the induced normal vector field , respectively. We characterize such the Hopf hypersurfaces of .


Introduction
Let MðcÞ, be a Sasakian space form equipped with the metric g. Let M be a hypersurface of MðcÞ where the structural vector field � is tangent to M. Let ϕ and N denote the contact structure and the locally unit normal vector field on MðcÞ and M, respectively. Then, À ϕðNÞ ¼ U is a tangent vector field to M, which is called the induced normal vector field on M. Now, we consider the hypersurface M with the metric structure ðF; g; �; η; U; uÞ, which is induced from the contact metric and the contact structure ϕ of MðcÞ. If the plane spanned by the structure vector field � and the induced normal vector field U turns out to be an invariant subspace by A, where A is the shape operator of M, the hypersurface M is called a Hopf hypersurface (Abedi et al., 2012). Hypersurfaces in the Sasakian space forms were studied in (Abedi and Ilmakchi, 2016;Abedi & Ilmakchi, 2015).
The induced operator L U on a hypersurface M of the 2 À form L U g is defined by ðL U gÞðX; YÞ ¼ gðL U X; YÞ for any vector fields X and Y on M, where L U denotes the Lie derivative operator with respect to the induced normal vector field U.
In this paper, at first we show that the operator L U gives L U ¼ FA À AF on M, and the induced normal vector field U is Killing if L U ¼ 0. Indeed, we show that: For the curvature tensor field R on a real hypersurface M, we define the Jacobi operator R X by R X ¼ Rð:; XÞX with respect to a unit vector field X (Kim et al., 2014). Also, we obtain: and η of type ð1; 1Þ; ð0; 1Þ and ð1; 0Þ, respectively, in which satisfy where I denotes the field of identity transformations of the tangent spaces at the all points. These conditions imply that ϕ� ¼ 0 and η � ϕ ¼ 0, where the endomorphism ϕ has the rank of 2m at the every point in e M 2mþ1 . A manifold e M 2mþ1 , which is equipped with an almost contact structure ðϕ; �; ηÞ, is called an almost contact manifold and denoted by ð e M 2mþ1 ; ðϕ; �; ηÞÞ. Suppose that e M 2mþ1 is a manifold carrying an almost contact structure. The Riemannian metric g on e M 2mþ1 which satisfies gðϕX; ϕYÞ ¼ gðX; YÞ À ηðXÞηðYÞ; for all the vector fields X and Y, is said to be compatible with the almost contact structure, and ðϕ; �; η; gÞ is an almost contact metric structure on e M 2mþ1 . Note that putting Y ¼ �, yields ηðXÞ ¼ gðX; �Þ; for all the vector fields X tangent to e M 2mþ1 , which shows η is the metric dual to the characteristic vector field �.
A manifold e M 2mþ1 is said to be a contact manifold if it carries the global one-form η such that η^ðdηÞ m �0; everywhere on M. The one-form η is called the contact form.
A submanifold M of a contact manifold e M 2mþ1 tangent to � is called an invariant (resp. anti-invariant) submanifold if ϕðT p MÞ � T p M; "p 2 M (resp. ϕðT p MÞ � T ? p M; "p 2 M). A submanifold M tangent to � of the Riemannian contact manifold e M 2mþ1 is called a contact CR-submanifold if there exists a pair of the orthogonal differentiable distributions D and D ? on M, such that: ( M; ϕ; �; η; e gÞ be a ð2n þ 1Þ-dimensional contact manifold such that then e M is called a Sasakian manifold. By taking into account that, the plane section π of T e M is called a ϕ À section if ϕπ x � π x for any x 2 e M, a Sasakian space form is the Sasakian manifold of the constant ϕ À sectional curvature. The Riemannian curvature tensor field e R of the Sasakian space form is given by (Blair, 1976)  for any X; Y; Z 2 χð e MÞ.

Hypersurfaces in the Sasakian space form
Let ðM; gÞ be a real hypersurface tangent to � of the Sasakian space form MðcÞ and let N be a unit normal vector field on M. Then, we have where D is a ϕ-invariant subspace and D ? is a onedimensional subspace, that is spanned by U ¼ À ϕðNÞ, and is the orthogonal component of D.
Moreover, it is clear that ϕTM � TM � SpanN. Hence, we have for any tangent vector field X the following decomposition in the tangent and the normal components: (3:1) It is easily shown that F is a skew-symmetric linear endomorphism that acts on T x M. Since the structure vector field � is tangent to M, (3.1) implies Next, by applying ϕ to (3.1) and using (3.2), we also have We denote by Ñ and Ñ the Levi-Civita connections on M and M, respectively. Then the Gauss formula is given by for any vector fields X; Y tangent to M. Here and in the sequel h denotes the second fundamental form and A is the shape operator corresponding to the normal vector field N. Therefore, Definition 3.1. (Abedi et al., 2012) Let A be the shape operator of the hypersurface M in e MðcÞ and the plane spanned by f�; Ug be an invariant subspace of A. Then, the hypersurface M is called a Hopf hypersurface of e MðcÞ.
By taking the covariant derivative of both sides of the Equation (3.1) and comparing the tangent and the normal parts, we have On the other hand, since the structure vector � is tangent to M, we get Ñ X � ¼ FX; (3:7) A� ¼ U: (3:10)

Proof of the main theorems
In this section, let M be an immersed hypersurface in the Sasakian space form MðcÞ and take into the account all the previous assumptions. From the Gauss Equation (3.9), the Jacobi operator structure R U is given by for any vector field X on M. By applying the Equation (3.6), we have ðL U gÞðX; YÞ ¼ gððFA À AFÞX; YÞ for any vector fields X and Y on M. Hence, the induced operator L U from L U g is given by (4:2) for any vector field X on M. The last equation substitute into the (3.10) and use the (3.2), we verify that À c þ 3 2 gðFX; YÞ þ 2gðFX; A 2 YÞ ¼ ðXαÞuðYÞ À ðYαÞuðXÞ þ 2αgðFX; AYÞ: (4:4) By putting X = U into the above equation and taking (3.2), we obtain Xα ¼ ðUαÞuðXÞ; (4:5) in which, by taking derivative and applying (3.5), we get ðYðUαÞÞuðXÞ À ðXðUαÞÞuðYÞ À 2ðUαÞgðFAX; YÞ ¼ 0: (4:6) Also, put X ¼ U into the Equation (4.6) and use (3.2), we see XðUαÞ ¼ ðUαÞuðXÞ; (4:7) where, use (4.5) and (4.7) in (4.6), gives for any vector field X on M, ðUαÞFAX ¼ 0. If ðUαÞ�0, we have FX ¼ 0 for any vector field X on M, which is a contradiction. Hence, Uα ¼ 0. From (4.5) for any vector field X on M we have Xα ¼ 0. Therefore, α is a constant.
Since A is self adjoint, D and spanf�; Ug are the invariant subspaces under A, there exist the locally orthonormal frames X 1 ; . . . ; X 2nÀ 2 ; and fW 1 ; W 2 g for D and the space that is spanned by spanf�; Ug, respectively where We set for some 0 < θ < π=2. Note that � and U can not be the eigenvectors of A hence, cos θ and sin θ can not vanish. Lemma 4.3. (Abedi & Ilmakchi, 2015) Under the above conditions, γ 1 γ 2 ¼ À 1. where the multiplicities of γ 1 ; γ 2 and λ 1 ; λ 2 are 1 and n À 1, respectively.
Proof. If we denote by λ the eigenvalue corresponding to the eigenvector of A, which is orthogonal to U and �, then from (3.3) and (4.4) that λ satisfies λ 2 À αλ À c þ 3 4 ¼ 0; and consequently with respect to the Lemma 4.2, the shape operator A has at most four constant eigenvalues ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi α 2 þ 4 p 2 ; λ 1;2 ¼ α � ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi α 2 þ c þ 3 p 2 ; c�1: If the shape operator A has exactly two constant eigenvalues ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi α 2 þ 4 p 2 ; whose multiplicities are n, because of AF ¼ FA.
Similarly, If the shape operator A has exactly four constant eigenvalues ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi α 2 þ 4 p 2 ; λ 1;2 ¼ α � ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi from AF ¼ FA, the multiplicities are 1 and n, respectively.
In the rest of this section, we add the following Theorem.
Combining from the special case and the results of proofs in (Kim & Pak, 2007), we have Theorem 4.6. Let M 2n be an immersed hypersurface in the unit sphere S 2nþ1 , where AF ¼ FA and the shape operator A has exactly two constant eigenvalues. Then M is a locally isometric to S 2n 1 þ1 ðr 1 Þ � S 2n 2 þ1 ðr 2 Þ ðr 2 1 þ r 2 2 ¼ 1Þ; for some integers n 1 ; n 2 with n 1 þ n 2 ¼ n À 1. Now, we suppose that the shape operator A has exactly four constant eigenvalues. Let Z ¼ λ 1 � þ U then Z ? ¼ � À λ 1 U. We consider the distribution D 0 ¼ D � spanfZ ? g therefore D 0 ? ¼ spanfZg. then, gð½X; Z ? �; ZÞ ¼ 0 and it implies that ½X; Z ? � 2 D � spanfZ ? g. This shows the distribution D � spanfZ ? g is involutive in M. Now, we consider the integral submanifold M of the distribution D 0 in M. On the other hand, because Z is an 1dimensional distribution, thus is involutive. Also, its integral manifold is the integral curve such that ζ is a geodesic, that is, Ñ ζ 0 ζ 0 ¼ 0 because of the assumption ζ 0 ¼ Z. Proof. Let TM 0 ¼ D � spanfZ ? g and A 0 is the shape operator corresponding to the normal vector field Z. If X 2 D, we have Proof. According to the above assumptions, it is sufficient to show that Because ζ is the geodesic curve, so Ñ Z Z ¼ 0.
Whereas M 0 is a totally geodesic manifold in M, so Ñ TM 0 Z ¼ 0. On the other hand, we have gðÑ Z ? Z ? ; ZÞ ¼ gðÑ �À λ 1 U ð� À λ 1 UÞ; λ 1 � þ UÞ ¼ 0; and from the Lemma 4.8 for X; Y in TM 0 we have gðÑ X Y; ZÞ ¼ 0. In the other words Ñ Z TM 0 � TM 0 . Also gðÑ Z X; ZÞ ¼ À gðX; Ñ Z ZÞ ¼ 0; for all X in TM 0 , so Ñ TM 0 TM 0 � TM 0 . Hence, by the de Rham decomposition theorem (De Rham, 1952), M is a locally isometric to the Riemannian product of the totally geodesic manifold M 0 and ζ. for all the vector field X in M. This equation shows that either β ¼ 0 or AF À FA ¼ 0. If β ¼ 0 then M is the Hoph hypersurface and obviously L U ¼ 0. In the case of AF À FA ¼ 0, the Equation (4.2) shows that L U ¼ 0. Therefore, the Theorem 1.1 gives the results.

PUBLIC INTEREST STATEMENT
In this paper we introduce the Characterizations of hypersurfaces in Sasakian space forms with commuting operators. Older, the other author introduces this condition and similar conditions for hypersurfaces in complex space forms. In the following these conditions can be introduced in the other spaces, similar; Kenmotsu space forms or generalized Sasakian space forms.