495–507 REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH COMMUTING STRUCTURE JACOBI OPERATOR Young Jin Suh and Hae Young Yang Reprinted from the Bulletin of the Korean Mathematical Society Vol

Bull. Korean Math. Soc. 45(2008), No. 3, pp. 495–507

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH COMMUTING STRUCTURE

JACOBI OPERATOR

Young Jin Suh and Hae Young Yang

Reprinted from the

Bulletin of the Korean Mathematical Society Vol. 45, No. 3, August 2008

⃝c2008 The Korean Mathematical Society

Bull. Korean Math. Soc. 45(2008), No. 3, pp. 495–507

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH COMMUTING STRUCTURE

JACOBI OPERATOR

Young Jin Suh and Hae Young Yang

Abstract. In this paper we give a complete classification of real hyper- surfaces in complex two-plane GrassmanniansG2(C^m+2) with commut- ing structure Jacobi operatorRξ and another geometric condition.

0. Introduction

In the geometry of real hypersurfaces in complex space formsMn(c) Kimura [6] has proved that Hopf real hypersurfaces M in a complex projective space P_n(C) with commuting Ricci tensor are locally congruent to of type (A), a tube over a totally geodesicP_k(C), of type (B), a tube over a complex quadricQ_n−1, cot²2r=n−2, of type (C), a tube overP₁(C)×P_(n−1)/2(C), cot²2r=_n−¹₂ and nis odd, of type (D), a tube over a complex two-plane GrassmannianG₂(C⁵), cot²2r= ³₅ andn= 9, of type (E), a tube over a Hermitian symmetric space SO(10)/U(5), cot²2r= ⁵₉ andn= 15.

The notion of Hopf real hypersurfaces means that the structure vector ξ defined by ξ =−J N satisfies Aξ =αξ, where J denotes a Kaehler structure of P_n(C),N andAa unit normal and the shape operator ofM in P_n(C). We say such a structure vectorξonM theReebvector field, and its flow theReeb flow on M.

In a quaternionic projective space QP^m P´erez [7] has classified real hyper- surfaces inQP^mwith commuting Ricci tensorSϕi=ϕiS, i= 1,2,3, whereS (resp. ϕi) denotes the Ricci tensor (resp. the structure tensor) ofM in QP^m, is locally congruent to of A1, A2-type, that is, a tube overQP^k with radius 0< r < ^π₂,k∈{0, . . ., m−1}.

A Jacobi field along geodesics of a given Riemannian manifold (M, g) is an important role in the study of differential geometry. It satisfies a well known

Received October 11, 2007; Revised January 23, 2008.

2000Mathematics Subject Classification. Primary 53C40; Secondary 53C15.

Key words and phrases. real hypersurfaces, complex two-plane Grassmannians, commut- ing structure Jacobi operator, Reeb flow.

This work was supported by grant Proj. No. KRF-2007-313-C00058 from Korea Research Foundation.

⃝c2008 The Korean Mathematical Society 495

differential equation which inspires Jacobi operators. The Jacobi operator is defined by (RX(Y))(p) = (R(Y, X)X)(p), whereRdenotes the curvature tensor of M and X, Y denote tangent vector fields on M. Then we see that RX is a self-adjoint endomorphism on the tangent space of M and is related to the differential equation, so called Jacobi equation, which is given by∇γ^′(∇^′γY) + R(Y, γ^′)γ^′ = 0 along a geodesic γ onM, whereγ^′ denotes the velocity vector alongγ onM.

When we study a real hypersurfaceM in a complex space formM_n(c),c̸=0, we will call the Jacobi operator on M with respect to the Reeb vector ξ the structure Jacobi operator onM and will denote it byRξ, where Rξ is defined byRξ(X) =R(X, ξ)ξfor the curvature tensorRofM and any tangent vector field X onM.

For a commuting problem in quaternionic space forms Berndt [2] has intro- duced the notion of normal Jacobi operator ¯R(X, N)N ∈EndTxM, x∈M for real hypersurfacesM in quaternionic projective spaceQP^mor in quaternionic hyperbolic spaceQH^m, where ¯Rdenotes the curvature tensor of a quaternionic projective space QP^mand a quaternionic hyperbolic space QH^m. He [2] also has shown that the curvature adaptedness, that is, the normal Jacobi operator commutes with the shape operatorA, is equivalent to the fact that the distri- butions DandD^⊥= Span{ξ₁, ξ₂, ξ₃}are invariant by the shape operatorAof M, whereT_xM =D⊕D^⊥,x∈M.

Now let us consider a complex two-plane Grassmannians G₂(C^m+2) which consists of all complex 2-dimensional linear subspaces in C^m+2. Then the situation for real hypersurfaces in G₂(C^m+1) with normal Jacobi operator ¯R_N is not so simple and will be quite different from the cases mentioned above.

The ambient spaceG2(C^m+2) is known to be the unique compact irreducible Riemannian symmetric space equipped with both a K¨ahler structure J and a quaternionic K¨ahler structure J not containing J (See Berndt and Suh [3]).

So, in G2(C^m+2) we have the two natural geometric conditions for real hyper- surfaces that [ξ] = Span {ξ} or D^⊥ = Span {ξ1, ξ2, ξ3} is invariant under the shape operator. By using such two conditions Berndt and Suh [3] have proved the following:

Theorem A. Let M be a connected real hypersurface in G2(C^m+2), m ≥3.

Then both[ξ]andD^⊥ are invariant under the shape operator of M if and only if either

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

(B) m is even, say m = 2n, and M is an open part of a tube around a totally geodesic QPⁿ inG2(C^m+2).

If theReebvector fieldξ of a real hypersurfaceM inG2(C^m+2) is invariant by the shape operator, M is said to be aHopf hypersurface. In such a case the integral curves of the Reeb vector field ξ are geodesics (See Berndt and Suh [4]). Moreover, the flow generated by the integral curves of the structure

vector fieldξfor Hopf hypersurfaces inG₂(C^m+2) is said to be ageodesic Reeb flow. Moreover, we say M is with non-vanishing geodesic Reeb flow if the corresponding principal curvature αis non-vanishing.

On the other hand, we say that the Reeb vector field is Killing if the Lie derivative along the direction of the structure vector field ξ vanishes, that is, Lξg = 0, where g denotes the Riemannian metric induced from G2(C^m+2).

Then this is equivalent to the fact that the structure tensorϕcommutes with the shape operatorAofM inG2(C^m+2). This condition also has the geometric meaning that the flow of Reeb vector field is isometric. Moreover, Berndt and Suh [4] have proved that real hypersurfaces in G2(C^m+2) with isometric flow is of a tube over a totally geodesicG2(C^m+1) inG2(C^m+2).

Now let us introduce a structure Jacobi operatorR_ξ in such a way that Rξ(X) =R(X, ξ)ξ

for the curvature tensor R(X, Y)Z of M in G2(C^m+2), where ξ denotes the structure vector,X, Y andZany tangent vector fields ofM inG2(C^m+2). Then the structure Jacobi operatorRξis said to becommutingif the structure Jacobi operator Rξ commutes with the structure tensorϕ, that is,Rξ◦ϕ=ϕ◦Rξ.

Recently, some geometric properties for such a structure Jacobi operatorRξ

of real hypersurfaces in complex space formsM_n(c) have been studied by many authors (See [5], [8], and [9]). Among them commuting and parallel properties of such a structure Jacobi operator was studied by Ki, P´erez, Santos and Suh [5]. Moreover,D-parallel or Lieξ-parallel of the structure Jacobi operator are studied by P´erez, Santos, and Suh (See [8] and [9]).

Now let us put the structure vector ξ =−J N into the curvature tensorR of a real hypersurfaceM inG2(C^m+2), whereN denotes a unit normal vector of M in G2(C^m+2). Then for any tangent vector field X on M in G2(C^m+2) we calculate the structure Jacobi operator Rξ in such a way that

Rξ(X) =R(X, ξ)ξ

=X−η(X)ξ−∑3 ν=1

{η_ν(X)ξ_ν−η(X)η_ν(ξ)ξ_ν + 3g(ϕνX, ξ)ϕνξ+ην(ξ)ϕνϕX}

+αAX−η(AX)Aξ, where αdenotes the function defined byg(Aξ, ξ).

Related to such a structure Jacobi operator Rξ, in this paper we give a classification of real hypersurfaces M in G2(C^m+2) with commuting structure Jacobi operator, that isR_ξ◦ϕ=ϕ◦R_ξ, as follows:

Theorem. LetM be a real hypersurface inG₂(C^m+2)with non-vanishing Reeb flow and commuting structure Jacobi operator. If the D component of the structure vectorξis invariant by the shape operator, then M is congruent to a tube over a totally geodesic G2(C^m+1)inG2(C^m+2).

1. Riemannian geometry of G₂(C^m+2)

In this section we summarize basic material about G2(C^m+2), for details we refer to [3] and [4]. By G2(C^m+2) we denote the set of all complex two- dimensional linear subspaces inC^m+2. The special unitary groupG=SU(m+ 2) acts transitively on G2(C^m+2) with stabilizer isomorphic to K=S(U(2)× U(m)) ⊂G. Then G2(C^m+2) can be identified with the homogeneous space G/K, which we equip with the unique analytic structure for which the natural action ofGonG2(C^m+2) becomes analytic. Denote bygandkthe Lie algebra of GandK, respectively, and bym the orthogonal complement ofkin gwith respect to the Cartan-Killing formBofg. Theng=k⊕mis anAd(K)-invariant reductive decomposition of g. We put o=eK and identify T_oG₂(C^m+2) with min the usual manner. SinceBis negative definite ong, its negative restricted tom×myields a positive definite inner product onm. ByAd(K)-invariance of B this inner product can be extended to aG-invariant Riemannian metricgon G₂(C^m+2). In this wayG₂(C^m+2) becomes a Riemannian homogeneous space, even a Riemannian symmetric space. For computational reasons we normalize g so that the maximal sectional curvature of (G2(C^m+2), g) is eight. Since G2(C³) is isometric to the three-dimensional complex projective space CP² with constant holomorphic sectional curvature eight we will assume m ≥ 2 from now on. Note that the isomorphismSpin(6)≃SU(4) yields an isometry between G2(C⁴) and the real Grassmann manifold G⁺₂(R⁶) of oriented two- dimensional linear subspaces of R⁶.

The Lie algebra k has the direct sum decomposition k =su(m)⊕su(2)⊕ R, where R denotes the center of k. Viewing k as the holonomy algebra of G2(C^m+2), the center R induces a K¨ahler structure J and the su(2)-part a quaternionic K¨ahler structureJ onG2(C^m+2). IfJ1 is any almost Hermitian structure in J, thenJ J1 =J1J, and J J1 is a symmetric endomorphism with (J J1)²=I and tr(J J1) = 0.This fact will be used frequently throughout this paper.

A canonical local basisJ₁, J₂, J₃ofJ consists of three local almost Hermit- ian structures J_ν in Jsuch that J_νJ_ν+1 =J_ν+2 =−J_ν+1J_ν, where the index is taken modulo three. Since Jis parallel with respect to the Riemannian con- nection ¯∇of (G₂(C^m+2), g), there exist for any canonical local basisJ₁, J₂, J₃ ofJ three local one-formsq1, q2, q3such that

(1.1) ∇¯XJ_ν=q_ν+2(X)J_ν+1−q_ν+1(X)J_ν+2 for all vector fieldsX onG2(C^m+2).

Let p ∈ G2(C^m+2) and W a subspace of TpG2(C^m+2). We say that W is a quaternionic subspace of TpG2(C^m+2) if J W ⊂ W for all J ∈ Jp. And we say that W is a totally complex subspace of TpG2(C^m+2) if there exists a one-dimensional subspace V of Jp such that J W ⊂ W for all J ∈ V and J W ⊥W for all J ∈V^⊥ ⊂Jp. Here, the orthogonal complement of V in Jp

is taken with respect to the bundle metric and orientation onJ for which any

local oriented orthonormal frame field of J is a canonical local basis of J. A quaternionic (resp. totally complex) submanifold ofG2(C^m+2) is a submanifold all of whose tangent spaces are quaternionic (resp. totally complex) subspaces of the corresponding tangent spaces ofG2(C^m+2).

The Riemannian curvature tensor ¯R ofG2(C^m+2) is locally given by

(1.2)

R(X, Y¯ )Z

=g(Y, Z)X−g(X, Z)Y + g(J Y, Z)J X

−g(J X, Z)J Y −2g(J X, Y)J Z +

∑3

ν=1

{g(JνY, Z)JνX−g(JνX, Z)JνY −2g(JνX, Y)JνZ}

+

∑3

ν=1

{g(J_νJ Y, Z)J_νJ X−g(J_νJ X, Z)J_νJ Y},

where J1, J2, J3 is any canonical local basis ofJ.

2. Some fundamental formulas for real hypersurfaces in G2(C^m+2) In this section we want to derive the normal Jacobi operator from the curva- ture tensor of complex two-plane Grassmannian G2(C^m+2) given in (1.2) and the equation of Gauss. Moreover, in this section we derive some basic formulae from the Codazzi equation for a real hypersurface in G₂(C^m+2) (See [3], [4], [11], [12], [14], and [15]).

Let M be a real hypersurface of G₂(C^m+2), that is, a hypersurface of G₂(C^m+2) with real codimension one. The induced Riemannian metric on M will also be denoted by g, and ∇ denotes the Riemannian connection of (M, g). Let N be a local unit normal field of M and A the shape operator of M with respect to N. The K¨ahler structure J ofG2(C^m+2) induces onM an almost contact metric structure (ϕ, ξ, η, g). Furthermore, let J1, J2, J3 be a canonical local basis of J. Then each Jν induces an almost contact metric structure (ϕν, ξν, ην, g) onM. Using the above expression for ¯R, the Codazzi equation becomes

(∇XA)Y −(∇YA)X =η(X)ϕY −η(Y)ϕX−2g(ϕX, Y)ξ +

∑3

ν=1

{η_ν(X)ϕ_νY −η_ν(Y)ϕ_νX−2g(ϕ_νX, Y)ξ_ν}

+

∑3

ν=1

{ην(ϕX)ϕνϕY −ην(ϕY)ϕνϕX}

+

∑3

ν=1

{η(X)ην(ϕY)−η(Y)ην(ϕX)} ξν .

The following identities can be proved in a straightforward method and will be used frequently in subsequent calculations:

(2.1)

ϕν+1ξν =−ξν+2, ϕνξν+1=ξν+2, ϕξ_ν=ϕ_νξ, η_ν(ϕX) =η(ϕ_νX), ϕ_νϕ_ν+1X =ϕ_ν+2X+η_ν+1(X)ξ_ν, ϕν+1ϕνX =−ϕν+2X+ην(X)ξν+1. Now let us put

(2.2) J X =ϕX+η(X)N, JνX=ϕνX+ην(X)N

for any tangent vector X of a real hypersurface M in G₂(C^m+2), where N denotes a normal vector ofM inG₂(C^m+2). Then from this and the formulae (1.1) and (2.1) we have that

(2.3) (∇Xϕ)Y =η(Y)AX−g(AX, Y)ξ, ∇Xξ=ϕAX, (2.4) ∇Xξν =qν+2(X)ξν+1−qν+1(X)ξν+2+ϕνAX,

(2.5) (∇Xϕ_ν)Y =−q_ν+1(X)ϕ_ν+2Y +q_ν+2(X)ϕ_ν+1Y +η_ν(Y)AX

−g(AX, Y)ξν.

Summing up these formulae, we find the following

(2.6)

∇X(ϕνξ) =∇X(ϕξν)

= (∇Xϕ)ξ_ν+ϕ(∇Xξ_ν)

=qν+2(X)ϕν+1ξ−qν+1(X)ϕν+2ξ+ϕνϕAX

−g(AX, ξ)ξν+η(ξν)AX.

Moreover, from J Jν=JνJ, ν= 1,2,3,it follows that (2.7) ϕϕνX=ϕνϕX+ην(X)ξ−η(X)ξν.

Then from (1.2) and the above formulae, the equation of Gauss is given by

(2.8)

R(X, Y)Z

=g(Y, Z)X−g(X, Z)Y

+ g(ϕY, Z)ϕX−g(ϕX, Z)ϕY −2g(ϕX, Y)ϕZ +∑3

ν=1{g(ϕνY, Z)ϕνX−g(ϕνX, Z)ϕνY −2g(ϕνX, Y)ϕνZ} +∑3

ν=1{g(ϕ_νϕY, Z)ϕ_νϕX−g(ϕ_νϕX, Z)ϕ_νϕY}

−∑3

ν=1{η(Y)ην(Z)ϕνϕX−η(X)ην(Z)ϕνϕY}

−∑3

ν=1{η(X)g(ϕνϕY, Z)−η(Y)g(ϕνϕX, Z)}ξν

+g(AY, Z)AX−g(AX, Z)AY.

3. Proof of our main theorem

LetM be a real hypersurface inG2(C^m+2) with commuting structure Jacobi operator, that is,Rξ◦ϕ=ϕ◦Rξ.

Now by the equation of Gauss (2.8), we define a structure Jacobi operator Rξ in such a way that

Rξ(X) =R(X, ξ)ξ

=X−η(X)ξ−∑3 ν=1

{ην(X)ξν−η(X)ην(ξ)ξν

+ 3g(ϕνX, ξ)ϕνξ+ην(ξ)ϕνϕX} +αAX−η(AX)Aξ.

Then it follows that

Rξ(ϕX) =ϕX−∑3 ν=1

{ην(ϕX)ξν−3ην(X)ϕνξ + 3η(ξν)η(X)ϕνξ−ην(ξ)ϕνX

+ην(ξ)η(X)ϕνξ} −η(AϕX)Aξ+αAϕX, ϕRξ(X) =ϕX−∑3

ν=1

{ην(X)ϕξν

−η(X)ην(ξ)ϕξν−3η(ϕνX)ξν+ 4ην(ξ)η(ϕνX)ξ

−ην(ξ)ϕνX+ην(ξ)η(X)ϕξν} +αϕAX−η(AX)ϕAξ.

Then the commuting Jacobi structure operator,R_ξ◦ϕ=ϕ◦R_ξ is given by (3.1) 4∑3

ν=1

{η_ν(ϕX)ξ_ν−η_ν(X)ϕξ_ν+η(X)η_ν(ξ)ϕξ_ν−η_ν(ξ)η(ϕ_νX)ξ}

=α(Aϕ−ϕA)X+η(AX)ϕAξ−η(AϕX)Aξ.

Since we have assumed thatM is Hopf, by differentiatingAξ=αξ(See Berndt and Suh [3]) we have

(3.2)

α(Aϕ+ϕA)X−2AϕAX+ 2ϕX

= 2∑3

ν=1{−ην(X)ϕξν−ην(ϕX)ξν

−ην(ξ)ϕνX+ 2ην(ξ)η(X)ϕξν+ 2ην(ξ)ην(ϕX)ξ}.

By the assumption that the structure vector ξ is principal in (3.1) and (3.2), we also have

(3.3) AϕAX−αAϕX−ϕX=∑3

ν=1{3η_ν(X)ϕξ_ν−η_ν(ϕX)ξ_ν +ην(ξ)ϕνX−4ην(ξ)η(X)ϕξν}. Now in this paper we prove the following

Proposition 3.1. Let M be a hypersurface in G₂(C^m+2) with commuting structure Jacobi operator and non-vanishing geodesic Reeb flow. If the Dcom- ponent of the structure vector ξ is invariant by the shape operator, then the structure vector ξbelongs to either the distribution Dor the distribution D^⊥. Proof. Let us put ξ = η(X0)X0+η(ξ1)ξ1 for some unit X0∈D and ξ1∈D^⊥. Since we have assumed M is Hopf,Aξ=αξ gives that

η(X₀)AX₀+η(ξ₁)Aξ₁=αη(X₀)X₀+αη(ξ₁)ξ₁. Then it follows that

(∗) Aξ1=αξ1, AX0=αX0 and η2(ξ) =η3(ξ) = 0.

Then putting X =X0 into (3.1) and usingg(X0, ϕνξ) =−g(ϕνX0, ξ) = 0, we have

(3.4) 4η(X₀)η₁(ξ)ϕξ₁=α(Aϕ−ϕA)X₀=αAϕX₀−α²ϕX₀. Also by puttingX =X0 into (3.3), we have

(3.5)

AϕAX₀−αAϕX₀−ϕX₀

=η1(ξ)ϕ1X0−4η1(ξ)η(X0)ϕξ1

=η1(ξ)ϕ1X0−4η1(ξ)η(X0)ϕξ1

=η₁(ξ)ϕ₁X₀−4η₁(ξ)η(X₀)²ϕ₁X₀

=η1(ξ)(1−4η(X0)²)ϕ1X0. Then from (3.5) and (∗) we have

(3.6) η(ξ1)²= 1−η(X0)²=g(ϕX0, ϕX0) =η1(ξ)²(1−4η(X0)²)². If η(ξ₁) = 0, then by (∗) we know that the structure vectorξ belongs to the distribution D. If η1(ξ)̸=0, then (3.6) gives 1−4η(X0)² = ±1. Then we consider the following two subcases.

Sub. 1) 1−4η(X₀)²= 1.

In such a subcase η(X₀) = 0. Then it follows thatξ=η(ξ₁)ξ₁∈D^⊥. Sub. 2) 1−4η(X0)²=−1.

It follows that η(X0) =η(ξ1) = ±^√1₂. Then without loss of generality we consider that

ξ= 1

√2X₀+ 1

√2ξ₁. Then (3.5) and (∗) give the following

(3.7) −ϕX₀=− 1

√2ϕ₁X₀. On the other hand, (3.4) implies the following

(3.8) AϕX0= (α+ 2

α)ϕX0.

Then by puttingX =ξ₁ into (3.1) and using (3.7), (3.8), we have

(3.9)

4∑3 ν=1

{ην(ϕξ1)ξν−ϕξ1+η(ξ1)²ϕ1ξ−η1(ξ)η(ϕ1ξ)}

=α(Aϕ−ϕA)ξ1

=α(Aϕ₁ξ−ϕAξ₁)

=α(Aη(X0)ϕ1X0−αη(X0)ϕ1X0)

=αη(X0)(Aϕ1X0−αϕ1X0)

=α {

((α+ 2

α)ϕX0−αϕX0) }

= 2ϕX₀.

On the other hand, the left side of (3.9) becomes 4∑3

ν=1

{η_ν(ϕξ₁)ξ_ν−ϕξ₁+η(ξ₁)²ϕ₁ξ−η₁(ξ)η(ϕ₁ξ)}

=−4η(X₀)³ϕ₁X₀=−2ϕX₀. From this, together with (3.9), we haveϕX0= 0. This gives

X₀=η(X₀)ξ= 1

√2( 1

√2X₀+ 1

√2ξ₁).

This impliesX0=ξ1, which makes a contradiction. So we complete the proof

of our proposition. ¤

4. Key propositions

In this section we want to give a complete proof of our main theorem. In order to do this, let us use Proposition 3.1. First we consider the case that ξ∈D^⊥. Accordingly, we may putξ=ξ1. Then (3.1) implies the following Proposition 4.1. Let us consider the same assumptions as in Proposition3.1.

If ξ∈D^⊥, then M is congruent to a tube over a totally geodesicG2(C^m+1) in G2(C^m+2).

Proof. Since we have assumedξ∈D^⊥, we may putξ=ξ1. Then from (3.1) we know that

α(Aϕ−ϕA)X= 4∑3 ν=1

{ην(ϕX)ξν−ην(X)ϕξν−ην(ξ)η(ϕνX)ξ}= 0

for any vector field X on M in G2(C^m+2). Then by a non-vanishing Reeb flow we know that the structure tensor ϕ commutes with the shape operator A of M in G2(C^m+2). This means that the Reeb flow is isometric. Then by a theorem due to Berndt and Suh [4]M is locally congruent to a tube over a

totally geodesic G2(C^m+1) in G2(C^m+2) ¤

On the other hand, we introduce the following derived from Aξ = αξ in Berndt and Suh [3]

(4.1)

αg((Aϕ+ϕA)X, Y)−2g(AϕAX, Y) + 2g(ϕX, Y)

= 2∑3

ν=1{ην(X)ην(ϕY)−ην(Y)ην(ϕX)

−g(ϕνX, Y)ην(ξ)−2η(X)ην(ϕY)ην(ξ) + 2η(Y)η_ν(ϕX)η_ν(ξ)}.

Next also by Proposition 3.1 we are able to consider the case thatξ∈D. Now we assert the following

Proposition 4.2. Let us consider the same assumption as in Proposition3.1.

If ξ∈D, theng(AD,D^⊥) = 0.

Proof. The formula (4.1) forξ∈Dgives the following (4.2) α(Aϕ+ϕA)−2AϕAX+ 2ϕX=−2∑3

ν=1{ην(X)ϕξν+ην(ϕX)ξν}. Moreover, from (3.1) andξ∈Dwe have

(4.3) 4∑3

ν=1η_ν(ϕX)ξ_ν−4∑3

ν=1η_ν(X)ϕξ_ν=α(Aϕ−ϕA)X,

where we have used thatM is with geodesic Reeb flow, that is,Aξ=αξ. Then from (4.2) and (4.3) we have the following

Lemma 4.3. Forξ∈Dwe have αAϕX−A²ϕX+ϕX =∑3

ν=1ην(ϕX)ξν−3∑3

ν=1ην(X)ϕξν

−4 α

∑3

ν=1ην(ϕX)Aξν+ 4 α

∑3

ν=1ην(X)Aϕξν. Now let us consider the maximal distribution hspanned by the orthogonal complement of the structure vector ξ of M in G2(C^m+2). Then by replacing ϕX ofX∈hwe have

(4.4)

A²X−αAX−X = −∑3

ν=1ην(X)ξν−3∑3

ν=1ην(ϕX)ϕξν

+4 α

∑3

ν=1ην(X)Aξν+ 4 α

∑3

ν=1ην(ϕX)Aϕξν. Then for any ξµ∈D^⊥ andξµ∈h, we have

(4.5) αA²ξµ−(α²+ 4)Aξµ= 0.

From this, let us take an inner product (4.4) withX∈D. Then it follows that

(4.6)

αg(A²ξµ, X)

=αg(ξµ, A²X)

=αg(ξ_µ, αAX+X−∑3

ν=1η_ν(X)ξ_ν−3∑3

ν=1η_ν(ϕX)ϕξ_ν +4

α

∑3

ν=1ην(X)Aξν+ 4 α

∑3

ν=1ην(ϕX)Aϕξν)

=α²g(ξµ, AX) + 4∑3

ν=1g(ξµ, Aϕξν)ην(ϕX).

For any X∈Dorthogonal toϕ1ξ, ϕ2ξandϕ3ξ we have αg(A²ξ_µ, X) =α²g(ξ_µ, AX).

From this, together with (4.5), it follows that

(4.7) 0 = αg(A²ξµ, X)−(α²+ 4)g(Aξµ, X)

= −4g(Aξµ, X) for any X∈Dorthogonal to ϕ₁ξ, ϕ₂ξandϕ₃ξ.

Let us put X in (4.6) byϕ_λξ∈D,λ= 1,2,3. Then it follows that αg(A²ξ_µ, ϕ_λξ) =α²g(ξ_µ, Aϕ_λξ) + 4∑3

ν=1g(ξ_µ, Aϕξ_ν)η_ν(ϕ²ξ_λ)

=α²g(ξ_µ, Aϕ_λξ)−4g(ξ_µ, Aϕξ_λ).

Comparing this one with the formula obtained from (4.5) by taking an inner product with ϕλξ gives

(α²+ 4)g(Aξ_µ, ϕ_λξ) =αg(A²ξ_µ, ϕ_λξ) =α²g(ξ_µ, Aϕ_λξ)−4g(ξ_µ, Aϕ_λξ).

From this it follows that

(4.8) 8g(Aξµ, ϕλξ) = 0.

Summing up (4.7) and (4.8), we conclude that g(Aξµ, X) = 0

for any X∈D. That is, g(AD,D^⊥) = 0. This completes the proof of our

Proposition 4.2. ¤

By Proposition 4.2 and a theorem due to Berndt and Suh [3] we know that M is congruent to a tube over a totally geodesic and totally real quaternionic projective space QPⁿ, n= 2m, inG₂(C^m+2).

It remains to check whether such kind of hypersurfaces satisfy commuting structure Jacobi operatoror not.

Let us recallR_ξ◦ϕ=ϕ◦R_ξ forξ∈Dandξis principal. Then we have (4.9) α(Aϕ−ϕA)X = 4∑3

ν=1{ην(ϕX)ξν−ην(X)ϕξν}. We introduce a proposition due to Berndt and Suh [3] as follows:

Proposition B. Let M be a connected real hypersurface of G₂(C^m+2). Sup- pose that AD ⊂D, Aξ =αξ, andξ is tangent to D. Then the quaternionic dimension m of G2(C^m+2) is even, say m= 2n, andM has five distinct con- stant principal curvatures

α=−2 tan(2r), β= 2 cot(2r), γ= 0 , λ= cot(r), µ=−tan(r) with some r∈(0, π/4). The corresponding multiplicities are

m(α) = 1 , m(β) = 3 =m(γ) , m(λ) = 4n−4 =m(µ) and the corresponding eigenspaces are

Tα=Rξ , Tβ =JJ ξ , Tγ=Jξ , Tλ , Tµ , where

Tλ⊕Tµ= (HCξ)^⊥ , JTλ=Tλ , JTµ=Tµ , J Tλ=Tµ .

In Proposition B we consider a vectorX∈T_λ such thatAX=λX = cotrX.

From this we have

α(AϕX−ϕAX) =α(µ−λ)ϕX= 0.

This means cot²r+ 1 = 0, which makes a contradiction. So a real hypersurface M in G2(C^m+2) with commuting structure Jacobi operator does not exist.

From this we give a complete proof of our theorem in the introduction.

Remark 4.1. A tube over a totally geodesic G₂(C^m+1) in G₂(C^m+2) in The- orem A has commuting shape operator on the distributions D and D^⊥. Of course, it is Hopf and naturally in Section 3 we have asserted that such a hypersurface satisfyRξ◦ϕ=ϕ◦Rξ.

Remark 4.2. A tube over a totally real totally geodesicQPⁿ inG2(C^m+2) has not commuting shape operator on the distributionsDandD^⊥. In Section 4 we have proved that such a hypersurface is Hopf but can not satisfyRξ◦ϕ=ϕ◦Rξ. Acknowledgments. The present authors would like to express their sincere gratitude to the referee for his careful reading of our manuscript and useful comments.

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Young Jin Suh

Department of Mathematics Kyungpook National University Taegu 702-701, Korea

E-mail address: yjsuh@knu.ac.kr

Hae Young Yang

Department of Mathematics Kyungpook National University Taegu 702-701, Korea

E-mail address: yang9973@naver.com