A characterization of ruled hypersurfaces in complex space forms in terms of the Lie derivative of shape operator

: In this paper, it is proved that if a non-Hopf real hypersurface in a nonﬂat complex space form of complex dimension two satisﬁes Ki and Suh’s condition (J. Korean Math. Soc., 32 (1995), 161–170), then it is locally congruent to a ruled hypersurface or a strongly 2-Hopf hypersurface. This extends Ki and Suh’s theorem to real hypersurfaces of dimension greater than or equal to three.


Introduction
A complete and simply connected complex space form of complex dimension n, denoted by M n (c), is complex analytically isometric to • a complex projective space CP n (c) if c > 0, • a complex Euclidean space C n if c = 0, • a complex hyperbolic space CH n (c) if c < 0, where the constant c is the holomorphic sectional curvature of a complex space form.Let M be a real hypersurface immersed in a complex space form.On M there exists a natural almost contact metric structure (see Section 2) induced from the complex structure on M n (c) and the normal vector field, respectively.Let ξ be the Reeb (or structure) vector field of the almost contact metric structure on M. If ξ is an eigenvector of the shape operator of a real hypersurface at each point, then the hypersurface is said to be Hopf.A real hypersurface is said to be non-Hopf if there exists at least one point on which ξ is not an eigenvector of the shape operator.For a real hypersurface, an eigenfunction of an eigenvector field of the shape operator is said to be a principal curvature.
The classification of real hypersurfaces in a nonflat complex space form M n (c), c 0, having constant principal curvatures is one of the most challenging problems in geometry of real hypersurfaces and is still an open question till now (the case of n = 2 has been settled completely in [20,22] for CP 2 and in [3] for CH 2 ), we refer the reader to [2,5,7] for some recent progress.Under certain additional geometric conditions, the above problem was considered a long time ago.Theorem 1.1.[11] Let M be a connected Hopf hypersurface of CP n (c).Then M has constant principal curvatures if and only if M is locally congruent to one of the following: (A 1 ) a geodesic sphere of radius r with 0 < r < π/ √ c; (A 2 ) a tube of radius r around a totally geodesic When the ambient space is the complex hyperbolic space CH n (c), the corresponding version of the above theorem is given as follows: Theorem 1.2.[1] Let M be a connected Hopf hypersurface of CH n (c).Then M has constant principal curvatures if and only if M is locally congruent to one of the following: (A 0 ) a self-tube, that is, a horosphere; (A 1,0 ) a geodesic hypersphere of radius r with 0 < r < ∞; (A 1,1 ) a tube of radius r around a totally geodesic complex hyperbolic hyperplane CH n−1 (c) with 0 < r < ∞; (A 2 ) a tube of radius r around a totally geodesic (B) a tube of radius r around a totally real totally geodesic hyperbolic space RH n (c/4) with 0 < r < ∞.
Applying the above Theorems 1.1 and 1.2, many characterization theorems of Hopf hypersurfaces having constant principal curvatures have been obtained (for examples see a great number of references in [4,17]).Among others, it has been proved in [21] that a real hypersurface in M n (c), c 0, can not be totally umbilical.Applying this, the shape operator (denoted by A throughout this paper) can not be a multiple of the metric tensor at each point of the hypersurface.Generalizing this, Ki, Kim and Lee in [9] presented a characterization theorem of type (A) hypersurfaces, where by type (A) hypersurfaces we mean those real hypersurfaces of type (A 1 ) or (A 2 ) in CP n , or of type (A 0 ), (A 1,0 ), (A 1,1 ), (A 2 ) in CH n .Specifically, Ki, Kim and Lee in [9] obtained that a real hypersurface in M n (c), c 0, satisfies for any vector fields X, Y orthogonal to ξ, then it is of type (A).
Let M be a non-Hopf hypersurface in M n (c), c 0. Assume that Ω of M is an open subset consisting those points on which the structure vector field ξ is not principal.One can set on Ω, where U is a unit vector field orthogonal to ξ and α = g(Aξ, ξ), and β is the length of Aξ − αξ.
Ki and Suh in [10, Theorem 2] presented a characterization theorem of ruled hypersurfaces.Where by ruled hypersurfaces we mean those real hypersurfaces having a foliation by totally geodesic complex hyperplanes (see [12]), or equivalently, the shape operator satisfies for any vector fields X, Y orthogonal to ξ (see [4]).In other words, Ki and Suh's result in [10, Theorem 2] can be rewritten as the following.
for any vector fields X, Y orthogonal to ξ and dα(ξ) 0, then M is locally congruent to a ruled real hypersurface.
In this paper, we aim to extend the above three theorems to real hypersurfaces of dimension three.
Theorem 1.5.Let M be a real hypersurface in CP 2 or CH 2 which is non-Hopf at every point.If M satisfies (1.5) for any vector fields X, Y orthogonal to ξ and dα(ξ) 0, then M is locally congruent to a ruled real hypersurface.
On a real hypersurface in a nonflat complex space form M n (c), c 0, one can define a distribution H = span{ξ, Aξ, A 2 ξ • • • } which is the smallest A-invariant distribution.A real hypersurface is said to be 2-Hopf if rank(H) = 2 and H is integrable (see [4,6,8]).In particular, a 2-Hopf hypersurface is said to be strongly 2-Hopf if in addition the spectrum of A| H is constant along the integral submanifolds of H (see [6]).By applying such a concept and deleting condition dα(ξ) 0 (it is not natural and unnecessary), we obtain a more comprehensive classification result.
Theorem 1.6.Let M be a real hypersurface in CP 2 or CH 2 which is non-Hopf at every point.Then M satisfies (1.5) for any vector fields X, Y orthogonal to ξ if and only if it is locally congruent to one of the following: • a ruled real hypersurface; • a strongly 2-Hopf hypersurface satisfying g(AX, Y) = ag(X, Y) for any vector fields X, Y orthogonal to ξ and a certain nowhere vanishing function a.
As an application of the proofs of the above results, we also extend Theorem 1.3 to real hypersurfaces of dimension three.
Theorem 1.7.A real hypersurface in CP 2 or CH 2 satisfies (1.2) for any vector fields X, Y orthogonal to ξ if and only if it is of type (A).

Preliminaries
Let M be a real hypersurface immersed in a complex space form M n (c) and N be a unit normal vector field of M. We denote by ∇ the Levi-Civita connection of the metric g of M n (c) and J the complex structure, respectively.Let g and ∇ be the induced metric from the ambient space and the Levi-Civita connection of g, respectively.Then the Gauss and Weingarten formulas are given respectively as the following: for any X, Y ∈ X(M), where A denotes the shape operator of M in M n (c) and X(M) is the set of all tangent vector fields.For any X ∈ X(M), we put where φ and η are two tensor fields of type (1, 1) and (1, 0), respectively.Thus, on M there exits an almost contact metric structure (φ, ξ, η, g) satisfying for any X, Y ∈ X(M).If the structure vector field ξ is principal, that is, Aξ = αξ at each point, where α = η(Aξ), then M is called a Hopf hypersurface and α is called a Hopf principal curvature.Moreover, applying the parallelism of the complex structure (i.e., ∇J = 0) of M n (c), and using (2.1) and (2.2) we have for any X, Y ∈ X(M).Let R be the Riemannian curvature tensor of M. Because M n (c) is of constant holomorphic sectional curvature c, the Gauss and Codazzi equations of M in M n (c) are given respectively as the following: for any X, Y ∈ X(M).

Proof of Theorem 1.5
Let M be a real hypersurface in CP 2 (c) or CH 2 (c) which is non-Hopf at every point.In what follows, working on Ω (in this context it is M), let us put e 1 = ξ, e 2 = U and e 3 = φU such that {e 1 , e 2 , e 3 } forms a local orthonormal basis of the tangent space at each point of the hypersurface.We need the following result that can be seen in [19,[24][25][26].where γ, δ, µ, κ i , i = {1, 2, 3}, are smooth functions on Ω.
By a direct calculation, we have for any vector field X orthogonal to ξ. Putting (2.6) into the above equation gives Now in terms of the skew-symmetry of φ, the above equation becomes From (3.10) we conclude that δ = 0 and µ = γ hold on each point of Ω. Next, we show that dα(ξ) 0 means that µ = γ = 0. Or equivalently, we assume that there exits a non-empty open subset Q of Ω on which µ = γ 0, and we aim to show that ξ(α) = 0 on Q.
(3.11) , with the aid of (3.15), we obtain e 1 (α) = 0 because of µ 0 on Q.However, this contradicts our assumption (dα(ξ) 0 on the hypersurface) and means that when ξ(α) 0 on Ω, then γ = µ = 0. Now, on Ω, the shape operator is given by with respect to the local orthonormal basis {e 1 , e 2 , e 3 }.In addition, it is easily seen that Eq (3.21) is equivalent to g(AX, Y) = 0 for any vector fields X and Y orthogonal to the structure vector field ξ.
That is, the hypersurface M is locally congruent to a ruled hypersurface.This completes the proof of Theorem 1.5.

Proof of Theorem 1.6
Let M be a real hypersurface in CP 2 (c) or CH 2 (c) which is non-Hopf at every point satisfying Ki and Suh's condition (1.5).Working on Ω (in this context it is M), according to the proof of Theorem 1.5, δ = 0 and λ = µ 0 are necessarily true.If µ = γ 0, we have proved [e 1 , e 2 ] = 0 and that α, β and µ are all invariant along {e 1 , e 2 } = H.This implies that the hypersurface M is locally congruent to a strongly 2-Hopf hypersurface (non-ruled).In addition, applying Lemma 3.1, δ = 0 and λ = µ 0 are equivalent to g(AX, Y) = λg(X, Y) for any vector fields X, Y orthogonal to the structure vector field ξ.If µ = λ = 0 and δ = 0, as discussed before it is easily seen that in this case the hypersurface is a ruled one.
Conversely, suppose that M is a non-Hopf real hypersurface in CP 2 or CH 2 .If M is a ruled hypersurface, applying directly (3.21), we have from (3.3)-(3.6)that (1.5) holds.Now assume that M satisfies g(AX, Y) = ag(X, Y) for any vector fields X, Y orthogonal to ξ and a is a non-vanishing function.In order to prove (1.5), following (3.3)-(3.6)we need only to prove e 1 (a) = 0. On such a hypersurface M, we construct a local orthonormal basis similar to that in Section three and adopt the same symbols.In Section three, e 1 (a) = 0 has been confirmed due to a = µ = γ 0. This completes the proof of Theorem 1.6.Remark 4.1.A three-dimensional real hypersurface satisfying that g(AX, Y) = ag(X, Y) for any vector fields X, Y orthogonal to the structure vector field ξ and a function a has been considered in [13] which adopted Ivey and Ryan's formula in [8].

Proof of Theorem 1.7
First of all, we prove that a real hypersurface M in CP 2 (c) or CH 2 (c) satisfying (1.2) for any vector fields X and Y orthogonal to ξ must be Hopf.Suppose that such a hypersurface M is non-Hopf, then Ω defined in Section one is non-empty.Working on Ω, by Lemma 3.1, both (3.4) and (3.5) in Section one are necessarily true in this context.If Eq (1.2) is valid for any vector fields X, Y orthogonal to ξ, it follows from (3.4) and (3.5) that e respectively.The subtraction of the above equation from the previous one gives This reduces to β = 0, and contradicts our assumption.Therefore, Ω is empty and M is Hopf.Recall that the Hopf principal curvature (α = g(Aξ, ξ)) of any Hopf hypersurface in a nonflat complex space form is a constant (see [4,17]).
Considering Y = ξ in the Codazzi Eq (2.8) and using (2.6), for any vector field X, we have Putting this into (3.2) we have for any vector fields X, Y orthogonal to ξ.If (1.2) is valid, it follows from the above equation that for any vector field X orthogonal to ξ.Let X be an eigenvector field of the shape operator orthogonal to ξ with eigenfunction λ.Since the dimension of the real hypersurface is three, then φX is also an eigenvector field of the shape operator whose eigenfunction is denoted by µ.It follows from (5.1) that and µ is also a root of the quadratic Eq (5.2).From (5.2), we observe that all principal curvatures of the hypersurface M are constant.On the other hand, from [17, Corollary 2.3], we have It follows immediately that λ = µ.In fact, if λ µ and hence λ = 1 2 α, applying the Vieta theorem, from (5.2) we obtain µ = 1 2 α and we arrive at a contradiction.For any Hopf hypersurface in CP 2 or CH 2 , the two principal curvatures on the holomorphic distribution ker η being the same is equivalent to Aφ = φA.Thus, the hypersurface is of type (A) (see [16,18]).
Conversely, if the hypersurface is of type (A), applying the equivalent condition φA = Aφ, from (3.2) we see that (1.2) is valid if and only if g((∇ ξ A)X, Y) = 0 for any vector fields X, Y orthogonal to ξ.Such an equation holds on any real hypersurfaces of type (A) (see [4,Theorem 8.120]).This completes the proof of Theorem 1.7.

Conclusions
The geometry of real hypersurfaces in nonflat complex space forms is determined completely by the shape operator.There exist a great number literature in the study of real hypersurfaces in nonflat complex space forms in terms of the shape operator.The present paper give some new characterizations for type (A) hypersurfaces, ruled hypersurfaces and strongly 2-Hopf hypersurfaces of dimension three by means of the Lie derivative of the shape operator.This can be regarded as extensions for real hypersurfaces of dimension greater than three which were obtained by Ki and Suh.In view of reuslts in this paper, ones are helpful to understand better the geometry of real hypersurfaces of dimension three.