When we consider Hermitian symmetric spaces of rank 2, we can usually give examples of Riemannian symmetric spaces SU_m+2/S(U₂U_m) and SU_{2, m}/S(U₂U_m), which are said to be complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians respectively (see [21, 22, 23] ). These are viewed as Hermitian symmetric spaces and quaternionic Kähler symmetric spaces equipped with the Kähler structure J and the quaternionic Kähler structure .

In the complex projective space ℂP^m+1 and the quaternionic projective space ℚP^m+1 some classifications related to commuting Ricci tensor were investigated by Kimura [5, 6], Pérez [13] and Pérez and Suh [14, 15] respectively. The classification problems of the complex 2-plane Grassmannian G₂(ℂ^m+2) = SU_m+2/S(U₂U_m) with various geometric conditions were discussed in Jeong, Kim and Suh [4], Pérez [13], and Suh [21, 22, 29], where the classification of contact hypersurfaces, parallel Ricci tensor, harmonic curvature and Jacobi operator of a real hypersurface in G₂(ℂ^m+2) were extensively studied.

Another example of Hermitian symmetric space with rank 2 having non-compact type different from the above ones, is the complex hyperbolic quadric SO2,m0/SO2SOm. It is a simply connected Riemannian manifold whose curvature tensor is the negative of the curvature tensor of the complex quadric Q^m (see Besse [2], Helgason [3], and Knapp [10]). The complex hyperbolic quadric also can be regarded as a kind of real Grassmann manifolds of non-compact type with rank 2. Accordingly, the complex hyperbolic quadric Q^m* admits two important geometric structures, a complex conjugation structure A and a Kähler structure J, which anti-commute with each other, that is, AJ = –JA. For m≥2 the triple (Q^m*, J, g) is a Hermitian symmetric space of non-compact type and its maximal sectional curvature is equal to −4 (see Klein [7], Kobayashi and Nomizu [11], and Reckziegel [16]).

Two last examples of different Hermitian symmetric spaces with rank 2 in the class of compact type or non-compact type, are the complex quadric Q^m = SO_m+2/SO_mSO₂ or the complex hyperbolic quadric Qm*=SO2,mo/SOmSO2, which are a complex hypersurface in complex projective space CP^m+1 or in complex hyperbolic space respectively(see Romero [17, 18], Suh [24, 25], and Smyth [19]). The complex quadric Q^m or the complex hyperbolic quadric Q^m* can be regarded as a kind of real Grassmann manifold of compact or non-compact type with rank 2 respectively(see Helgason [3], Kobayashi and Nomizu [11]). Accordingly, the complex quadric Q^m and the complex hyperbolic quadric Q^m* both admit two important geometric structures, a complex conjugation structure A and a Kähler structure J, which anti-commute with each other, that is, AJ = –JA (see Klein [7] and Reckziegel [16]).

Now let us introduce a complex hyperbolic quadric Qm*=SOm,2o/SO2SOm, which can be regarded as a Hermitian symmetric space with rank 2 of noncompact type. Montiel and Romero [12] proved that the complex hyperbolic quadric Q^m* can be immersed in the indefinite complex hyperbolic space CH1m+1(-c), c > 0, by interchanging the Kähler metric with its opposite. Changing the Kähler metric of CPn-sn+1 with its opposite, we have that Qn-sn endowed with its opposite metric g′ = –g is also an Einstein hypersurface of CHs+1n+1(-c). When s = 0, we know that (Qnn, g′ = –g) can be regarded as the complex hyperbolic quadric Qm*=SOm,2o/SO2SOm, which is immersed in the indefinite complex hyperbolic quadric CH1m+1(-c), c > 0 as a space-like complex Einstein hypersurface.

Apart from the complex structure J there is another distinguished geometric structure on Q^m*, namely a parallel rank two vector bundle which contains a S¹- bundle of real structures. Note that these real structures are complex conjugations A on the tangent spaces of the complex hyperbolic quadric Q^m*. This geometric structure determines a maximal -invariant subbundle of the tangent bundle TM of a real hypersurface M in the complex hyperbolic quadric 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 Q^m*. There are two types of singular tangent vectors for the complex hyperbolic quadric Q^m* as follows:

• 1. If there exists a conjugation such that W ∈ V (A), then W is singular. Such a singular tangent vector is called -principal.

• 2. If there exist a conjugation and orthonormal vectors X, Y ∈ V (A) such that W/‖W‖=(X+JY)/2, then W is singular. Such a singular tangent vector is called -isotropic.

Here V (A) = {X ∈ T_[z]Q^m* : AX = X} and JV (A) = {X ∈ T_[z]Q^m* : AX = – X}, [z] ∈ Q^m*, are the (+1)-eigenspace and (−1)-eigenspace for the involution A on T_[z]Q^m*, [z] ∈ Q^m*.

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 cases depending on whether N is -isotropic or -principal (see [27, 28, 30, 31]). In the first case where N is -isotropic, we have shown in [27] that M is locally congruent to a tube over a totally geodesic complex hyperbolic space ℂH^k in the complex hyperbolic quadric Q^2k*. In the second case, when the unit normal N is -principal, we proved that a contact hypersurface M in the complex hyperbolic quadric Q^m* is locally congruent to a tube over a totally geodesic and totally real submanifold ℝH^m in Q^m* or a horosphere (see Suh [9], and Suh and Hwang [30]).

Usually, Jacobi fields along geodesics of a given Riemannian manifold M̄ satisfy a well known differential equation. Naturally the classical differential equation inspires the so-called Jacobi operator. That is, if R̄ is the curvature operator of M̄, the Jacobi operator with respect to X at z∈M, is defined by

(R¯XY) (z)=(R¯(Y,X)X) (z)

for any Y ∈T_zM̄. Then R̄_X∈End(T_zM̄ ) becomes a symmetric endomorphism of the tangent bundle T M̄ of M̄. Clearly, each tangent vector field X to M̄ provides a Jacobi operator with respect to X.

From such a view point, in the complex hyperbolic quadric Q^m* the normal Jacobi operator R̄_N is defined by

R¯N=R¯(·,N)N∈End (TzM),         z∈M

for a real hypersurface M in the complex hyperbolic quadric Q^m* with unit normal vector field N, where R̄ denotes the curvature tensor of the complex hyperbolic quadric Q^m*. Of course, the normal Jacobi opeartor R̄_N is a symmetric endomorphism of M in the complex hyperbolic quadric Q^m*.

The normal Jacobi operator R̄_N of M in the complex hyperbolic quadric Q^m* is said to be Lie invariant if the operator R̄_N satisfies

0=(LXR¯N)Y

for any X, Y ∈T_zM, z∈M, where the Lie derivative is defined by

(1.1)(LXR¯N)Y=[X,R¯N(Y)]-R¯N([X,Y])=∇X(R¯N(Y))-∇R¯N(Y)X-R¯N(∇XY-∇YX)=(∇XR¯N)Y-∇R¯N(Y)X+R¯N(∇YX).

For real hypersurfaces in the complex quadric Q^m we investigated the notions of parallel Ricci tensor, harmonic curvature and commuting Ricci tensor, which are respectively given by ∇Ric = 0, δRic = 0 and Ric φ = φ Ric (see Suh [25], [26], and Suh and Hwang [29]). But from the assumption of Ricci parallel or harmonic curvature, it was difficult for us to derive the fact that either the unit normal vector field N is -isotropic or -principal. So in [25] and [26] we gave a classification with the further assumption of -isotropic. Also in the study of complex hyperbolic quadric Q^m* we also have some obstructions to get the fact that the unit normal N is singular.

In the paper due to Suh [27] we investigate this problem of isometric Reeb flow for the complex hyperbolic quadric Qm*=SO2,mo/SOmSO2. In view of the previous results, naturally, we expected that the classification might include at least the totally geodesic Q^m−1* ⊂ Q^m*. But, the results are quite different from our expectations. The totally geodesic submanifolds of the above type are not included. Now we introduce the classification as follows:

Theorem 1.1

Let M be a real hypersurface of the complex hyperbolic quadricQm*=SO2,mo/SOmSO2, m ≥ 3. The Reeb flow on M is isometric if and only if m is even, say m = 2k, and M is an open part of a tube around a totally geodesic ℂH^k ⊂ Q^2k*or a horosphere whose center at infinity is-isotropic singular.

But fortunately, when we consider Lie invariant normal Jacobi operator, that is., ℒ_XR̄_N = 0 for any tangent vector field X on M in Q^m*, we can assert that the unit normal vector field N becomes either -isotropic or -principal as follows:

Theorem 1.2

Let M be a Hopf real hypersurface in the complex hyperbolic quadric Q^m*, m≥3, with Lie invariant normal Jacobi operator. Then the unit normal vector field N is singular, that is, N is-isotropic or-principal.

Then motivated by Theorem 1.1 and Theorem 1.2, we can give a complete classification for real hypersurfaces in Q^m* with invariant normal Jacobi operator as follows:

Theorem 1.3

Let M be a Hopf real hypersurface in the complex hyperbolic quadric Q^m*, m≥3 with Lie invariant normal normal Jacobi operator. Then M is locally congruent to a tube of radius r over a totally geodesic CH^kin Q^2k*or a horosphere whose center at infinity is-isotropic singular.

In this section, let us introduce a new known result of the complex hyperbolic quadric Q^m* different from the complex quadric Q^m. This section is due to Klein and Suh [9], and Suh [28].

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 Besse [2], and Helgason [3]).

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 [20, 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 [7] and [16]. Another related result by Smyth, [20, 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 SO2,m0/SO2SOm. As Q^1* is isomorphic to the real hyperbolic space ℝH2=SO1,20/SO2, and Q^2* is isomorphic to the Hermitian product of complex hyperbolic spaces ℂH¹×ℂH¹, we suppose m ≥ 3 in the sequel and throughout this paper. Let G:=SO2,m0 be the transvection group of Q^m* and K := SO₂SO_m be the isotropy group of Q^m* at the “origin” p₀ := eK ∈ Q^m*. Then

σ:G→G,g↦sgs-1         with         s:=(-1      -1      1      1      ⋱      1)

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 SO₂, and therefore Q^m* is in fact a Hermitian symmetric space.

The Lie algebra of G is given by

g={X∈gl(m+2,ℝ):Xt·s=-s·X}

(see [10, p. 59]). In the sequel we will write members of as block matrices with respect to the decomposition ℝ^m+2 = ℝ² ⊕ ℝ^m, i.e. in the form

X=(X11X12X21X22),

where X₁₁, X₁₂, X₂₁, X₂₂ are real matrices of the dimension 2 × 2, 2 × m, m × 2 and m × m, respectively. Then

g={(X11X12X21X22):X11t=-X11,X12t=X21,X22t=-X22}.

The linearisation σ_L = Ad(s) : of the involutive Lie group automorphism σ induces the Cartan decomposition , where the Lie subalgebra

k=Eig(σ*,1)={X∈g:sXs-1=X}={(X1100X22):X11t=-X11,X22t=-X22}≅so2⊕som

is the Lie algebra of the isotropy group K, and the 2m-dimensional linear subspace

m=Eig(σ*,-1)={X∈g:sXs-1=-X}={(0X12X210):X12t=X21}

is canonically isomorphic to the tangent space T_p0Q^m*. Under the identification , the Riemannian metric g of Q^m* (where the constant factor of the metric is chosen so that the formulae become as simple as possible) is given by

g(X,Y)=12tr(Yt·X)=tr(Y12·X21)         for         X,Y∈m.

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

JX=Ad(j)X         for         X∈m,         where         j:=(01    -10      1      1      ⋱      1)∈K.

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 with 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 the sequel.

Like for the complex quadric (again compare [7, 8, 16]), there is another important structure on the tangent bundle of the complex quadric besides the Riemannian metric and the complex structure, namely an S¹-bundle of real structures. The situation here differs from that of the complex quadric in that for Q^m*, the real structures in cannot be interpreted as the shape operator of a complex hypersurface in a complex space form, but as the following considerations will show, still plays an important role in the description of the geometry of Q^m*.

Let

a0:=(1      -1      1      1      ⋱      1).

Note that we have a₀ ∉ K, but only a₀ ∈ O₂SO_m. However, Ad(a₀) still leaves invariant, and therefore defines an ℝ-linear map A₀ on the tangent space . A₀ turns out to be an involutive orthogonal map with A₀ ◦ J = –J ◦ A₀ (i.e. A₀ is anti-linear with respect to the complex structure of T_p0Q^m*), and hence a real structure on T_p0Q^m*. But A₀ commutes with Ad(g) not for all g ∈ K, but only for g ∈ SO_m ⊂ K. More specifically, for g = (g₁, g₂) ∈ K with g₁ ∈ SO₂ and g₂ ∈ SO_m, say g1=(cos(t)-sin(t)sin(t)cos(t))

with t ∈ ℝ (so that Ad(g₁) corresponds to multiplication with the complex number μ := e^it), we have

A0∘Ad(g)=μ-2·Ad(g)∘A0.

This equation shows that the object which is Ad(K)-invariant and therefore geometrically relevant is not the real structure A₀ by itself, but rather the “circle of real structures”

Ap0:={λA0∣λ∈S1}.

is Ad(K)-invariant, and therefore generates an Ad(G)-invariant S¹-subbundle of the endomorphism bundle End(TQ^m*), consisting of real structures on the tangent spaces of Q^m*. For any , the tangent line to the fibre of through A is spanned by JA.

For any p ∈ Q^m* and , the real structure A induces a splitting

TpQm*=V(A)⊕JV(A)

into two orthogonal, maximal totally real subspaces of the tangent space T_pQ^m*. Here V (A) resp. JV (A) are the (+1)-eigenspace resp. the (−1)-eigenspace of A. For every unit vector Z ∈ T_pQ^m* there exist t∈[0,π4], and orthonormal vectors X, Y ∈ V (A) so that

Z=cos(t)·X+sin(t)·JY

holds; see [16, Proposition 3]. Here t is uniquely determined by Z. The vector Z is singular, i.e. contained in more than one Cartan subalgebra of , if and only if either t = 0 or t=π4 holds. The vectors with t = 0 are called -principal, whereas the vectors with t=π4 are called -isotropic. If Z is regular, i.e. 0<t<π4 holds, then also A and X, Y are uniquely determined by Z.

Like for the complex quadric, the Riemannian curvature tensor R̄ of Q^m* can be fully described in terms of the “fundamental geometric structures” g, J and . In fact, under the correspondence , the curvature R̄(X, Y )Z corresponds to – [[X, Y ], Z] for , see [11, Chapter XI, Theorem 3.2(1)]. By evaluating the latter expression explicitly, one can show that one has

(2.1)R¯(X,Y)Z=-g(Y,Z)X+g(X,Z)Y-g(JY,Z)JX+g(JX,Z)JY+2g(JX,Y)JZ-g(AY,Z)AX+g(AX,Z)AY-g(JAY,Z)JAX+g(JAX,Z)JAY

for arbitrary . Therefore the curvature of Q^m* is the negative of that of the complex quadric Q^m, compare [16, 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.

Let M be a real hypersurface in complex hyperbolic quadric Q^m* and denote by (φ, ξ, η, g) the induced almost contact metric structure on M and by ∇ the induced Riemannian connection on M. Note that ξ = –JN, where N is a (local) unit normal vector field of M. The vector field ξ is known as the Reeb vector field of M. If the integral curves of ξ are geodesics in M, the hypersurface M is called a Hopf hypersurface. The integral curves of ξ are geodesics in M if and only if ξ is a principal curvature vector of M everywhere. The tangent bundle TM of M splits orthogonally into , where is the maximal complex subbundle of TM and ℱ = ℝξ. The structure tensor field φ restricted to coincides with the complex structure J restricted to , and we have φξ = 0. We denote by νM the normal bundle of M.

We first introduce some notations. For a fixed real structure and X ∈ T_[z]M we decompose AX into its tangential and normal component, that is,

AX=BX+ρ(X)N

where BX is the tangential component of AX and

ρ(X)=g(AX,N)=g(X,AN)=g(X,AJξ)=g(JX,Aξ).

Since JX = φX + η(X)N and Aξ = Bξ + ρ(ξ)N we also have

ρ(X)=g(φX,Bξ)=η(X)ρ(ξ)=η(BφX)=η(X)ρ(ξ).

We also define

δ=g(N,AN)=g(JN,JAN)=-g(JN,AJN)=-g(ξ,Aξ).

At each point [z] ∈ M we define

Q[z]={X∈T[z]M:AX∈T[z]M for all A∈A[z]},

which is the maximal -invariant subspace of T_[z]M. Then by using the same method for real hypersurfaces in complex hyperbolic quadric Q^m* as in Berndt and Suh [1] we get the following

Lemma 2.1

Let M be a real hypersurface in complex hyperbolic quadric Q^m*. Then the following statements are equivalent:

• (i) The normal vector N_[z]of M is-principal,

• (ii) ,

• (iii) There exists a real structuresuch that AN_[z] ∈ ℂν_[z]M.

Assume now that the normal vector N_[z] of M is not -principal. Then there exists a real structure such that

N[z]=cos(t)Z1+sin(t)JZ2

for some orthonormal vectors Z₁, Z₂ ∈ V (A) and 0<t<π4. This implies

(2.2)AN[z] =cos(t)Z1-sin(t)JZ2,ξ[z] =sin(t)Z2-cos(t)JZ1,Aξ[z] =sin(t)Z2+cos(t)JZ1,

and therefore is strictly contained in . Moreover, we have

Aξ[z]=Bξ[z]   and   ρ(ξ[z])=0.

We have

g(Bξ[z]+δξ[z],N[z]) =0,g(Bξ[z]+δξ[z],ξ[z]) =0,g(Bξ[z]+δξ[z],Bξ[z]+δξ[z]) =sin2(2t),

where the function δ denotes δ = –g(ξ, Aξ) = – (sin²t – cos²t) = cos2t. Therefore

U[z]=1sin(2t)(Bξ[z]+δξ[z])

is a unit vector in and

C[z]=Q[z]⊕[U[z]]   (orthogonal direct sum).

If N_[z] is not -principal at [z], then N is not -principal in an open neighborhood of [z], and therefore U is a well-defined unit vector field on that open neighborhood. We summarize this in the following

Lemma 2.2

Let M be a real hypersurface in complex hyperbolic quadric Q^m*whose unit normal N_[z]is not-principal at [z]. Then there exists an open neighborhood of [z] in M and a section A inon that neighborhood consisting of real structures such that

• (i) Aξ = Bξ and ρ(ξ) = 0,

• (ii) U = (Bξ + δξ)/||Bξ + δξ|| is a unit vector field tangent to,

• (iii) .

Let M be a real hypersurface in the complex hyperbolic quadric Q^m* and denote by (φ, ξ, η, g) the induced almost contact metric structure. Note that ξ = –JN, where N is a (local) unit normal vector field of M and η the corresponding 1-form defined by η(X) = g(ξ, X) for any tangent vector field X on M. The tangent bundle TM of M splits orthogonally into , where is the maximal complex subbundle of TM. The structure tensor field φ restricted to coincides with the complex structure J restricted to , and φξ = 0.

At each point z ∈ M we define a maximal -invariant subspace of T_zM, z∈M as follows:

Qz={X∈TzM:AX∈TzM for all A∈Az}.

Then we want to introduce an important lemma which will be used in the proof of our main Theorem in the introduction.

Lemma 3.1.([24])

For each z ∈ M we have

• (i) If N_zis-principal, then.

• (ii) If N_zis not-principal, there exist a conjugationand orthonormal vectors X, Y ∈ V (A) such that N_z = cos(t)X+sin(t)JY for some t ∈ (0, π/4]. Then we have.

From the explicit expression of the Riemannian curvature tensor of the complex hyperbolic quadric Q^m* we can easily derive the Codazzi equation for a real hypersurface M in Q^m*:

(3.1)g((∇XS)Y-(∇YS)X,Z)=-η(X)g(φY,Z)+η(Y)g(φX,Z)+2η(Z)g(φX,Y)-ρ(X)g(BY,Z)+ρ(Y)g(BX,Z)+η(BX)g(BY,φZ)+η(BX)ρ(Y)η(Z)-η(BY)g(BX,φZ)+η(BY)ρ(X)η(Z).

We now assume that M is a Hopf hypersurface. Then the shape operator S of M in Q^m* satisfies

Sξ=αξ

with the smooth function α = g(Sξ, ξ) on M. Inserting Z = ξ into the Codazzi equation leads to

g((∇XS)Y-(∇YS)X,ξ)=2g(φX,Y)-2ρ(X)η(BY)+2ρ(Y)η(BX).

On the other hand, we have

(3.2)g((∇XS)Y-(∇YS)X,ξ)=g((∇XS)ξ,Y)-g((∇YS)ξ,X)=dα(X)η(Y)-dα(Y)η(X)+αg((Sφ+φS)X,Y)-2g(SφSX,Y).

Comparing the previous two equations and putting X = ξ yields

dα(Y)=dα(ξ)η(Y)+2δρ(Y),

where the function δ = –g(Aξ, ξ) and ρ(Y) = g(AN, Y ) for any vector field Y on M in Q^m*.

Reinserting this into the previous equation yields

g((∇XS)Y-(∇YS)X,ξ)=-2δη(X)ρ(Y)+2δρ(X)η(Y)+αg((φS+Sφ)X,Y)-2g(SφSX,Y).

Altogether this implies

(3.3)0=2g(SφSX,Y)-αg((φS+Sφ)X,Y)+2g(φX,Y)-2δρ(X)η(Y)-2ρ(X)η(BY)+2ρ(Y)η(BX)+2δη(X)ρ(Y)=g((2SφS-α(φS+Sφ)+2φ)X,Y)-2ρ(X)η(BY+δY)+2ρ(Y)η(BX+δX)=g((2SφS-α(φS+Sφ)+2φ)X,Y)-ρ(X)η(BY+δY)+2ρ(Y)η(BX+δX)=g((2SφS-α(φS+Sφ)+2φ)X,Y)-2ρ(X)g(Y,Bξ+δξ)+2g(X,Bξ+δξ)ρ(Y).

If AN = N we have ρ = 0, otherwise we can use Lemma 2.2 to calculate ρ(Y) = g(Y, AN) = g(Y, AJξ) = –g(Y, JAξ) = –g(Y, JBξ) = –g(Y, φBξ). Thus we have proved

Lemma 3.2

Let M be a Hopf hypersurface in Q^m*, m ≥ 3. Then we have

(2SφS-α(φS+Sφ)+2φ)X=2ρ(X) (Bξ+δξ)+2g(X,Bξ+δξ)φBξ.

If the unit normal vector field N is -principal, we can choose a real structure such that AN = N. Then we have ρ = 0 and φBξ = –φξ = 0, and therefore

2SφS-α(φS+Sφ)=-2φ.

If N is not -principal, we can choose a real structure as in Lemma 2.2 and get

 ρ(X) (Bξ+δξ)+g(X,Bξ+δξ)φBξ=-g(X,φ(Bξ+δξ)) (Bξ+δξ)+g(X,Bξ+δξ)φ(Bξ+δξ)=‖Bξ+δξ‖2(g(X,U)φU-g(X,φU)U)=sin2(2t) (g(X,U)φU-g(X,φU)U),

which is equal to 0 on and equal to sin²(2t)φX on . Altogether we have proved:

Lemma 3.3

Let M be a Hopf hypersurface in Q^*m, m ≥ 3. Then the tensor field

2SφS-α(φS+Sφ)

leavesandinvariant and we have

2SφS-α(φS+Sφ)=-2φ   on   Q

and

2SφS-α(φS+Sφ)=-2δ2φ   on   C⊙Q,

where δ = cos 2t as in section 3.

By the curvature tensor R̄ of (2.1) for a real hypersurface in the complex hyperbolic quadric Q^m* in section 2, the normal Jacobi operator R̄_N is defined in such a way that

R¯N(X)=R¯(X,N)N=-X-g(JN,N)JX+g(JX,N)JN+2g(JX,N)JN-g(AN,N)AX+g(AX,N)AN-g(JAN,N)JAX+g(JAX,N)JAN

for any tangent vector field X in T_zM and the unit normal N of M in T_zQ^m*, z∈Q^m*. Then the normal Jacobi operator R̄_N becomes a symmetric operator on the tangent space T_zM, z∈M, of Q^m*. From this, by the complex structure J and the complex conjugations , together with the fact that g(Aξ, N) = 0 and ξ = –JN in section 3, the normal Jacobi operator R̄_N is given by

(4.1)R¯N(Y)=-Y-3η(Y)ξ-g(AN,N)AY+g(AY,N)AN+g(AY,ξ)Aξ

for any Y ∈T_zM, z∈M. Then the derivative of R̄_N is given by

(4.2)(∇XR¯N)Y=∇X(R¯N(Y))-R¯N(∇XY)=-3(∇Xη) (Y)ξ-3η(Y)∇Xξ-{g(∇¯X(AN),N)+g(AN,∇¯XN)}AY-g(AN,N) {∇¯X(AY)-A∇XY}+{g(∇¯X(AY)-A∇XY,N)+g(AY,∇¯XN)}AN+g(AY,N)∇¯X(AN)+{g(∇¯X(AY)-A∇XY,ξ)Aξ+g(AY,∇¯Xξ)}Aξ+g(AY,ξ)∇¯X(Aξ),

where the connection ∇̄ on the complex hyperbolic quadric Q^m* is given by

∇¯X(AY)-A∇XY=(∇¯XA)Y+A∇¯XY-A∇XY=q(X)JAY+Aσ(X,Y)=q(X)JAY+g(SX,Y)AN.

From this, together with the invariance of ℒ_XR̄_N = 0 in (1.1), it follows that

(4.3)(∇R¯N(Y)X-R¯N(∇YX)=(∇XR¯N)Y=-3g(φSX,Y)ξ-3η(Y)φ(SX)-{q(X)g(JAN,N)-g(ASX,N)-g(AN,SX)}AY-g(AN,N) {q(X)JAY+g(SX,Y)AN}+{q(X)g(JAY,N)+g(SX,Y)g(AN,N)}AN-g(AY,SX)AN+g(AY,N){(∇¯XA)N+A∇¯XN}+g((∇¯XA)Y,ξ)Aξ+g(AY,φSX+σ(X,ξ))Aξ+g(AY,ξ)∇¯X(Aξ),

where we have used the equation of Gauss ∇̄_Xξ = ∇_Xξ + σ(X, ξ), σ(X, ξ) denotes the normal bundle T^⊥M valued second fundament tensor on M in Q^m*. Fromthis, putting Y = ξ and using (∇̄_XA)Y = q(X)JAY, and ∇̄_XN = –SX we have

(4.4)∇R¯N(ξ)X-R¯N(∇ξX)=(∇XR¯N)ξ=-3φSX-{q(X)g)(JAN,N)-g(ASX,N)-g(AN,SX)}Aξ-g(AN,N) {q(X)JAξ+g(SX,ξ)AN}+{q(X)g(JAξ,N)+g(SX,ξ)g(AN,N)}AN-g(Aξ,SX)AN+g(q(X)JAξ,ξ)Aξ+g(Aξ,φSX+σ(X,ξ))Aξ+g(Aξ,ξ){q(X)JAξ+AφSX+g(SX,ξ)}AN.

From this, by taking the inner product with the unit normal N, we have

(4.5)-g(Aξ,SX)g(AN,N)+g(Aξ,ξ) {q(X)g(JAξ,N)+g(AφSX,N)+g(SX,ξ)g(AN,N)}=0.

Then by putting X = ξ and using the assumption of Hopf, we have

(4.6)q(ξ)g(Aξ,ξ)2=0.

This gives that q(ξ) = 0 or g(Aξ, ξ) = 0. The latter case implies that the unit normal N is -isotropic. Now we only consider the case q(ξ) = 0.

We put Y = ξ in (4.1). Then it follows that

R¯N(ξ)=-4ξ-{g(AN,N)-g(Aξ,ξ)}Aξ=-4ξ-2g(AN,N)Aξ,

where we have used that g(Aξ, ξ) = g(AJN, JN) = –g(JAN, JN) = –g(AN, N).

Differentiating this one, it follows that

(4.7)∇R¯N(ξ)X-R¯N(∇ξX)=(∇XR¯N)ξ=-4∇Xξ-2{g(∇¯X(AN),N)Aξ+g(AN,∇¯XN)Aξ}-2g(AN,N)∇¯X(Aξ).

Then, by putting Y = ξ, and taking the inner product of (4.7) with the unit normal N, we have

g(AN,N) {q(ξ)g(Aξ,ξ)-αg(Aξ,ξ)}=0.

From this, together with q(ξ) = 0, it follows that

(4.8)αg(Aξ,ξ)g(AN,N)=0.

Then from (4.8) we can assert the following lemma.

Lemma 4.1

Let M be a Hopf real hypersurface in the complex hyperbolic quadric Q^m*, m≥3, with parallel normal Jacobi operator. Then the unit normal vector field N is-principal or-isotropic.

In this section let us consider a real hypersurface M in the complex hyperbolic quadric Q^m* with -principal unit normal vector field. Then the unit normal vector field N satisfies AN = N for a complex conjugation . This also implies that Aξ = –ξ for the Reeb vector field ξ = –JN.

Then the normal Jacobi operator R̄_N in section 4 becomes

(5.1)R¯N(X)=-X-3η(X)ξ-AX+η(X)ξ=-X-2η(X)ξ-AX,

where we have used that AN = N and

g(AX,ξ)Aξ=g(AX,JN)AJN=g(X,AJN)AJN=g(X,JAN)JAN=g(X,JN)JN=η(X)ξ.

On the other hand, we can put

AY=BY+ρ(Y)N,

where BY denotes the tangential component of AY and ρ(Y) = g(AY, N) = g(Y, AN) = g(Y, N) = 0. So it becomes always AY = BY for any vector field Y on M in Q^m*. Then by differentiating (5.1) along any direction X, we have

(5.2)(∇XR¯N)Y=∇X(R¯N(Y))-R¯N(∇XY)=-2(∇Xη) (Y)-2η(Y)∇Xξ-(∇XB)Y.

Now let us consider that the normal Jacobi operator R̄_N is invariant, that is, ℒ_XR̄_N = 0. This is given by

0=(ℒXR¯N)Y =ℒX(R¯NY)-R¯N(ℒXY) =[X,R¯NY]-R¯N[X,Y] =∇X(R¯NY)-∇R¯N(Y)X-R¯N(∇XY-∇YX) =∇X(R¯NY)-∇R¯N(Y)X+R¯N(∇YX).

Then it follows that

-2g(φSX,Y)ξ-2η(Y)φSX-(∇XB)Y={∇YX+2η(∇YX)ξ+A∇YX}-{∇YX+2η(Y)∇ξX+∇AYX}.

From this putting Y = ξ and using Aξ = –ξ, it follows that

(5.3)-2φSX-(∇XB)ξ=2η(∇ξX)ξ+A∇ξX+∇ξX=-2φSX-{q(X)JAξ-σ(X,Aξ)+η(SX)N}.

where we have used the following

(∇XA)ξ=∇X(Aξ)-A∇Xξ =∇¯X(Aξ)-A∇Xξ ={(∇¯XA)ξ+A∇¯Xξ}-AφSX =q(X)JAξ+AφSX-σ(X,Aξ)+g(SX,ξ)AN-AφSX =q(X)JAξ-σ(X,Aξ)+αη(X)N.

Then by taking the inner product of (5.3) with the unit normal N, we have

q(X)=2αη(X).

This implies q(ξ) = 2α, and the 1-form q is given by

(5.4)q(X)=q(ξ)η(X).

On the other hand, in section 4 from the Lie invariance of the normal Jacobi operator we have calculated the following

(5.5)∇R¯N(ξ)X-R¯N(∇ξX)=(∇XR¯N)ξ=-3φSX-{q(X)g(JAN,N)-g(ASX,N)-g(AN,SX)}Aξ-g(AN,N) {q(X)JAξ+g(SX,ξ)AN}+{q(X)g(JAξ,N)+g(SX,ξ)g(AN,N)}AN-g(Aξ,SX)AN+g(q(X)JAξ,ξ)Aξ+g(Aξ,φSX+σ(X,ξ))Aξ+g(Aξ,ξ) {q(X)JAξ+AφSX+g(SX,ξ)AN}.

From this, by taking the inner product with the unit normal N, we have

(5.6)-g(Aξ,SX)g(AN,N)+g(Aξ,ξ) {q(X)g(JAξ,N)+g(Aξ,ξ) {g(JAξ,N)+g(AφSX,N)+g(SX,ξ)g(AN,N)}=0.

Then by putting X = ξ and using the assumption of Hopf, we have

(5.7)q(ξ)g(Aξ,ξ)2=0.

From this, together with (5.4) and Aξ = –ξ, it follows that the 1-form q vanishes identically on M.

On the other hand, we know that the complex hyperbolic quadric Q^m* can be immersed into the indefinite complex hyperbolic space CH1m+1 in C2m+2 (see Montiel and Romero [12], and Kobayashi and Nomizu [11]). Then the same 1-form q appears in the Weingarten formula

∇˜Xz¯=-Az¯X+q(X)Jz¯

for unit normal vector fields {z̄, J z̄} on the complex hyperbolic quadric Q^m* which can be immersed in indefinite complex hyperbolic space CH1m+1 as a space-like complex hypersurface, where ∇̃ denotes the Riemannian connection on CH1m+1 induced from the Euclidean connection on C2m+2 (see Smyth [19] and [20]). But the 1-form q never vanishes on Q^m*. This gives a contradiction (see Smyth [19]). This means that there do not exist any real hypersurfaces in the complex hyperbolic quadric Q^m* with invariant normal Jacobi operator, that is, ℒ_XR̄_N = 0 for the -principal unit normal vector field N.

In this section let us assume that the unit normal vector field N is -isotropic. Then the normal vector field N can be put

N=12(Z1+JZ2)

for Z₁, Z₂∈V (A), where V (A) denotes a (+1)-eigenspace of the complex conjugation . Then it follows that

AN=12(Z1-JZ2),AJN=-12(JZ1+Z2),and JN=12(JZ1-Z2).

From this, together with (2.2) and the anti-commuting AJ = –JA, it follows that

g(ξ,Aξ)=g(JN,AJN)=0,g(ξ,AN)=0 and g(AN,N)=0.

By virtue of these formulas for the -isotropic unit normal, the normal Jacobi operator R̄_N in section 4 is given by

R¯N(Y)=-Y-3η(Y)ξ+g(AY,N)AN+g(AY,ξ)Aξ.

Then the derivative of the normal Jacobi operator R̄_N on M is given as follows:

(6.1)(∇XR¯N)Y=-3(∇Xη) (Y)ξ-3η(Y)∇Xξ+g(∇X(AN),Y)AN+g(AN,Y)∇X(AN)+g(Y,∇X(Aξ))Aξ+g(Aξ,Y)∇X(Aξ).

On the other hand, the Lie invariance (4.1) gives that

(6.2)(∇XR¯N)Y=∇X(R¯N(Y))-R¯N(∇XY)=∇R¯N(Y)X-R¯N(∇YX).

Then by putting Y = ξ in (6.1) and (6.2), and using R̄_N(ξ) = 4ξ, we have

(6.3)-3φSX-g(AN,φSX)AN-g(φSX,Aξ)Aξ=-4∇ξX+{∇ξX+3η(∇ξX)ξ-g(A∇ξX,N)AN-g(A∇ξX,ξ)Aξ}

From this, taking the inner product (6.3) with the vector field AN, it follows that

4g(φSX,AN)=4g(∇ξX,AN).

Then from this, together with (6.3), we get

(6.4)φSX=∇ξX-η(∇ξX)ξ.

For any X∈ξ^⊥, where ξ^⊥ denotes the orthogonal complement of the Reeb vector field ξ in the tangent space T_zM, z∈M, we know that ∇_ξX is orthogonal to the Reeb vector field ξ, that is, η(∇_ξX) = –g(∇_ξξ, X) = 0. Then the formula (6.4) becomes for any tangent vector field X∈ξ^⊥

(6.5)φSX=∇ξX.

When we consider the -isotropic unit normal, the vector fields Aξ and AN belong to the distribution in section 3.

On the other hand, by virtue of Lemma 3.1, we prove the following for a Hopf hypersurface in Q^m* with -isotropic unit normal vector field as follows:

Lemma 6.1

Let M be a Hopf real hypersurface in the complex hyperbolic quadric Q^m*, m≥3, with-isotropic unit normal vector field. Then

(6.6)SAN=0,         and         SAξ=0.