In the class of Hermitian symmetric spaces of rank 2, usually we can give examples of Riemannian symmetric spaces G₂(ℂ^m+2) = SU_m+2/S(U₂U_m) and G2*(ℂm+2)=SU2,m/S(U2Um), which are said to be complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians, respectively (see [9]). These are viewed as Hermitian symmetric spaces and quaternionic Kähler symmetric spaces equipped with the Kähler structure J and the quaternionic Kähler structure . There are exactly two types of singular tangent vectors X of SU_m+2/S(U₂U_m) and SU_2,m/S(U₂U_m) which are characterized by the geometric properties JX ∈ X and JX ⊥ X respectively. Hereafter let G2m+2(c) be the compact complex Grassmannian G₂(ℂ^m+2), m ≥ 3, and the noncompact complex Grassmannian G2*(ℂm+2), m ≥ 3, of rank two for c > 0 and c < 0 respectively, where c is a scaling factor for the Riemannian metric g. The Riemannian curvature tensor R̃ of G2m+2(c) is locally given by

for all X, Y and Z∈TxG2m+2(c), x∈G2m+2(c). Actually, in the previous studies for G2m+2(c) (e.g. [1], [2], [3], [5], [7], [11] etc.), the scaling factor c was given by 1 and -12 for G₂(ℂ^m+2) and G2*(ℂm+2), respectively.

For real hypersurfaces M in G2m+2(c), we have the following two natural geometric conditions: the 1-dimensional distribution = Span{ξ} and the 3-dimensional distribution = Span{ξ₁, ξ₂, ξ₃} are invariant under the shape operator A of M. Here the almost contact structure vector field ξ defined by ξ = −JN is said to be a Reeb vector field, where N denotes a local unit normal vector field of M in G2m+2(c). The almost contact 3-structure vector fields ξ₁, ξ₂, ξ₃ spanning the 3-dimensional distribution of M in G2m+2(c) are defined by ξ_ν = −J_νN (ν = 1, 2, 3), where J_ν denotes a canonical local basis of the quaternionic Kaehler structure , such that T_xM = ⊕ = ⊕ , x ∈ M. Under these invariant conditions for two kinds of distributions and in TxG2m+2(c) Berndt and Suh gave the complete classifications for real hypersurfaces in complex Grassmannians G2m+2(c) of rank 2, respectively (see [3] and [5]).

The Reeb vector field ξ is said to be Hopf if it is invariant under the shape operator A. The 1-dimensional foliation of M by the integral curves of the Reeb vector field ξ is said to be a Hopf foliation of M. We say that M is a Hopf hypersurface in G2m+2(c) if and only if the Hopf foliation of M is totally geodesic. By the almost contact metric structure (φ, ξ, η, g) and the formula ∇_Xξ = φAX for any X ∈ TM, it can be easily checked that M is Hopf if and only if the Reeb vector field ξ is Hopf. On the other hand, when the distribution of a hypersurface M in G2m+2(c) is invariant under the shape operator, we say that M is a -invariant hypersurface. Moreover, we say that the Reeb flow on M in G2m+2(c) is isometric, when the Reeb vector field ξ on M is Killing. This means that the metric tensor g is invariant under the Reeb flow of ξ on M, that is, ℒ_ξg = 0 where ℒ_ξ is the Lie derivative along the flow of ξ. For the complex two-plane Grassmannian G₂(ℂ^m+2) the following result is known.

Theorem A.([4])

The Reeb flow on a real hypersurface in G₂(ℂ^m+2) is isometric if and only if M is an open part of a tube around a totally geodesic G₂(ℂ^m+1) in G₂(ℂ^m+2).

Moreover, in [13] Suh has proved:

Theorem B

Let M be a connected orientable real hypersurface in the complex hyperbolic two-plane Grassmannians G2*(ℂm+2)=SU2,m/S(U2Um), m ≥ 3. The the Reeb flow on M is isometric if and only if M is an open part of a tube around some totally geodesic SU_2,m−1/S(U₂U_m−1) in SU_2,m/S(U₂U_m) or a horosphere whose center at infinity is singular.

Usually, any submanifold in Kaehler manifolds has many kinds of connections. Among them, we consider two connections, namely, Levi-Civita and Tanaka-Webster connections for real hypersurfaces M in G2m+2(c). In fact, G2m+2(c) is a Riemannian symmetric space for real hypersurfaces in a Kaehler manifold, we consider an affine connection ∇̂^(k) which is called by the k-th generalized Tanaka-Webster connection (in short, the g-Tanaka-Webster connection). It becomes a generalization of the well-known connection defined by Tanno [16]. Besides, it coincides with Tanaka-Webster connection if a real hypersurface in Kaehler manifolds satisfies φA+Aφ = 2kφ for a non-zero real number k. The Tanaka-Webster connection is defined as the canonical affine connection on a non-degenerate, pseudo-Hermitian CR-manifold ([6], [15] and [17]). Using the k-th generalized Tanaka-Webster connection ∇̂^(k) defined in such a way that (*)∇^X(k)Y=∇XY+g(φAX,Y)ξ-η(Y)φAX-kη(X)φY

for any X, Y tangent to M, where ∇ denotes the Levi-Civita connection on M and k is a non-zero real number. The latter part of the k-th generalized Tanaka-Webster connection g(φAX, Y )ξ − η(Y )φAX − kη(X)φY is denoted by F^X(k)Y. Here the operator F^X(k) is a kind of (1,1)-type tensor and said to be Tanaka-Webster operator. Recently, there are many results for the classification problem of real hypersurfaces in G2m+2(c) related to the k-th generalized Tanaka-Webster connection ∇̂^(k). In particular, [7] was given the result about the shape operator as follows:

Theorem C

Let M be a Hopf hypersurface in complex two-plane Grassmannians G₂(ℂ^m+2), m ≥ 3. If M satisfies (∇ξA)Y=(∇^ξ(k)A)Y for all tangent vector field Y on M, then M is locally congruent to a tube of radius r over a totally geodesic G₂(ℂ^m+1) in G₂(ℂ^m+2).

Motivated by this result, in this paper we study a real hypersurface M in G2m+2(c) whose Lie derivative coincides with k-th generalized Tanaka-Webster derivative for the shape operator of M, that is,

for arbitrary tangent vector field X and Y on M. Thus we assert the following:

Main Theorem

Let M be a Hopf hypersurface in complex Grassmannians G2m+2(c), c ≠ 0 and m ≥ 3. If M satisfies (C-1), then M is locally congruent one of the following :

We use some references [2, 7, 8, 10, 12, 14] to recall the Riemannian geometry of complex Grassmannians of rank two G2m+2(c), c ≠ 0, m ≥ 3, and some fundamental formulas including the Codazzi and Gauss equations for a real hypersurface in G2m+2(c). We can derive some facts from our assumption that M is a real hypersurface in G2m+2(c) with geodesic Reeb flow, that is, Aξ = αξ where α = g(Aξ, ξ). Among them, we introduce a lemma which is induced from the equation of Codazzi [11, 12].

Lemma A

If M is a connected orientable real hypersurface in G2m+2(c) with geodesic Reeb flow, then

and

for any tangent vector field X on M in G2m+2(c).

As mentioned in Section 1, the complete classifications of real hypersurfaces in G2m+2(c), c ≠ 0, m ≥ 3, with two kinds of A-invariant for the distributions = Span{ξ} and = Span{ξ₁, ξ₂, ξ₃} was given in [3, 5], respectively. Here we introduce these results as follows.

Theorem D.([3])

Let M be a connected real hypersurface in G₂(ℂ^m+2), m ≥ 3. Then bothandare invariant under the shape operator of M if and only if

• (M_A) M is an open part of a tube around a totally geodesic G₂(ℂ^m+1) in G₂(ℂ^m+2),

  or

• (M_B) m is even, say m = 2n, and M is an open part of a tube around a totally geodesic ℍPⁿin G₂(ℂ^m+2).

Theorem E.([5])

Let M be a connected real hypersurface in G2*(ℂm+2), m ≥ 3. Then bothandare invariant under the shape operator of M if and only if M is congruent to an open part of one of the following hypersurfaces:

• ( ) a tube around a totally geodesic SU_2,m−1/S(U₂U_m−1) in SU_2,m/S(U₂U_m);

• ( ) a tube around a totally geodesic ℍHⁿin SU_2,m/S(U₂U_m), m = 2n;

• (ℋ_A) a horosphere in SU_2,m/S(U₂U_m) whose center at infinity is singular and of type JX ∈ X;

• (ℋ_B) a horosphere in SU_2,m/S(U₂U_m) whose center at infinity is singular and of type JX⊥ X;

  or the following exceptional case holds:

• (ℰ) The normal bundleνM of M consists of singular tangent vectors of type JX⊥ X. Moreover, M has at least four distinct principal curvatures, three of which are given by

  α=2,         γ=0,         λ=12

  with corresponding principal curvature spaces

  Tα=(C∩Q)⊥,         Tγ=JQ⊥,         Tλ⊂C∩Q∩JQ.

  Ifμis another (possibly nonconstant) principal curvature function, then we have T_μ ⊂ ∩ ∩ J, JT_μ ⊂ T_λandT_μ ⊂ T_λ.

In particular, let us observe the structure of the model spaces, (M_A), ( ) and (ℋ_A), of Type (A) which are mentioned in Theorems D and E, respectively. In [1, 3], [5] the authors gave the characterization of the singular tangent vector N of M in G2m+2(c): There are two types of singular tangent vector, those N for which JN⊥ N, and those for which JN ∈ N. In other words, it means that ξ ∈ or ξ ∈ , since JN = −ξ and N = Span{ξ₁,ξ₂,ξ₃} = where TM = ⊕ . The following two propositions tell us that the normal vector field N on these model spaces is singular of type of JN ∈ N, that is, ξ ∈ .

Proposition A

Let M be a connected real hypersurface of G₂(ℂ^m+2). Suppose that A ⊂ , Aξ = αξ, andξis tangent to. Let J₁ ∈ be the almost Hermitian structure such that JN = J₁N. Then M has the following three (if r=π/28) or four (otherwise) distinct constant principal curvaturesα,β,λandμwith some r∈(0,π/8). Here ℍN denotes quaternionic span of the structure vector fieldξ.

Type	Eigenvalues	Eigenspace	Multiplicity
(MA)	α=8cot(8r)	Tα=	1
	β=2cot(2r)	Tβ= ⊖	2
	λ=-2tan(2r)	Tλ={X | X⊥ℍN, JX=J1X}	2m−2
	μ=0	Tμ={X |X⊥ℍN, JX=−J1X}	2m−2

Proposition B

Let M be a connected real hypersurface of G2*(ℂm+2). Assume that the maximal complex subbundleof TM and the maximal quaternionic subbundleof TM are both invariant under the shape operator of M. If JN ∈ N, then one of the following statements holds:

• ( ) M has exactly four distinct constant principal curvaturesα,β,λ₁andλ₂. The principal curvature spaces T_λ1and T_λ2are complex (with respect to J) and totally complex (with respect to).

• (ℋ_A) M has exactly three distinct constant principal curvaturesα,βandλ.

The eigenvalues and its corresponding eigenspaces and multiplicities are given as follows.

Type	Eigenvalues	Eigenspace	Multiplicity
( )	α=2 coth(2r)	Tα=	1
	β=coth(r)	Tβ= ⊖	2
	λ1=tanh(r)	Tλ1=E−1	2m−2
	λ2=0	Tλ2=E+1	2m−2
(ℋA)	α=2	Tα=	1
	β=1	Tβ=( ⊖ )⊕E−1	2m
	λ=0	Tλ=E+1	2m−2

On, we have (φφ₁)² = I and Tr(φφ₁) = 0. Let E₊₁and E₋₁be the eigenbundles ofwith respect to the eigenvalues +1 and −1, respectively.

We will prove that on a real hypersurface M satisfying the conditions given in Main Theorem of Section 1, the shape operator A and the structure tensor φ commute with each other, that is, the Reeb flow of M becomes isometric. Then by virtue of Theorems A and B we assert our main theorem in Section 1.

In order to do this, first we calculate the squared norm of symmetric operator (Aφ − φA) of a real hypersurface M in G2m+2(c), c ≠ 0, m ≥ 3.

Lemma 3.1

Let M be a real hypersurface in G2m+2(c), m ≥ 3. Then the squared norm of a symmetric operator (Aφ − φA) of M is given:

Lemma 3.2

Let M be a Hopf hypersurface in G2m+2(c), c ≠ 0, m ≥ 3. The condition (LξA)Y=(∇^ξ(k)A)Y for all tangent vector fields Y on M is equivalent that the Reeb flow on M is isometric, that is, Aφ = φA. Furthermore, M is locally congruent to the model space of Type (A).

Hereafter we will give a proof of Corollary introduced in Section 1. From now on, assume that M is a Hopf hypersurface in G2m+2(c), c ≠ 0, m ≥ 3 with

for all vector fields X and Y are tangent to M. It is trivial that the condition of (C-1) is weaker than (C-2). So, M satisfies the condition (C-1), naturally. Hence we see that if M satisfies our assumptions in Corollary, then M is of Type (A) by virtue of Lemma 3.2.

Now let us check the converse problem: whether the shape operator of model spaces of Type (A) in G2m+2(c) satisfies the condition (C-2) or not? In order to do this, suppose that the model spaces of Type (A), that is, (M_A), ( ), and (ℋ_A), in G2m+2(c) satisfies our conditions given in Corollary. Since

and

the condition (C-2) can be rewritten as

for all vector fields X, Y ∈ TM^* where M^* denotes the model space of Type (A). From Propositions A and B we see that ξ = ξ₁, furthermore q_ν(X) = 2g(Aξ_ν,X) for ν = 2, 3. If we put X = ξ₂ ∈ T_β and Y = ξ₃ ∈ T_β in (4.1), then it becomes

where we have used φξ₃ = ξ₂, φξ₂ = −ξ₃ and

From this, we see that β = 0 or α = β.

For the case of (M_A), since α=8cot(8r) and β=2cot(2r) where r∈(0,π8), it makes an contradiction. Hence the model space of (M_A) does not satisfy our condition (C-2).

Moreover, for ( ) (and (ℋ_A), resp.) we obtain the contradiction, since α = 2 coth(2r) (and α = 2, resp.) and β = coth(r) (and β = 1, resp.) where r ∈ (0,∞).

Summing up these observations, we assert the corollary in Section 1.

This work was supported by grant Proj. Nos. NRF-2016-R1A6A3A-11931947 and NRF-2017-R1A2B-4005317 from National Research Foundation of Korea.