REAL HYPERSURFACES IN THE COMPLEX HYPERBOLIC QUADRIC WITH REEB INVARIANT RICCI TENSOR

. We ﬁrst give the notion of Reeb invariant Ricci tensor for real hypersurfaces M in the complex quadric Q m ∗ = SO 02 ,m /SO 2 SO m , which is deﬁned by L ξ Ric = 0, where Ric denotes the Ricci tensor of M in Q m ∗ , and L ξ the Lie derivative along the direction of the Reeb vector ﬁeld ξ = − JN . Next we give a complete classiﬁcation of real hypersurfaces in the complex hyperbolic quadric Q m ∗ = SO 02 ,m /SO 2 SO m with Reeb invariant Ricci tensor.


Introduction
Since the late 20th century there have been many studies for real hypersurfaces in the complex projective space CP m (see [6], [8], [18], [19], [20]) and the complex hyperbolic space CH m (see Berndt [1], Montiel and Romero [17]), which can be regarded as the class of Hermitian symmetric spaces of rank 1.
In the class of another Hermitian symmetric space of non-compact type with rank 2, we can give the example of complex hyperbolic quadric Q m * . It is also said to be of type (B) in Hermitian symmetric spaces. By using the method given in Kobayashi  Hermitian symmetric space SO 0 2,m /SO 2 SO m of rank 2 and also can be regarded as a kind of real Grassmann manifold of all oriented space-like 2-dimensional subspaces in indefinite flat Riemannian space R m+2 2 (see Montiel and Romero [15,16]). Accordingly, the complex hyperbolic quadric admits both a complex conjugation structure A and a Kähler structure J, which anti-commute with each other, that is, AJ = −JA. Then for m ≥ 2 the triple (Q m * , J, g) is a Hermitian symmetric space of compact type with rank 2 and its maximal sectional curvature is equal to −4.
Montiel and Romero [15] proved that the complex hyperbolic quadric Q m * can be immersed in the indefinite complex hyperbolic space CH m+1 1 (−c), c > 0, by interchanging the Kähler metric with its opposite. Because if we change the Kähler metric of CP m+1 m−s by its opposite, we have that Q m m−s endowed with its opposite metric g = −g is also an Einstein hypersurface of CH m+1 s+1 (−c). When s = 0, we know that (Q m m , g = −g) can be regarded as the complex hyperbolic quadric Q m * = SO o m,2 /SO 2 SO m , which is immersed in the indefinite complex hyperbolic quadric CH m+1 1 (−c), c > 0, as a space-like complex Einstein hypersurface. In the paper [35] due to Suh and Hwang, we investigated the problem of commuting Ricci tensor, Ric φ = φ Ric, for real hypersurfaces in the complex quadric Q m = SO m+2 /SO m SO 2 and obtained the following result.
Theorem A. Let M be a Hopf real hypersurface in the complex quadric Q m , m ≥ 4, with commuting Ricci tensor. If the shape operator commutes with the structure tensor on the distribution Q ⊥ , then M is locally congruent to an open part of a tube around totally geodesic CP k in Q 2k , m = 2k or M has 3 distinct constant principal curvatures given by with corresponding principal curvature spaces respectively Remark 1.1. Besides the complex structure J, there is another distinguished geometric structure on the complex quadric Q m , namely a parallel rank 2 vector bundle A which contains an S 1 -bundle of real structures, that is, complex conjugations A on the tangent spaces of the complex quadric Q m (see Reckziegel [23]). This geometric structure determines a maximal A-invariant subbundle Q, which is mentioned in the assumption of Theorem A, of the tangent bundle T M of a real hypersurface M in Q m .
Recall that a nonzero tangent vector W ∈ T [z] Q m * is called singular if it is tangent to more than one maximal flat in the complex hyperbolic quadric Q m * . There are two types of singular tangent vectors for the complex hyperbolic quadric Q m * : 1. If there exists a conjugation A ∈ A such that W ∈ V (A), then W is singular. Such a singular tangent vector is called A-principal.

If there exist a conjugation A ∈ A and orthonormal vectors
When we consider a hypersurface M in the complex hyperbolic quadric Q m * , under the assumption of some geometric properties the unit normal vector field N of M in Q m can be divided into two classes if either N is A-isotropic or A-principal (see [3], [30], [33], and [34]). In the first case where N is A-isotropic, that is, [33] has shown that a real hypersurface M in Q m * with isometric Reeb flow is locally congruent to a tube over a totally geodesic CH k in Q 2k or a horosphere with A-isotropic center at the infinity. In the second case, when the unit normal N is A-principal, that is, AN = N for a conjugation A ∈ A, we proved that a contact hypersurface M in Q m * is locally congruent to a tube over a totally geodesic and totally real submanifold RH m in Q m * (see Klein and Suh [11]).
Also motivated by Theorem A, Suh and Hwang [36] gave a complete classification for real hypersurfaces in the complex hyperbolic quadric Q m * with commuting Ricci tensor, that is, Ric · φ = φ · Ric as follows.
Theorem B (Suh and Hwang [36]). Let M be a Hopf real hypersurface with commuting Ricci tensor in the complex hyperbolic quadric Q m * = SO 0 2,m /SO 2 SO m , m ≥ 3. Then M is locally congruent to an open part of the following manifolds: i) a tube around totally geodesic CH k ⊂ Q * 2k ; ii) a horosphere whose center at infinity is A-isotropic singular; iii) a hypersurface with A-isotropic unit normal and 3 distinct constant principal curvatures given by with corresponding principal curvature spaces respectively iv) a hypersurface with A-principal unit normal vector field and at most 4 distinct roots λ 1 , λ 2 , µ 1 , and µ 2 satisfying the equation with corresponding principal curvature spaces T λ1 , T λ2 , T µ1 , and T µ2 such that Remark 1.2. In Theorem B, cases i), ii), and iii) can be applied when the unit normal vector field N is A-isotropic, and case iv) corresponds to the A-principal unit normal vector field N in the complex hyperbolic quadric Q m * . Now let us consider the notion of Reeb invariant Ricci tensor for real hypersurfaces M in Q m * = SO 0 2,m /SO 2 SO m , which is given by L ξ Ric = 0, where Ric and L ξ respectively denote the Ricci tensor of M in Q m * and the Lie derivative along the Reeb direction ξ = −JN for the Kähler structure J and the unit normal vector field N of M in Q m * . Then motivated by such a notion and the results mentioned above, by the help of Theorem B, we want to give a complete classification for real hypersurfaces in the complex hyperbolic quadric Q m * with Reeb invariant Ricci tensor as follows.

Main Theorem. Let M be a Hopf real hypersurface with Reeb invariant Ricci tensor in the complex hyperbolic quadric
Then M is locally congruent to an open part of the following manifolds: i) a tube around totally geodesic CH k ⊂ Q * 2k ; ii) a horosphere whose center at infinity is A-isotropic singular; iii) a hypersurface with A-isotropic unit normal and 3 distinct constant principal curvatures given by iv) a hypersurface with A-principal unit normal and at most 4 distinct roots λ 1 , λ 2 , µ 1 , and µ 2 satisfying the equation with corresponding principal curvature spaces T λ1 , T λ2 , T µ1 , and T µ2 such that Our paper is composed as follows. In Section 2 we present basic material about the complex quadric Q m * , motivated by the recent work due to Klein and Suh [11]. In Section 3, we study the geometry of the complex subbundle Q for real hypersurfaces in Q m * and some equations including Codazzi's and fundamental formulas related to the vector fields ξ, N , Aξ, and AN , where the operator A denotes the complex conjugation of M in the complex hyperbolic quadric Q m * , which is explicitly constructed in Section 2 by the Lie algebraic method.
In Section 4, the first step is to derive the formula of Ricci tensor for M in Q m * and in the next step we can show the formula of Reeb invariant Ricci tensor from the equation of Gauss for real hypersurfaces M in Q m * . Moreover, we give an important Lemma 4.2 which shows that the unit normal vector field N is either A-isotropic or A-principal.
In Section 5, a complete proof of our Main Theorem with A-isotropic unit normal vector field will be given. In this section we prove that a real hypersurface in Q m * , m = 2k, with invariant Ricci tensor is locally congruent to a tube over a totally geodesic CH k in Q 2k * or a horosphere whose center at infinity is A-isotropic singular.
Finally, in Section 6 we give a complete proof of our Main Theorem with A-principal unit normal vector field. The first part of this proof is devoted to studying some fundamental formulas from the Reeb invariant Ricci tensor and A-principal unit normal vector field. Then in the latter part we will use some trace formulas given by Tr(φS − Sφ)(φ · Ric − Ric · φ) and Tr(φ · Ric − Ric · φ) 2 . Then as a result we will get the formula of commuting Ricci tensor, that is, Ric · φ = φ · Ric.

The complex hyperbolic quadric
In this section, let us introduce a new known result of the complex hyperbolic quadric Q m * different from the complex quadric Q m which is mentioned in [3], [11], and [32]. The m-dimensional complex hyperbolic quadric Q m * is the non-compact dual of the m-dimensional complex quadric Q m , which is a kind of Hermitian symmetric space of non-compact type with rank 2 (see [5], [2], and [7]).
The complex hyperbolic quadric Q m * cannot be realized as a homogeneous complex hypersurface of the complex hyperbolic space CH m+1 . In fact, Smyth [26, Theorem 3 (ii)] has shown that every homogeneous complex hypersurface in CH m+1 is totally geodesic. This is in marked contrast to the situation for the complex quadric Q m , which can be realized as a homogeneous complex hypersurface of the complex projective space CP m+1 in such a way that the shape operator for any unit normal vector to Q m is a real structure on the corresponding tangent space of Q m ; see [10] and [23]. Another related result by Smyth [26,Theorem 1], which states that any complex hypersurface CH m+1 for which the square of the shape operator has constant eigenvalues (counted with multiplicity) is totally geodesic, also precludes the possibility of a model of Q m * as a complex hypersurface of CH m+1 with the analogous property for the shape operator.
Therefore we realize the complex hyperbolic quadric Q m * as the quotient manifold SO 0 2,m /SO 2 SO m . As Q 1 * is isomorphic to the real hyperbolic space RH 2 = SO 0 1,2 /SO 2 , and Q 2 * is isomorphic to the Hermitian product of complex hyperbolic spaces CH 1 × CH 1 , we suppose m ≥ 3 in what follows and throughout this paper. Let G := SO 0 2,m be the transvection group of Q m * and K := SO 2 SO m be the isotropy group of Q m * at the "origin" p 0 := eK ∈ Q m * . Then is an involutive Lie group automorphism of G with Fix(σ) 0 = K, and therefore Q m * = G/K is a Riemannian symmetric space. The center of the isotropy group K is isomorphic to SO 2 , and therefore Q m * is in fact a Hermitian symmetric space. The Lie algebra g := so 2,m of G is given by [12, p. 59]). In what follows we will write members of g as block matrices with respect to the decomposition R m+2 = R 2 ⊕ R m , i.e. in the form The linearization σ L = Ad(s) : g → g of the involutive Lie group automorphism σ induces the Cartan decomposition g = k ⊕ m, where the Lie subalgebra is the Lie algebra of the isotropy group K, and the 2m-dimensional linear subspace Under the identification T p0 Q m * ∼ = m, the Riemannian metric g of Q m * (where the constant factor of the metric is chosen so that the formulas become as simple as possible) is given by g is clearly Ad(K)-invariant, and therefore corresponds to an Ad(G)-invariant Riemannian metric on Q m * . The complex structure J of the Hermitian symmetric space is given by Because j is in the center of K, the orthogonal linear map J is Ad(K)-invariant, and thus defines an Ad(G)-invariant Hermitian structure on Q m * . By identifying the multiplication by the unit complex number i with the application of the linear map J, the tangent spaces of Q m * thus become m-dimensional complex linear spaces, and we will adopt this point of view in what follows. For the complex quadric, the Riemannian curvature tensorR of Q m * can be fully described in terms of the "fundamental geometric structures" g, J, and A.

COMPLEX HYPERBOLIC QUADRIC WITH REEB INVARIANT RICCI TENSOR 391
Ortega [9] and Smyth [25]). Therefore the curvature of Q m * is the negative of that of the complex quadric Q m ; cf. [23,Theorem 1]. This confirms that the symmetric space Q m * which we have constructed here is indeed the non-compact dual of the complex quadric.
For any p ∈ Q m * and A ∈ A p , the real structure A induces a splitting

Lemma 3.1 (See [33]). For each [z] ∈ M we have
We now assume that M is a Hopf hypersurface. Then for the Reeb vector field ξ the shape operator S becomes Sξ = αξ with the smooth function α = g(Sξ, ξ) on M . When we consider a transform JX of the Kähler structure J on the complex hyperbolic quadric Q m * for any vector field X on M in Q m * , we may put On the other hand, at each point [z] ∈ M we can choose A ∈ A z such that for some orthonormal vectors Z 1 , Z 2 ∈ V (A) and 0 ≤ t ≤ π 4 (see Proposition 3 in [23]). Since ξ = −JN , we have This implies g(ξ, AN ) = 0. From the property JAξ = −AJξ = −AN , we obtain: ([14] and [33]). Let M be a Hopf hypersurface in the complex hyperbolic quadric Q m * with (local) unit normal vector field N . For each point in z ∈ M we choose A ∈ A z such that N z = cos(t)Z 1 + sin(t)JZ 2 holds for some orthonormal vectors Z 1 , Z 2 ∈ V (A) and 0 ≤ t ≤ π 4 . Then holds for all vector fields X and Y on M .
Then from (2.1) and the equation of Gauss, the curvature tensor R of M in the complex hyperbolic quadric Q m * is defined so that where (AX) T and S denote the tangential component of the vector field AX and the shape operator of M in Q m * , respectively.

Reeb invariance and a key lemma
Now we consider that M is a real hypersurface in the complex hyperbolic quadric Q m * . Then we may put g(AX, N ), where BX and ρ(X)N respectively denote the tangential and normal component of the vector field AX. Then Aξ = Bξ + ρ(ξ)N and ρ(ξ) = g(Aξ, N ) = 0. It follows that The equation gives g(AN, N ) = −η(Bξ) and g(AN, ξ) = 0. From this, together with the curvature tensor (3.2) for M in Q m * in Section 3, the Ricci tensor is given by On the other hand, it can be easily checked that the Ricci tensor is Reeb invariant, that is, L ξ Ric = 0 if and only if (φS − Sφ) · Ric = Ric · (φS − Sφ).
(4.2) Remark 4.1. Let M be a real hypersurface over a totally geodesic CH k ⊂ Q 2k * , m = 2k or a horosphere with A-isotropic center at the infinity. Then by a theorem due to Suh [33] the structure tensor commutes with the shape operator, that is, Sφ = φS. Moreover, the unit normal vector field N becomes A-isotropic. This gives η(Bξ) = g(Aξ, ξ) = 0. So it naturally satisfies the formula (4.2), i.e., it is Reeb invariant.
On the other hand, from (4.2) we assert the following important lemma.

Lemma 4.2.
Let M be a Hopf real hypersurface in the complex hyperbolic quadric Q m * , m ≥ 3, with Reeb invariant Ricci tensor. Then the unit normal vector field N becomes singular, that is, N is A-isotropic or A-principal.
The second case gives that Similarly, we also know that On the other hand, by taking the inner product of (4.4) with the tangent vector field Aξ we know that SφAξ = φSAξ = 0.
This gives that SAξ = αη(Aξ)ξ. From this, together with (4.6), we have for a non-vanishing Reeb function α = 0 When the Reeb function α is vanishing, by the first formula in Lemma 3.2, that is, Since in the second case we have assumed that N is not A-isotropic, we know that g(ξ, Aξ) = 0. So it follows that (AN ) T = 0. This means that This gives g(AN, N ) = ±1, that is, we can take the unit normal N such that AN = N . So the unit normal N is A-principal, that is, AN = N .
In order to prove our Main Theorem in the introduction, by virtue of Lemma 4.2 we are able to consider two classes of hypersurfaces in Q m * , with the unit normal N either A-principal or A-isotropic. For M a real hypersurface in Q m * with A-isotropic normal vector field, in Section 5 we will give the proof in detail; in Section 6 we will give the remaining proof for the case that M has a A-principal normal vector field.

Proof of Main Theorem with A-isotropic unit normal vector field
In this section we want to prove our Main Theorem for real hypersurfaces M in Q m * with commuting Ricci tensor when the unit normal vector field becomes A-isotropic.
Since we assumed that the unit normal N is A-isotropic, by the definition in Section 3 we know that t = π 4 . Then by the expression of the A-isotropic unit normal vector field, (3.1) This implies that g(Aξ, ξ) = 0. Then the Ricci tensor (4.1) for a real hypersurface M in the complex quadric Q m * reduces to where the function h denotes the trace of the shape operator S of M in Q m * . Then substracting (5.2) from (5.1) gives On the other hand, we know that the Reeb invariant Ricci tensor L ξ Ric = 0 is equivalent to By using the formula (5.4) and taking the trace in (5.3), we have On the other hand, the final term in (5.5) becomes From this, by taking the trace, the first two terms become Tr(φS) 2 · Ric − Tr φS · Ric · φS = Tr(φS) 2 Ric − Tr(φS) 2 Ric = 0.
Then taking the trace of the next two terms gives Tr φS · Ric · Sφ = Tr φS 2 φ · Ric .
Moreover, by using our assumption of N being A-isotropic, that is, g(AN, N ) = 0 and g(Aξ, ξ) = 0, the third equality becomes From this we conclude that the Ricci tensor Ric commutes with the structure tensor φ in the case where the unit normal N is A-isotropic. Then by Theorem B due to Suh and Hwang [36], we give a complete classification in our Main Theorem in the introduction.

Proof of Main Theorem with A-principal normal vector field
In this section we want to prove our Main Theorem for real hypersurfaces in the complex hyperbolic quadric Q m * with commuting Ricci tensor and A-principal unit normal vector field. By the Ricci tensor given in the formula (4.1) for A-principal unit normal, that is, AN = N , we have where the function h denotes the trace of the shape operator S of M in Q m * . When we consider that the unit normal N is A-principal, the unit normal N is invariant under the complex conjugation A in A, that is, AN = N and Aξ = −ξ. By using such properties into (6.1) and (6.2), we have From this, together with L ξ Ric = 0, which is equivalent to (φS − Sφ) · Ric = Ric · (φS − Sφ), we have Tr(φ · Ric − Ric · φ) 2 = h Tr(φS − Sφ)(φ · Ric − Ric · φ) − Tr(φS 2 − S 2 φ)(φ · Ric − Ric · φ) + Tr(φA − Aφ)(φ · Ric − Ric · φ).
On the other hand, since the complex conjugation is involutive and anti-commuting, such that AJ = −JA, and the unit normal N is A-invariant, it follows that φA = −Aφ.