Ruled real hypersurfaces in the complex hyperbolic quadric

1 Introduction

In the class of Hermitian symmetric spaces with rank 2 of noncompact type, we can consider the example of complex hyperbolic quadric Q n ∗ = S O 2 , n o ∕ S O 2 S O n , which is a simply connected Riemannian manifold whose curvature tensor is the negative of the curvature tensor of the complex quadric Q n = S O n + 2 ∕ S O 2 S O n (see [1–5]). The complex hyperbolic quadric Q n ∗ can be regarded as a kind of real Grassmann manifold of noncompact type with rank 2. Accordingly, Q n ∗ admits two important geometric structures, a complex conjugation (or real structure) C , and a Kähler structure (or complex structure) J , which anti-commute with each other, i.e., C J = − J C . Then, for n ≥ 2 , the triple ( Q n ∗ , J , g ) is a Hermitian symmetric space of noncompact type, and its minimal sectional curvature is equal to − 4 (see [6–8]).

In particular, Kimura-Ortega [9] and Montiel-Romero [10] proved that Q n ∗ can be immersed in the indefinite complex hyperbolic space C H 1 n + 1 ( − c ) , c > 0 , by interchanging the Kähler metric with its opposite. Indeed, if we change the Kähler metric of C P n − s n + 1 by its opposite, we have that Q n − s n endowed with its opposite metric g ′ = − g is also an Einstein hypersurface of C H s + 1 n + 1 ( − c ) . In the case of s = 0 , ( Q n n , g ′ = − g ) can be regarded as Q n ∗ = S O 2 , n o ∕ S O 2 S O n , which is immersed in the indefinite complex hyperbolic space C H 1 n + 1 ( − c ) , c > 0 as a complex Einstein hypersurface.

Apart from the complex structure J , there is another distinguished geometric structure on Q n ∗ , namely a parallel rank 2 vector bundle A , which contains an S 1 -bundle of real structures on the tangent spaces of Q n ∗ , i.e., A = { λ C ∣ λ ∈ S 1 } . This geometric structure determines a maximal A -invariant subbundle Q of the tangent bundle T M of a real hypersurface M in Q n ∗ .

In this article, we consider a classification problem of real hypersurfaces in the complex hyperbolic quadric Q n ∗ , n ≥ 3 . Let ζ be a unit normal vector field of a real hypersurface M in Q n ∗ . As a typical classification of real hypersurfaces in Q n ∗ , we introduce the following result, which was given by Suh [11].

Here, if the structure tensor ϕ commutes with the shape operator A of M , i.e., A ϕ = ϕ A , we say that M has the commuting shape operator (i.e., M has isometric Reeb flow). This result motivates us to study the weaker notion of η -commuting property of the shape operator. So, we define η -commuting property and η -parallelism of the shape operator A of M as follows:

A complete classification of real hypersurfaces in the complex quadric Q n with such two notions for shape operator was given in Kimura et al. [12]. By virtue of this classification, a new characterization of ruled real hypersurfaces foliated by complex totally geodesic hyperplanes Q n − 1 in Q n was given in the same article. For the complex projective space C P n , Kimura [13] and Loknherr and Reckziegel [14] gave some examples of ruled real hypersurfaces. The characterizations of ruled real hypersurfaces in C P n were investigated in [15–18] and so on. Recently, the ruled real hypersurfaces in the indefinite complex projective space C P p n have been introduced by Moruz et al. [19]. Moreover, they gave a classification of all minimal ruled real hypersurfaces in C P p n .

Motivated by these results, in this article, we will give a classification of real hypersurfaces in the complex hyperbolic quadric Q n ∗ regarding η -parallel and η -commuting shape operator. When the Reeb vector field ξ of M in Q n ∗ is principal, a real hypersurface M is said to be Hopf. As another kind of real hypersurfaces in Q n ∗ , we deal with a family of ruled real hypersurfaces in Q n ∗ , which are not Hopf. Indeed, a ruled real hypersurface is foliated by totally geodesic complex hypersurfaces Q n − 1 ∗ in Q n ∗ . More details on this family are given in Section 4. Then, by virtue of Theorems A, 4.2, and 6.4, we assert the following theorem:

Now, let us consider the notion of integrability of the holomorphic distribution C of a real hypersurface M in the complex hyperbolic quadric Q n ∗ , where C is given by C = { X ∈ T M ∣ X ⊥ ξ } . Kimura and Maeda [15] considered this notion for a real hypersurface in the complex projective space C P n . They gave a characterization of ruled real hypersurface in C P n . Motivated by such a result, for Q n ∗ , we obtain the following theorem:

As will be discussed in detail in Section 6, we know that if the shape operator of a real hypersurface M in Q n ∗ satisfies the conditions of η -commutativity and η -parallelism, then M is either Hopf or ruled (see Lemma 6.2). Now, let us focus our attention on the case that M is Hopf. Then, the η -commuting property is equivalent to the Reeb flow being isometric. By using this fact, we obtain a characterization of ruled real hypersurfaces in Q n ∗ (see Theorem 1.1). From this point of view, it is necessary to consider Hopf real hypersurfaces with η -parallel shape operator. So, as a final result, we want to give a complete classification of Hopf real hypersurfaces in Q n ∗ with η -parallel shape operator as follows:

2 The complex hyperbolic quadric

In this section, we introduce the complex hyperbolic quadric Q n ∗ . This section is due to Klein and Suh (see [11,21]).

The n -dimensional complex hyperbolic quadric Q n ∗ is the noncompact dual of the n -dimensional complex quadric Q n , i.e., the simply connected Riemannian symmetric space whose curvature tensor is the negative of the curvature tensor of Q n . It cannot be realized as a homogeneous complex hypersurface of the complex hyperbolic space C H n + 1 . In fact, Smyth [3, Theorem 3(ii)] has shown that every homogeneous complex hypersurface in C H n + 1 is totally geodesic. This is in marked contrast to the situation for the complex quadric Q n , which can be realized as a homogeneous complex hypersurface of the complex projective space C P n + 1 in such a way that the shape operator for any unit normal vector to Q n has a real structure on the corresponding tangent space of Q n (see [8,21,22]). Another related result by Smyth, [4, Theorem 1], which states that any complex hypersurface in C H n + 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 n ∗ as a complex hypersurface of C H n + 1 with the analogous property for the shape operator. Therefore, we realize the complex hyperbolic quadric Q n ∗ as the quotient manifold S O 2 , n o ∕ S O 2 S O n .

As Q 1 ∗ is isomorphic to the real hyperbolic space R H 2 = S O 2 , 1 o ∕ S O 2 , and Q 2 ∗ is isomorphic to the Hermitian product of complex hyperbolic spaces C H 1 × C H 1 , we suppose n ≥ 3 in the sequel and throughout this article. Let G ≔ S O 2 , n be the transvection group of Q n ∗ and K ≔ S O 2 S O n be the isotropy group of Q n ∗ at the “origin” p 0 ≔ e K ∈ Q n ∗ . Then,

is an involutive Lie group automorphism of G with Fix ( σ ) 0 = K , and therefore, Q n ∗ = G ∕ K is a Riemannian symmetric space. The center of the isotropy group K is isomorphic to S O 2 , and therefore, Q n ∗ is in fact a Hermitian symmetric space.

The Lie algebra g ≔ so 2 , n of G is given as follows:

(see [23, p. 59]). In the sequel, we will write members of g as block matrices with respect to the decomposition R n + 2 = R 2 ⊕ R n , i.e., in the form

where X 11 , X 12 , X 21 , and X 22 are real matrices of dimension 2 × 2 , 2 × n , n × 2 , and n × n , respectively. Then,

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 2 n -dimensional linear subspace

is canonically isomorphic to the tangent space T p 0 Q n ∗ . Under the identification T p 0 Q n ∗ ≅ n , the Riemannian metric g of Q n ∗ (where the constant factor of the metric is chosen so that the formulae become as simple as possible) is given as follows:

where g is clearly Ad ( K ) -invariant and therefore corresponds to an Ad ( G ) -invariant Riemannian metric on Q n ∗ . The complex structure J of the Hermitian symmetric space is given as follows:

As 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 n ∗ . By identifying the multiplication by the unit complex number i with the application of the linear map J , the tangent spaces of Q n ∗ thus become n -dimensional complex linear spaces, and we will adopt this point of view in the sequel.

As for the complex quadric (again compare [8] with [21] and [11]), there is another important structure on the tangent bundle of the complex quadric besides the Riemannian metric and the complex structure, namely an S 1 -bundle A of real structures. The situation in this case is distinct from that of the complex quadric, as the real structures in A cannot be construed as the shape operator of a complex hypersurface in a complex space form, but as the following considerations will show, A still plays a fundamental role in the description of the geometry of Q n ∗ .

Let

Note that we have a 0 ∉ K , but only a 0 ∈ O 2 S O n . However, Ad ( a 0 ) still leaves m invariant and therefore defines an R -linear map C 0 on the tangent space m ≅ T p 0 Q n ∗ . C 0 turns out to be an involutive orthogonal map with C 0 ∘ J = − J ∘ C 0   (i.e., C 0 is anti-linear with respect to the complex structure of T p 0 Q n ∗ ), and hence a real structure on T p 0 Q n ∗ . But C 0 commutes with Ad ( g ) not for all g ∈ K , but only for g ∈ S O n ⊂ K . More specifically, for g = ( g 1 , g 2 ) ∈ K with g 1 ∈ S O 2 and g 2 ∈ S O n , say g 1 = cos ( t ) − sin ( t ) sin ( t ) cos ( t ) with t ∈ R (so that Ad ( g 1 ) corresponds to multiplication with the complex number μ ≔ e i t ), we have

This equation shows that the object that is Ad ( K ) -invariant and therefore geometrically relevant is not the real structure C 0 by itself but rather the “circle of real structures”

A p 0 is Ad ( K ) -invariant and therefore generates an Ad ( G ) -invariant S 1 -subbundle A of the endomorphism bundle End ( T Q n ∗ ) , consisting of real structures on the tangent spaces of Q n ∗ . For any C V ∈ A , the tangent line to the fiber of A through C is spanned by J C .

For any p ∈ Q n ∗ and C ∈ A p , the complex conjugation (real structure) C induces a splitting

into two orthogonal, maximal totally real subspaces of the tangent space T p Q n ∗ . Here, V ( C ) respectively J V ( C ) are the ( + 1 ) -eigenspace respectively the ( − 1 ) -eigenspace of C . For every unit vector Z ∈ T p Q n ∗ , there exist t ∈ [ 0 , π 4 ] , C ∈ A p , and orthonormal vectors X , Y ∈ V ( C ) so that

holds (see [8, Proposition 3]). Here, t is uniquely determined by Z . The vector Z is singular, i.e., contained in more than one maximal flat in Q n ∗ if and only if either t = 0 or t = π 4 holds. The vectors with t = 0 are called A -principal, whereas the vectors with t = π 4 are called A -isotropic. If Z is regular, i.e., 0 < t < π 4 holds, then also C , X , and Y are also uniquely determined by Z .

The Riemannian curvature tensor R ¯ of Q n ∗ can be fully described in terms of the “fundamental geometric structures” g , J , and A as follows:

for arbitrary C ∈ A . Therefore, the curvature of Q n ∗ is the negative sign of that of the complex quadric Q n , compare [8, Theorem 1]. This confirms that the symmetric space Q n ∗ , which we have constructed here, is indeed the noncompact dual of the complex quadric.

It is well known that Q n ∗ becomes a Kähler manifold, i.e., the complex structure J is parallel, ∇ ¯ J = 0 , where ∇ ¯ is the Levi-Civita connection of Q n ∗ . Finally, because the S 1 -subbundle A of the endomorphism bundle End ( T Q m ∗ ) is Ad ( G ) -invariant, it is also parallel with respect to the same covariant derivative ∇ ¯ induced by ∇ ¯ on End ( T Q n ∗ ) . Because the tangent line of the fiber of A through some C p ∈ A is spanned by J C p , this means precisely that, for any section C of A , there exists a real-valued 1-form q : T Q n ∗ → R so that

3 Some general equations

Let M be a real hypersurface in the complex hyperbolic quadric Q n ∗ and ζ be a local unit normal vector field of M . Any vector field X tangent to M satisfies

The tangential component of equation (3.1) defines on M as a skew-symmetric tensor field ϕ of type (1,1), named the structure tensor. The structure vector field ξ is defined by ξ = − J ζ and is called the Reeb vector field. The 1-form η is given by η ( X ) = g ( ξ , X ) for any vector field X tangent to M . So, on M , an almost contact metric structure ( ϕ , ξ , η , g ) is defined. The tangent bundle T M of M splits orthogonally into T M = C ⊕ R ξ , where C = ker ( η ) is the maximal complex subbundle of T M . The structure tensor field ϕ restricted to C coincides with the complex structure J restricted to C , and ϕ ξ = 0 .

We assume that M is a Hopf hypersurface. Then, the Reeb vector field ξ = − J ζ satisfies the following:

where A denotes the shape operator of the real hypersurface M for a smooth function α = g ( A ξ , ξ ) on M . Now, we consider the equation of Codazzi:

Putting Z = ξ in equation (3.2), we obtain

On the other hand, we have

Comparing the previous two equations and putting X = ξ yield

Reinserting this into the previous equation yields

Altogether, this implies

At each point z ∈ M , we can choose C ∈ A z such that

for some orthonormal vectors Z 1 , Z 2 ∈ V ( C ) and 0 ≤ t ≤ π 4 (see Proposition 3 in [8]). Note that t is a function on M . First of all, since ξ = − J ζ , we have

This implies g ( ξ , C ζ ) = 0 and hence

4 Ruled real hypersurfaces

In this section, we define a ruled real hypersurface in the complex hyperbolic quadric Q n ∗ and give the form of its shape operator. From this fact, we give some characterizations of ruled real hypersurfaces M in Q n ∗ . Moreover, we will introduce the example due to Berndt [24].

Let M be a real hypersurface in the complex hyperbolic quadric Q n ∗ . If the Reeb vector field ξ = − J ζ of M is principal, M is said to be Hopf. Now, let us introduce another kind of real hypersurfaces, ruled real hypersurfaces in the complex hyperbolic quadric Q n ∗ , which are not Hopf, as follows:

Note. The above (c) can be rewritten as follows: when M is foliated by the integrable totally geodesic complex hyperplane Q n − 1 ∗ in Q n ∗ , then M can be given by M = { p ∈ Q n − 1 ∗ ( t ) ∣ t ∈ I } . In such a case, we say that M is a ruled real hypersurface in Q n ∗ .

From this result, we can compute a detailed description of the shape operator A of a ruled real hypersurface M in Q n ∗ . In fact, it can be seen that this property is also true on ruled real hypersurfaces of nonflat complex space forms and complex quadric Q n (see [12,13,25]). So, as a characterization of ruled real hypersurfaces in Q n ∗ , we have:

It holds that g ( ∇ X Y , ξ ) = − g ( Y , ∇ X ξ ) = − g ( Y , ϕ A X ) = g ( ϕ Y , A X ) for any X , Y ∈ C . By virtue of Theorem 4.2, it implies that ∇ X Y ∈ C . From this, we assert that the shape operator A of a ruled real hypersurface M is η -parallel, i.e., g ( ( ∇ X A ) Y , Z ) = 0 for any X , Y , Z ∈ C . By linearization, it becomes g ( ( ∇ X A ) X , X ) = 0 for any X ∈ C . Then, this is equivalent to the constancy of g ( A γ ′ , γ ′ ) 2 = g ¯ ( ∇ ¯ γ ′ γ ′ , ∇ ¯ γ ′ γ ′ ) , where γ is a geodesic on M . Here, g ¯ and ∇ ¯ denote, respectively, the Riemannian metric and the Riemannian connection of the complex hyperbolic quadric Q n ∗ . This means that every geodesic γ : I → M in Q n ∗ , which is orthogonal to the Reeb vector field ξ , i.e., γ ′ ( 0 ) ⊥ ξ p , and γ ( 0 ) = p , has constant first curvature.