Comparison of Differential Operators with Lie Derivative of Three-Dimensional Real Hypersurfaces in Non-Flat Complex Space Forms

In this paper, three-dimensional real hypersurfaces in non-flat complex space forms, whose shape operator satisfies a geometric condition, are studied. Moreover, the tensor field P = φA− Aφ is given and three-dimensional real hypersurfaces in non-flat complex space forms whose tensor field P satisfies geometric conditions are classified.


Introduction
A real hypersurface is a submanifold of a Riemannian manifold with a real co-dimensional one.Among the Riemannian manifolds, it is of great interest in the area of Differential Geometry to study real hypersurfaces in complex space forms.A complex space form is a Kähler manifold of dimension n and constant holomorphic sectional curvature c.In addition, complete and simply connected complex space forms are analytically isometric to complex projective space CP n if c > 0, to complex Euclidean space C n if c = 0, or to complex hyperbolic space CH n if c < 0. The notion of non-flat complex space form refers to complex projective and complex hyperbolic space when it is not necessary to distinguish between them and is denoted by M n (c), n ≥ 2.
Let J be the Kähler structure and ∇ the Levi-Civita connection of the non-flat complex space form M n (c), n ≥ 2. Consider M a connected real hypersurface of M n (c) and N a locally defined unit normal vector field on M. The Kähler structure induces on M an almost contact metric structure (φ, ξ, η, g).The latter consists of a tensor field of type (1, 1) φ called structure tensor field, a one-form η, a vector field ξ given by ξ = −JN known as the structure vector field of M and g, which is the induced Riemannian metric on M by G.Among real hypersurfaces in non-flat complex space forms, the class of Hopf hypersurfaces is the most important.A Hopf hypersurface is a real hypersurface whose structure vector field ξ is an eigenvector of the shape operator A of M .
Takagi initiated the study of real hypersurfaces in non-flat complex space forms.He provided the classification of homogeneous real hypersurfaces in complex projective space CP n and divided them into five classes (A), (B), (C), (D) and (E) (see [1][2][3]).Later, Kimura proved that homogeneous real hypersurfaces in complex projective space are the unique Hopf hypersurfaces with constant principal curvatures, i.e., the eigenvalues of the shape operator A are constant (see [4]).Among the above real hypersurfaces, the three-dimensional real hypersurfaces in CP 2 are geodesic hyperspheres of radius r, 0 < r < π 2 , called real hypersurfaces of type (A) and tubes of radius r, 0 < r < π 4 , over the complex quadric called real hypersurfaces of type (B).Table 1 includes the values of the constant principal curvatures corresponding to the real hypersurfaces above (see [1,2]).

Type
The study of Hopf hypersurfaces with constant principal curvatures in complex hyperbolic space CH n , n ≥ 2, was initiated by Montiel in [5] and completed by Berndt in [6].They are divided into two types: type (A), which are open subsets of horospheres (A 0 ), geodesic hyperspheres (A 1,0 ), or tubes over totally geodesic complex hyperbolic hyperplane CH n−1 (A 1,1 ) and type (B), which are open subsets of tubes over totally geodesic real hyperbolic space RH n .Table 2 includes the values of the constant principal curvatures corresponding to above real hypersurfaces for n = 2 (see [6]).
The Levi-Civita connection ∇ of the non-flat complex space form M n (c), n ≥ 2 induces on M a Levi-Civita connection ∇.Apart from the last one, Cho in [7,8] introduces the notion of the k-th generalized Tanaka-Webster connection ∇(k) on a real hypersurface in non-flat complex space form given by ∇(k) for all X, Y tangent to M , where k is a nonnull real number.The latter is an extension of the definition of generalized Tanaka-Webster connection for contact metric manifolds given by Tanno in [9] and satisfying the relation The following relations hold: In particular, if the shape operator of a real hypersurface satisfies φA + Aφ = 2kφ, the generalized Tanaka-Webster connection coincides with the Tanaka-Webster connection.
The k-th Cho operator on M associated with the vector field X is denoted by F(k) X and given by for any Y tangent to M.Then, the torsion of the k-th generalized Tanaka-Webster connection ∇(k) is given by for any X, Y tangent to M. Associated with the vector field X, the k-th torsion operator T (k) X is defined and given by The existence of Levi-Civita and k-th generalized Tanaka-Webster connections on a real hypersurface implies that the covariant derivative can be expressed with respect to both connections.Let K be a tensor field of type (1, 1); then, the symbols ∇K and ∇(k) K are used to denote the covariant derivatives of K with respect to the Levi-Civita and the k-th generalized Tanaka-Webster connection, respectively.Furthermore, the Lie derivative of a tensor field K of type (1, 1) with respect to Levi-Civita connection LK is given by for all X, Y tangent to M .Another first order differential operator of a tensor field K of type (1, 1) with respect to the k-th generalized Tanaka-Webster connection L(k) K is defined and it is given by for all X, Y tangent to M .Due to the existence of the above differential operators and derivatives, the following questions come up 1. Are there real hypersurfaces in non-flat complex space forms whose derivatives with respect to different connections coincide?2. Are there real hypersurfaces in non-flat complex space forms whose differential operator L(k) coincides with derivatives with respect to different connections?
The first answer is obtained in [10], where the classification of real hypersurfaces in complex projective space CP n ,n ≥ 3, whose covariant derivative of the shape operator with respect to the Levi-Civita connection coincides with the covariant derivative of it with respect to the k-th generalized Tanaka-Webster connection is provided, i.e., ∇ X A = ∇(k) X A, where X is any vector field on M. Next, in [11], real hypersurfaces in complex projective space CP n , n ≥ 3, whose Lie derivative of the shape operator coincides with the operator L(k) are studied, i.e., L X A = L(k) X A, where X is any vector field on M. Finally, in [12], the problem of classifying three-dimensional real hypersurfaces in non-flat complex space forms M 2 (c), for which the operator L(k) applied to the shape operator coincides with the covariant derivative of it, has been studied, i.e., L(k) X A = ∇ X A, for any vector field X tangent to M. In this paper, the condition , where X is any vector field on M is studied in the case of three-dimensional real hypersurfaces in M 2 (c).
The aim of the present paper is to complete the work of [11] in the case of three-dimensional real hypersurfaces in non-flat complex space forms M 2 (c).The equality X .Thus, the eigenspaces of A are preserved by the k-th torsion operator T (k) X , for any X tangent to M .First, three-dimensional real hypersurfaces in M 2 (c) whose shape operator A satisfies the following relation: for any X orthogonal to ξ are studied and the following Theorem is proved: Theorem 1.There do not exist real hypersurfaces in M 2 (c) whose shape operator satisfies relation (5).
Next, three-dimensional real hypersurfaces in M 2 (c) whose shape operator satisfies the following relation are studied: and the following Theorem is provided.
Theorem 2. Every real hypersurface in M 2 (c) whose shape operator satisfies relation ( 6) is locally congruent to a real hypersurface of type (A).
As an immediate consequence of the above theorems, it is obtained that Corollary 1.There do not exist real hypersurfaces in M 2 (c) such that L(k) X A = L X A, for all X ∈ TM.
Next, the following tensor field P of type (1, 1) is introduced: for any vector field X tangent to M. The relation P = 0 implies that the shape operator commutes with the structure tensor φ.Real hypersurfaces whose shape operator A commutes with the structure tensor φ have been studied by Okumura in the case of CP n , n ≥ 2, (see [13]) and by Montiel and Romero in the case of CH n , n ≥ 2 (see [14]).The following Theorem provides the above classification of real hypersurfaces in M n (c), n ≥ 2.
Theorem 3. Let M be a real hypersurface of M n (c), n ≥ 2.Then, Aφ = φA, if and only if M is locally congruent to a homogeneous real hypersurface of type (A).More precisely: In the case of CP n (A 1 ) a geodesic hypersphere of radius r , where 0 < r < π 2 , (A 2 ) a tube of radius r over a totally geodesic CP k ,(1 ≤ k ≤ n − 2), where 0 < r < π 2 .
In the case of CH n , (A 0 ) a horosphere in CH n , i.e., a Montiel tube, (A 1 ) a geodesic hypersphere or a tube over a totally geodesic complex hyperbolic hyperplane CH n−1 , (A 2 ) a tube over a totally geodesic CH k (1 Remark 1.In the case of three-dimensional real hypersurfaces in M 2 (c), real hypersurfaces of type (A 2 ) do not exist.
It is interesting to study real hypersurfaces in non-flat complex spaces forms, whose tensor field P satisfies certain geometric conditions.We begin by studying three-dimensional real hypersurfaces in M 2 (c) whose tensor field P satisfies the relation for any vector fields X, Y tangent to M. First, the following Theorem is proved: Theorem 4. Every real hypersurface in M 2 (c) whose tensor field P satisfies relation (7) for any X orthogonal to ξ and Y ∈ TM is locally congruent to a real hypersurface of type (A).
Next, we study three-dimensional real hypersurfaces in M 2 (c) whose tensor field P satisfies relation (7) for X = ξ, i.e., for any vector field Y tangent to M.Then, the following Theorem is proved: Theorem 5. Every real hypersurface in M 2 (c) whose tensor field P satisfies relation ( 8) is a Hopf hypersurface.
In the case of CP 2 , M is locally congruent to a real hypersurface of type (A) or to a real hypersurface of type (B) with α = −2k and in the case of CH 2 M is a locally congruent either to a real hypersurface of type (A) or to a real hypersurface of type (B) with α = 4 k .
This paper is organized as follows: in Section 2, basic relations and theorems concerning real hypersurfaces in non-flat complex space forms are presented.In Section 3, analytic proofs of Theorems 1 and 2 are provided.Finally, in Section 4, proofs of Theorems 4 and 5 are given.

Preliminaries
Throughout this paper, all manifolds, vector fields, etc. are considered of class C ∞ and all manifolds are assumed to be connected.
The non-flat complex space form M n (c), n ≥ 2 is equipped with a Kähler structure J and G is the Kählerian metric.The constant holomorphic sectional curvature c in the case of complex projective space CP n is c = 4 and in the case of complex hyperbolic space CH n is c = −4.The Levi-Civita connection of the non-flat complex space form is denoted by ∇.
Let M be a connected real hypersurface immersed in M n (c), n ≥ 2, without boundary and N be a locally defined unit normal vector field on M. The shape operator A of the real hypersurface M with respect to the vector field N is given by The Levi-Civita connection ∇ of the real hypersurface M satisfies the relation The Kähler structure of the ambient space induces on M an almost contact metric structure (φ, ξ, η, g) in the following way: any vector field X tangent to M satisfies the relation The tangential component of the above relation defines on M a skew-symmetric tensor field of type (1, 1) denoted by φ known as the structure tensor.The structure vector field ξ is defined by ξ = −JN and the 1-form η is given by η(X) = g(X, ξ) for any vector field X tangent to M. The elements of the almost contact structure satisfy the following relation: for all tangent vectors X, Y to M. Relation (9) implies φξ = 0, η(X) = g(X, ξ).
Because of ∇J = 0, it is obtained for all X, Y tangent to M.Moreover, the Gauss and Codazzi equations of the real hypersurface are respectively given by and for all vectors X, Y, Z tangent to M, where R is the curvature tensor of M.
The tangent space T p M at every point p ∈ M is decomposed as where D = ker η = {X ∈ T p M : η(X) = 0} and is called (maximal) holomorphic distribution (if n ≥ 3).Next, the following results concern any non-Hopf real hypersurface M in M 2 (c) with local orthonormal basis {U, φU, ξ} at a point p of M. Lemma 1.Let M be a non-Hopf real hypersurface in M 2 (c).The following relations hold on M: where α, β, γ, δ, µ, κ 1 , κ 2 , κ 3 are smooth functions on M and β = 0.
Remark 2. The proof of Lemma 1 is included in [15].
The Codazzi equation for X ∈ {U, φU} and Y = ξ implies, because of Lemma 1, the following relations: and for X = U and Y = φU The following Theorem refers to Hopf hypersurfaces.In the case of complex projective space CP n , it is given by Maeda [16], and, in the case of complex hyperbolic space CH n , it is given by Ki and Suh [17] (see also Corollary 2.3 in [18]).Theorem 6.Let M be a Hopf hypersurface in M n (c), n ≥ 2.Then, )φW.
(iii) If the vector field W satisfies AW = λW and AφW = νφW, then Remark 3. Let M be a three-dimensional Hopf hypersurface in M 2 (c).Since M is a Hopf hypersurface relation Aξ = αξ, it holds when α = constant.At any point p ∈ M, we consider a unit vector field W ∈ D such that AW = λW.Then, the unit vector field φW is orthogonal to W and ξ and relation AφW = νφW holds.Therefore, at any point p ∈ M, we can consider the local orthonormal frame {W, φW, ξ} and the shape operator satisfies the above relations.

Proofs of Theorems 1 and 2
Suppose that M is a real hypersurface in M 2 (c) whose shape operator satisfies relation (5), which because of the relation of k-th generalized Tanaka-Webster connection (1) becomes for any X ∈ D and for all Y ∈ TM.
Let N be the open subset of M such that The inner product of relation ( 20) for Y = ξ with ξ due to relation ( 13) implies δ = 0 and the shape operator on the local orthonormal basis {U, φU, ξ} becomes Aξ = αξ + βU, AU = γU + βξ and AφU = µφU.
Differentiation of γ = k with respect to φU taking into account that k is a nonzero real number implies (φU)γ = 0. Thus, relation (18) results, because of δ = µ = 0, in κ 1 = −β.Furthermore, relations ( 14)- (17) due to δ = 0 and relation (22 The inner product of Codazzi equation (11) for X = U and Y = ξ with U and ξ implies because of δ = 0 and relation (21), The Lie bracket of U and ξ satisfies the following two relations: A combination of the two relations above taking into account relations of Lemma 1 and ( 27) yields Suppose that k = κ 3 , then (φU)β = 0 and relation (26 Differentiation of the last one with respect to φU results, taking into account relation (25), in κ 3 = −α.The Riemannian curvature satisfies the relation Therefore, on M, 10) for X = U, Y = φU and Z = U because of relations ( 22) and ( 26) and A combination of the latter with relation (23) implies Differentiation of the above relation with respect to φU gives, due to relation (26) and k 2 = −αk − 3c 2 , If the ambient space is the complex projective space CP 2 with c = 4, then the above relation leads to a contradiction.If the ambient space is the complex hyperbolic space CH 2 with c = −4, combination of the latter relation with β 2 + k 2 = − 5c 8 yields c = 0, which is a contradiction.Thus, N is empty and the following proposition is proved: Proposition 1.Every real hypersurface in M 2 (c) whose shape operator satisfies relation ( 5) is a Hopf hypersurface.
Since M is a Hopf hypersurface, Theorem 6 and remark 3 hold.Relation (20) for X = W and for X = φW implies, respectively, Combination of the above relations results in If ν = α, then λ = α and relation (19) implies that ν is also constant.Therefore, the real hypersurface is locally congruent to a real hypersurface of type (B).Substitution of the values of for any Y tangent to M.
Let N be the open subset of M such that N = {p ∈ M : β = 0, in a neighborhood of p}.
The inner product of relation (34) for Y = ξ implies, taking into account relation ( 13), β = 0, which is impossible.Thus, N is empty and the following proposition has been proved Proposition 4. Every real hypersurface in M 2 (c) whose tensor field P satisfies relation ( 8) is a Hopf hypersurface.
Since M is a Hopf hypersurface, Theorems 6 and 3 hold.Relation (34) for Y = W implies, due to AW = λW and AφW = νφW, (λ We have two cases: Case I: Supposing that λ = ν, then the above relation implies ν + λ = 2k.Relation (19) implies, due to the last one, that λ, ν are constant.Thus, M is locally congruent to a real hypersurface with three distinct principal curvatures.Therefore, it is locally congruent to a real hypersurface of type (B).
Thus, in the case of CP 2 , substitution of the eigenvalues of real hypersurface of type (B) in ν + λ = 2k implies α = −2k.In the case of CH 2 , substitution of the eigenvalues of real hypersurface of type (B) in ν + λ = 2k yields α = 4 k .
Case II: Supposing that λ = ν, then the structure tensor φ commutes with the shape operator A, i.e., Aφ = φA and, because of Theorem 3, M is locally congruent to a real hypersurface of type (A) and this completes the proof of Theorem 5.
As a consequence of Theorems 4 and 5, the following Corollary is obtained: A real hypersurface M in M 2 (c) whose tensor field P satisfies relation (7) is locally congruent to a real hypersurface of type (A).

Conclusions
In this paper, we answer the question if there are three-dimensional real hypersurfaces in non-flat complex space forms whose differential operator L (k) of a tensor field of type (1, 1) coincides with the Lie derivative of it.First, we study the case of the tensor field being the shape operator A of the real hypersurface.The obtained results complete the work that has been done in the case of real hypersurfaces of dimensions greater than three in complex projective space (see [11]).In Table 3 all the existing results and also provides open problems are summarized.Next, we study the above geometric condition in the case of the tensor field being P = Aφ − φA, which is introduced here.In Table 4, we summarize the obtained results.

Table 1 .
Principal curvatures of real hypersurfaces in CP 2 .

Table 2 .
Principal curvatures of real hypersurfaces in CH 2 .

Table 3 .
Results on condition L(k)X A = L X A. A = L X A, X ∈ TM does not exist does not exist open X

Table 4 .
Results on condition L(k)X P = L X P.