As examples of some Hermitian symmetric spaces of rank 2, usually we can consider 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 [1], [2], [3], [6], [7], [8], [16], [17] and [23]). Those are said to be Hermitian symmetric spaces and quaternionic Kähler symmetric spaces equipped with the Kähler structure J and the quaternionic Kähler structure = Span{J₁, J₂, J₃} on SU_2,m/S(U₂U_m). The rank of SU_2,m/S(U₂U_m) is 2 and there are exactly two types of singular tangent vectors X of SU_2,m/S(U₂U_m) which are characterized by the geometric properties and respectively.

As another example of Hermitian symmetric space with rank 2 of compact type different from the above ones, we could give a complex quadric Q^m = SO_m+2/SO₂SO_m, which is a complex hypersurface in complex projective space CP^m (see Reckziegel [14], Suh [19], [20], [22] and Smyth [15]). The complex quadric also can be regarded as a kind of real Grassmann manifolds of compact type with rank 2 (see Kobayashi and Nomizu [10]). Accordingly, the complex quadric admits two important geometric structures that 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 (see Klein [9] and Reckziegel [14]).

About the latter part of twentieth century, many geometers have investigated some real hypersurfaces in Hermitian symmetric spaces of rank 1 like the complex projective space ℂP^m or the complex hyperbolic space ℂH^m. For the complex projective space ℂP^m and the quaternionic projective space ℚP^m a characterization with isometric Reeb flow was obtained by Okumura [11], -parallel shape operator = 0 by Pérez [12], and -parallel curvature tensor = 0 by Pérez and Suh [13], respectively, where = Span{ξ₁, ξ₂, ξ₃}, ξ_i = −J_iN, i = 1, 2, 3.

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. Here we consider a real hypersurface M in Q^m* with shape operator of Codazzi type, that is, (∇_XS)Y = (∇_Y S)X for the shape operator S and any vector fields X and Y on M in Q^m*. In Suh [18] we gave a non-existence property for real hypersurfaces of Codazzi type in the complex quadric Q^m* as follows:

Theorem A

There do not exist any Hopf real hypersurfaces in complex quadric Q^m*, m ≥ 3, with shape operator of Codazzi type.

Next we have considered the notion of parallel shape operator for real hypersurfaces M in Q^m*. Usually, parallelism is naturally included in the notion of shape operator of Codazzi type. So by Theorem A we can assert the following

Theorem B

There do not exist any Hopf real hypersurfaces in complex quadric Q^m*, m ≥ 3, with parallel shape operator.

Apart from the complex structure J there is another distinguished geometric structure on Q^m*. Namely, a vector subbundle = {A_λz̄|λ ∈ S¹⊂C}, [z] ∈ Q^m*, which consists of all complex conjugations defined on the complex quadric Q^m*. The vector bundle contains a S¹-bundle of real structures, that is, complex conjugations A on the tangent spaces of Q^m* and becomes a parallel rank 2-subbundle of of End TQ^m*. This geometric structure determines a maximal -invariant sub-bundle of the tangent bundle TM of a real hypersurface M in Q^m*.

Recall that a nonzero tangent vector W ∈ T_zQ^m* is called singular if it is tangent to more than one maximal flat in Q^m*. Here maximal flat means a 2-dimensional curvature flat maximal totally geodesic submanifold in Q^m*. Such a maximal flat always exists, because the rank of Q^m* is 2. There are two types of singular tangent vectors for the complex quadric Q^m* as follows:

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

• 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 we note that the unit normal N is said to be -principal if N is invariant under the complex conjugation A, that is, AN = N.

Moreover, the derivative of the complex conjugation A on Q^m* is defined by

for any vector fields X and Y on M and q denotes a certain 1-form defined on 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 quadric Q^m*:

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

• 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.

The shape operator S of M in Q^m* is said to be Killing if the operator S satisfies

for any X, Y ∈ T_zM, z ∈ M. The equation is equivalent to (∇_XS)X = 0 for any X ∈ T_zM, z ∈ M, because of linearization. Moreover, we can give the geometric meaning of Killing Jacobi tensor as follows:

When we consider a geodesic γ with initial conditions such that γ(0) = z and γ̇ (0) = X. Then the transformed vector field Sγ̇ is Levi-Civita parallel along the geodesic γ of the vector field X (see Blair [5] and Tachibana [24]).

In the study of real hypersurfaces in the complex quadric Q^m we considered the notion of parallel Ricci tensor, that is, ∇Ric = 0 (see Suh [19]). But from the assumption of Ricci being parallel, it was difficult for us to derive the fact that either the unit normal N is -isotropic or -principal. So in [19] we gave a classification with the further assumption of -isotropic. But fortunately, when we consider Killing shape operator for real hypersurfaces in the complex hyperbolic quadric Q^m*, first we can assert that the unit normal vector field N becomes either -isotropic or -principal as follows:

Main Theorem 1

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

Then motivated by such a result, next we give a complete classification for real hypersurfaces in the complex hyperbolic quadric Q^m* with Killing shape operator as follows:

Main Theorem 2

Let M be a real hypersurface in the complex hyperbolic quadric Q^m*, m≥4, with Killing shape operator. Then M is locally congruent to a horosphere or has 6 distinct constant principal curvatures given by

and

with corresponding principal curvature spaces respectively

and

Let us denote by C1m+2 an indefinite complex Euclidean space ℂ^m+2, on which the indefinite Hermitian product

is negative definite.

The homogeneous quadratic equation z12+…+zm2-zm+12-zm+22=0 consists of the points in ℂ1m+2 defines a noncompact complex hyperbolic quadric Q*m=SO2,mo/SO2SOm which can be immersed in the (m + 1)-dimensional in complex hyperbolic space ℂH^m+1 = SU_1,m+1/S(U_m+1U₁). The complex hypersurface Q^m* in ℂH^m+1 is known as the m-dimensional complex hyperbolic quadric. The complex structure J on ℂH^m+1 naturally induces a complex structure on Q^*m which we will denote by J as well. We equip Q^m* with the Riemannian metric g which is induced from the Begerman metric on ℂH^m+1 with constant holomorphic sectional curvature 4. For m ≥ 2 the triple (Q^m*, J, g) is a Hermitian symmetric space of rank two and its minimal sectional curvature is equal to −4. The 1-dimensional quadric Q^1* is isometric to the 2-dimensional real hyperbolic space ℝH2=SO1,2o/SO1SO2. The 2-dimensional complex quadric Q^2* is isometric to the Riemannian product of complex hyperbolic spaces ℂH¹× ℂH¹. We will assume m ≥ 3 for the main part of this paper.

For a nonzero vector z∈ℂ1m+2 we denote by [z] the complex span of z, that is, [z] = {λz | λ ∈ ℂ}. Note that by definition [z] is a point in ℂH^m+1. As usual, for each [z] ∈ ℂH^m+1 we identify T_[z]ℂH^m+1 with the orthogonal complement ℂ1m+2⊖[z] of [z] in ℂ1m+2. For [z] ∈ Q^m* the tangent space T_[z]Q^m* can then be identified canonically with the orthogonal complement ℂ1m+2⊖([z]⊕[z¯]) in ℂ1m+2. Note that z̄ ∈ ν_[z]Q^m* is a unit normal vector of Q^m* in ℂH^m+1 at the point [z].

We denote by A_z̄ the shape operator of Q^m* in ℂH^m+1 with respect to z̄. Then we have A_z̄w = w for all w ∈ T_[z]Q^m*, that is, A_z̄ is just complex conjugation restricted to T_[z]Q^m*. The shape operator A_z̄ is an antilinear involution on the complex vector space T_[z]Q^*m and

where V(Az¯)=ℝ1m+2∩T[z]Qm* is the (+1)-eigenspace and JV(Az¯)=iℝ1m+2∩T[z]Qm* is the (−1)-eigenspace of A_z̄. Geometrically this means that the shape operator A_z̄ defines a real structure on the complex vector space T_[z]Q^m*. Recall that a real structure on a complex vector space V is by definition an antilinear involution A : V → V. Since the normal space ν_[z]Q^m* of Q^m* in ℂH1m+1 at [z] is a complex subspace of T_[z]ℂH^m+1 of complex dimension one, every normal vector in ν_[z]Q^m* can be written as λz̄ with some λ ∈ ℂ. The shape operators A_λz̄ of Q^m* define a rank two vector subbundle of the endomorphism bundle End(TQ^m*). Since the second fundamental form of the embedding Q^m* ⊂ ℂH^m+1 is parallel (see e.g. [15]), is a parallel subbundle of End(TQ^m*). For λ ∈ S¹ ⊂ ℂ we again get a real structure A_λz̄ on T_[z]Q^m* and we have V (A_λz̄) = λV (A_z̄). We thus have an S¹-subbundle of consisting of real structures on the tangent spaces of Q^m*.

The Gauss equation for the complex hypersurface Q^m* ⊂ ℂH^m+1 implies that the Riemannian curvature tensor R̄ of Q^m* can be expressed in terms of the Riemannian metric g, the complex structure J and a generic real structure A in :

Note that the complex structure J anti-commutes with each endomorphism , that is, AJ = −JA.

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 quadric Q^m*:

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

• If there exist a real structure 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.

Basic complex linear algebra shows that for every unit tangent vector W ∈ T_[z]Q^m* there exist a real structure and orthonormal vectors X, Y ∈ V (A) such that

for some t ∈ [0, π/4]. The singular tangent vectors correspond to the values t = 0 and t = π/4.

Let M be a real hypersurface in 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 = ker(η) 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,

where BX is the tangential component of AX and

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

We also define

At each point [z] ∈ M we define

which is the maximal -invariant subspace of T_[z]M. Then by using the same method for real hypersurfaces in Q^*m as in Berndt and Suh [4] we get the following

Lemma 2.1

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

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

• ,

• 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

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

and therefore = T_[z]Q^m ⊝ ([Z₁] ⊕ [Z₂]) is strictly contained in . Moreover, we have

We have

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

is a unit vector in and

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 the 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

• Aξ = Bξ and ρ(ξ) = 0,

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

• ⊕ [U].

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 ⊂ Q^*m:

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

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

On the other hand, we have

Comparing the previous two equations and putting X = ξ yields

where δ denotes δ = −g(Aξ, ξ). Reinserting this into the previous equation yields

Altogether this implies

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 3.1

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

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

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

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

Lemma 3.2

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

leavesandinvariant and we have

and

where δ = −g(Aξ, ξ) = cos2t as in Section 3.

Let us put AX = BX + ρ(X)N for any vector field X ∈ T_zQ^m*, z ∈ M, ρ(X) = 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. Then it follows that

The shape operator S of M in the complex hyperbolic quadric Q^m* is said to be Killing if the operator S satisfies

for any X, Y ∈ T_zM, z ∈ M.

From (4.1), together with the equation of Codazzi, it follows that

Since we have assumed the real hypersurface M in Q^m* is Hopf, then Sξ = αξ. This gives

From this, let us put Y = ξ in (4.2) and use g(Aξ,N) = 0, we see that

Here, let us put X = ξ in (4.3) and also use g(ξ,AN) = 0, we have

From this we get ξα = 0. Then the derivative Y α in Section 3 becomes

From this, together with (4.3), it follows that

Then by putting Z = ξ into (4.3), we have

Since g(Aξ,N) = 0, (4.6) gives that

Then we have g(Aξ, ξ) = 0 or (AN)^T = 0, where (AN)^T denotes the tangential part of the vector AN.

Summing up above discussions, we conclude the following

Lemma 4.1

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

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

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

From this, together with anti-commuting AJ = −JA, it follows that

Then (4.3) gives the following for any X,Z ∈ T_zM, z ∈ M

Since Aξ,AN ∈ T_zM, z ∈ M, it implies

On the other hand, we have the following from Lemma 3.1 in Section 3 a Hopf real hypersurface M with -isotropic unit normal N

Then by virtue of (5.2) and (5.3), first we have

Then naturally it follows that

We know that the tangent space T_zM, z ∈ M is decomposed as follows:

where = Span[Aξ,AN].

Lemma 5.1

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

Theorem 5.2

Let M be a real hypersurface in the complex hyperbolic quadric Q^m* with-isotropic unit normal vector field. Then M is locally congruent to a horosphere or M has 4 distinct constant principal curvatures given by

with corresponding principal curvature spaces respectively

and

In this section let us consider a real hypersurface M in the complex hyperbolic quadric Q^m* with Killing shape operator for the case that the unit normal N is -principal. In this case the Killing shape operator (4.3) gives that

where we have used g(ξ,Aξ) = −1 and JAX = φAX + η(AX)N. Then it follows that

Since the unit normal vector field N is -principal, Aξ = −ξ. Then differentiating this and using Gauss equation give

where q denotes a certain 1-form defined on M as in the introduction. From this, together with ∇_X(Aξ) = −∇_Xξ = −φSX, it follows that

By taking the inner product of (6.3) with the unit normal N, we have q(X) = 2αη(X). From this, we know

So when we consider SX = λX for and λ = 0, (6.4) becomes AφX = −φX, where the distribution denotes the orthogonal complement of the Reeb vector field ξ. When the principal curvature λ = 0, SX = 0 in (6.1) gives that

Accordingly, for any cases we know that

Then we have

But TrA = 0, because T_zQ^m = V (A) ⊕ JV (A), where V (A) = {X ∈ T_zQ^m*|AX = X} and JV (A) = {X ∈ T_zQ^m|AX = −X}. This leads to a contradiction, which implies another theorem as follows:

Theorem 6.1

There do not exist any real hypersurface in the complex hyperbolic quadric Q^m* with Killing shape operator if the unit normal vector field is-principal.

Summing up all of discussions including Sections 4 and 5, by Lemma 4.1, Theorems 5.2 and 6.1, we give a complete proof of our Main Theorem 2 in the introduction.

This work was supported by grant Proj. No. NRF-2015-R1A2A1A-01002459 from National Research Foundation of Korea. The first author was supported by NRF-2017-R1A2B4005317.