First we introduce the notion of parallel Ricci tensor for real hypersurfaces in the complex quadricQm=SOm+2/SOmSO2

PARALLEL RICCI TENSOR

YOUNG JIN SUH*

Abstract. First we introduce the notion of parallel Ricci tensor for real hypersurfaces in the complex quadricQ^m=SO_m+2/SO_mSO₂ . Next, according to theA-principal or theA-isotropic unit normalN, we give a complete classification of real hypersurfaces in Q^m=SO_m+2/SO_mSO₂ with parallel Ricci tensor.

1. Introduction

When we consider some Hermitian symmetric spaces of rank 2, usually we can give ex- amples of Riemannian symmetric spacesSUm+2/S(U2Um) andSU2,m/S(U2Um), which are said to be complex two-plane Grassmannians and complex hyperbolic two-plane Grass- mannians respectively (see [1], [2], [4] and [13]). Those are said to be Hermitian symmetric spaces and quaternionic K¨ahler symmetric spaces equipped with the K¨ahler structure J and the quaternionic K¨ahler structureJonSU_2,m/S(U₂U_m). The rank ofSU_2,m/S(U₂U_m) is 2 and there are exactly two types of singular tangent vectorsX of G₂(C^m+2) which are characterized by the geometric properties J X ∈JX and J X ⊥JX respectively.

As another kind of Hermitian symmetric space with rank 2 of compact type different from the above ones, we can give an example of complex quadricQ^m =SO_m+2/SO_mSO₂, which is a complex hypersurface in complex projective space CP^m+1 (see Berndt and Suh [3], and Smyth [11]). The complex quadric also can be regarded as a kind of real Grassmann manifolds of compact type with rank 2 (see Kobayashi and Nomizu [7]).

Accordingly, the complex quadric admits two important geometric structures, that is, a complex conjugation structureAand a K¨ahler structureJ, which anti-commute with each other, that is, AJ =−J A. 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 [5] and Reckziegel [10]).

For the complex projective space CP^m+1 and the quaternionic projective spaceQP^m+1 some classifications related to parallel Ricci tensor were respectively investigated by Kimura [6], and P´erez and Suh [8]. For the complex 2-plane Grassmannian G₂(C^m+2) = SU_m+2/S(U_mU₂) the classification was obtained by Berndt and Suh [1]. By using such a classification P´erez and Suh [9] proved a non-existence property for Hopf hypersurfaces in

* Corresponding author. E-mail address: yjsuh@knu.ac.kr

2010Mathematics Subject Classification: Primary 53C40. Secondary 53C55.

Key words: parallel Ricci tensor, A-isotropic,A-principal, K¨ahler structure, complex conjugation, com- plex quadric.

This work was supported by grant Proj. No. NRF-2012-R1A2A2A-01043023 from National Research Foundation of Korea.

YOUNG JIN SUH*

G₂(C^m+2) with parallel and commuting Ricci tensor. Suh [12] extended this result to hy- persurfaces inG₂(C^m+2) with parallel Ricci tensor. Moreover, Suh and Woo [15] studied another non-exitence property for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians SU_2,m/S(U₂U_m) with parallel Ricci tensor.

It is known that 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) ⊂ G₂(C^m+2) in [2]. Moreover, in [13] we have asserted that the Reeb flow on a real hypersurface in SU_2,m/S(U₂U_m) is isometric if and only if M is an open part of a tube around a totally geodesic SU_2,m−1/S(U₂U_m−1) ⊂ SU_2,m/S(U₂U_m). Here, the Reeb flow on real hypersurfaces in SUm+2/S(UmU2) or SU2,m/S(U2Um) is said to be isometric if the shape operator commutes with the structure tensor. In the paper [3] due to Berndt and Suh, we investigated this problem for the complex quadric Q^m =SO_m+2/SO_mSO₂ and obtained the follwoing result:

Let M be a real hypersurface of the complex quadric Q^m, m ≥3. The Reeb flow on M is isometric if and only if m is even, saym= 2k, andM is an open part of a tube around a totally geodesic CP^k ⊂Q^2k.

When we consider a hypersurfaceM in the complex quadricQ^m, the unit normal vector field N of M in Q^m can be divided into two classes that N is A-isotropic or A-principal (See [3], [4], and [14]). In the first class where M has an A-isotropic unit normal N, we have asserted in [3] that M is locally congruent to a tube over a totally geodesic CP^k in Q^2k. In the second class forN isA-principal we have proved that a contact hypersurface M inQ^m is locally congruent to a tube over a totally geodesic and totally real submanifold S^m in Q^m (see [4]). From such a view point, in this paper let us consider the notion of Ricci parallel for hypersurfaces in Q^m , that is, ∇Ric= 0. Then motivated by a theorem due to the A-principal normal vector field [4] for contact hypersurfaces in Q^m we assert the following

Theorem 1. There does not exist any Hopf hypersurfaces in the complex quadric Q^m with parallel Ricci tensor and A-principal normal vector field.

In section 4 we will give this proof in detail. Now at each pointz ∈M let us consider a maximal A-invariant subspace Qz of T_zM, z∈M, defined by

Qz ={X∈T_zM |AX ∈T_zM for all A∈A_z}

ofT_zM,z∈M. Thus for a case where the unit normal vector fieldN isA-isotropic it can be easily checked that the orthogonal complementQ^⊥z =Cz⊖Qz,z∈M, of the distribution Q in the complex subbundleC, becomesQ^⊥z = Span [Aξ, AN]. Here it can be easily checked that the vector fields Aξ and AN belong to the tangent space T_zM, z∈M if the unit normal vector field N becomes A-isotropic. Then motivated by the above result, in this paper we give another theorem for real hypersurfaces in the complex quadric Q^m with parallel Ricci tensor andA-isotropic unit normal as follows:

Theorem 2. Let M be a Hopf real hypersurface in the complex quadric Q^m, m≥4, with parallel Ricci tensor andA-isotropic unit normal N. If the shape operator commutes with the structure tensor on the distributionQ^⊥, then M has 3distinct constant principal curvatures, three of which are given by

α = 0, γ = 0, λ =−√

3, and µ=√ 3

or

α=√

2(m−3), γ = 0, λ= 0, and µ=− 2

√2(m−3) with corresponding principal curvature spaces

T_α = [ξ], T_γ = [Aξ, AN], ϕ(T_λ) = T_µ,dim T_λ =dim T_µ=m−2.

Our paper is organized as follows. In section 2 we present basic material about the complex quadric Q^m, including its Riemannian curvature tensor and a description of its singular vectors ofQ^mlikeA-principal orA-isotropic unit normal vector filed. Apart from the complex structureJ there is another distinguished geometric structure onQ^m, namely a parallel rank two vector bundleAwhich contains anS¹-bundle of real structures, that is, complex conjugationsAon the tangent spaces ofQ^m. This geometric structure determines a maximal A-invariant subbundle Qof the tangent bundle T M of a real hypersurfaceM in Q^m. Accordingly, in section 3, we investigate the geometry of this subbundle Q for hypersurfaces in Q^m and some equations including Codazzi and fundamental formulas related to the vector fieldsξ, N, Aξ, andAN for the complex conjugation Aof M inQ^m. Finally, in sections 4 and 5, we present the proof of Theorems 1 and 2, respectively.

In section 4, the first step is to derive the Ricci tensor from the equation of Gauss for real hypersurfacesM inQ^m and to get some formulas by the assumption of parallel Ricci tensor andA-principal normal vector field, and show that a real hypersurface inQ^m which is a tube over an m-dimensional unit sphere S^m does not admit a parallel Ricci tensor.

The next step is to show that there do not exist any real hypersurfaces in the complex quadric Q^m with parallel Ricci tensor andA-principal normal vector field.

In section 5, we give a complete proof of Theorem 2. The first part of this proof is devoted to give some fundamental formulas from Ricci parallel and A-isotropic unit normal vector field. Then in the latter part of the proof we will use the expression of the shape operator for real hypersurfaces in Q^m when the shape operator and the structure tensor commutes on the distribution Q^⊥, where Q^⊥ denotes an orthogonal complement of the maximalA-invariant subspace Qin the complex subbundle C of T_zM,z∈M inQ^m.

2. The complex quadric

For more details in this section we refer to [3], [4], [5], [7], [10] and [14]. The complex quadric Q^m is the complex hypersurface in CP^m+1 which is defined by the equationz₀²+ . . .+z_m+1² = 0, wherez₀, . . . , z_m+1 are homogeneous coordinates onCP^m+1. We equipQ^m with the Riemannian metricgwhich is induced from the Fubini-Study metric ¯gonCP^m+1 with constant holomorphic sectional curvature 4. The Fubini-Study metric ¯g is defined by

¯

g(X, Y) = Φ(J X, Y) for any vector fieldsXand Y onCP^m+1 and a globally closed (1,1)- form Φ given by Φ =−4i∂∂¯logf_j on an open set U_j ={[z⁰, z¹,· · ·, z^m+1]∈CP^m+1|z^j̸=0}, where the function f_j denotes f_j =∑_m+1

k=0t^k_j¯t^k_j, and t^k_j = ^z_z^kj for j, k = 0,· · ·, m+ 1. Then naturally the K¨ahler structure onCP^m+1 induces canonically a K¨ahler structure (J, g) on the complex quadric Q^m.

The complex projective spaceCP^m+1is a Hermitian symmetric space of the special uni- tary groupSU_m+2, namelyCP^m+1 =SU_m+2/S(U_m+1U₁). We denote byo= [0, . . . ,0,1]∈

YOUNG JIN SUH*

CP^m+1 the fixed point of the action of the stabilizer S(U_m+1U₁). The special orthogonal group SO_m+2 ⊂SU_m+2 acts on CP^m+1 with cohomogeneity one. The orbit containing o is a totally geodesic real projective space RP^m+1 ⊂ CP^m+1. The second singular orbit of this action is the complex quadric Q^m = SO_m+2/SO_mSO₂. This homogeneous space model leads to the geometric interpretation of the complex quadricQ^m as the Grassmann manifold G⁺₂(R^m+2) of oriented 2-planes in R^m+2. It also gives a model of Q^m as a Her- mitian symmetric space of rank 2. The complex quadric Q¹ is isometric to a sphere S² with constant curvature, andQ² is isometric to the Riemannian product of two 2-spheres with constant curvature. For this reason we will assume m ≥3 from now on.

For a nonzero vector z ∈ C^m+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 CP^m+1. As usual, for each [z] ∈CP^m+1 we identify T_zCP^m+1 with the orthogonal complement C^m+2 ⊖[z] of [z] in C^m+2. For [z]∈ Q^m the tangent spaceT_zQ^m can then be identified canonically with the orthogonal complement C^m+2⊖([z]⊕[¯z]) of [z]⊕[¯z] inC^m+2 (see Kobayashi and Nomizu [7]). Note that ¯z∈ν_zQ^m is a unit normal vector of Q^m in CP^m+1 at the point [z].

We denote byA_¯zthe shape operator ofQ^m inCP^m+1with respect to the unit normal ¯z.

It is defined byA_¯zw=∇wz¯=wfor a complex Euclidean connection∇and allw∈T_zQ^m. That is, the shape operatorA_¯z is just complex conjugation restricted toT_zQ^m. The shape operator A_z¯ is an anti-commuting involution such that A²_¯z = I and AJ = −J A on the complex vector spaceT_zQ^m and

T_zQ^m =V(A_¯z)⊕J V(A_z¯),

where V(A_z¯) = R^m+2 ∩T_zQ^m is the (+1)-eigenspace and J V(A_z¯) = iR^m+2 ∩T_zQ^m is the (−1)-eigenspace of A_z¯. That is, A_z¯X =X and A_¯zJ X =−J X, respectively, for any X∈V(A_z¯).

Geometrically this means that the shape operator A_z¯ defines a real structure on the complex vector space T_zQ^m, or equivalently, is a complex conjugation on T_zQ^m. Since the real codimension of Q^m inCP^m+1 is 2, this induces anS¹-subbundle A of the endo- morphism bundle End(T Q^m) consisting of complex conjugations.

There is a geometric interpretation of these conjugations. The complex quadricQ^m can be viewed as the complexification of the m-dimensional sphere S^m. Through each point z ∈ Q^m there exists a one-parameter family of real forms of Q^m which are isometric to the sphere S^m. These real forms are congruent to each other under action of the center SO₂ of the isotropy subgroup of SO_m+2 at z. The isometric reflection of Q^m in such a real formS^m is an isometry, and the differential at z of such a reflection is a conjugation on T_zQ^m. In this way the family A of conjugations on T_zQ^m corresponds to the family of real forms S^m of Q^m containingz, and the subspaces V(A)⊂T_zQ^m correspond to the tangent spaces T_zS^m of the real forms S^m of Q^m.

The Gauss equation for Q^m ⊂CP^m+1 implies that the Riemannian curvature tensor ¯R ofQ^m can be described in terms of the complex structureJ and the complex conjugations A∈A:

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 +g(AY, Z)AX−g(AX, Z)AY +g(J AY, Z)J AX −g(J AX, Z)J AY.

Note that J and each complex conjugationA anti-commute, that is, AJ =−J Afor each A∈A.

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. There are two types of singular tangent vectors for the complex quadric Q^m:

1. If there exists a conjugation A ∈ A such that W ∈ V(A), then W is singular.

Such a singular tangent vector is calledA-principal.

2. If there exist a conjugation A ∈ A and orthonormal vectors X, Y ∈ V(A) such thatW/||W||= (X+J Y)/√

2, thenW is singular. Such a singular tangent vector is calledA-isotropic.

For every unit tangent vector W ∈ T_zQ^m there exist a conjugation A ∈ A and or- thonormal vectorsX, Y ∈V(A) such that

W = cos(t)X+ sin(t)J Y

for some t ∈ [0, π/4]. The singular tangent vectors correspond to the values t = 0 and t=π/4. If 0 < t < π/4 then the unique maximal flat containingW isRX⊕RJ Y. Later we will need the eigenvalues and eigenspaces of the Jacobi operator R_W =R(·, W)W for a singular unit tangent vector W.

1. If W is an A-principal singular unit tangent vector with respect to A ∈ A, then the eigenvalues of R_W are 0 and 2 and the corresponding eigenspaces are RW ⊕ J(V(A)⊖RW) and (V(A)⊖RW)⊕RJ W, respectively.

2. If W is an A-isotropic singular unit tangent vector with respect to A ∈ A and X, Y ∈ V(A), then the eigenvalues of R_W are 0, 1 and 4 and the corresponding eigenspaces are RW ⊕C(J X +Y), T_zQ^m⊖(CX⊕CY) and RJ W, respectively.

3. Some general equations

LetM be a real hypersurface inQ^mand denote by (ϕ, ξ, η, g) the induced almost contact metric structure. Note that ξ = −J N, 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 T M of M splits orthogonally into T M = C ⊕Rξ, whereC = ker(η) is the maximal complex subbundle of T M. The structure tensor field ϕ restricted to C coincides with the complex structureJ restricted toC, and ϕξ= 0.

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

Qz ={X ∈T_zM |AX ∈T_zM for all A∈A_z}. Lemma 3.1. (see [14]) For each z ∈M we have

(i) If N_z isA-principal, then Qz =Cz.

(ii) IFN_z is notA-principal, there exist a conjugationA∈Aand orthonormal vectors X, Y ∈V(A) such that N_z = cos(t)X+ sin(t)J Y for some t∈ (0, π/4]. Then we have Qz =Cz⊖C(J X+Y).

We now assume thatM is a Hopf hypersurface. Then we have Sξ =αξ

YOUNG JIN SUH*

with the smooth function α =g(Sξ, ξ) on M. When we consider a transform J X of the Kaehler structure J onQ^m for any vector field X on M inQ^m, we may put

J X =ϕX +η(X)N

for a unit normal N toM. Then we now consider the Codazzi equation

g((∇XS)Y −(∇YS)X, Z) = η(X)g(ϕY, Z)−η(Y)g(ϕX, Z)−2η(Z)g(ϕX, Y) +g(X, AN)g(AY, Z)−g(Y, AN)g(AX, Z) +g(X, Aξ)g(J AY, Z)−g(Y, Aξ)g(J AX, Z).

(3.1) Putting Z =ξ in (3.1) we get

g((∇XS)Y −(∇YS)X, ξ) = −2g(ϕX, Y)

+g(X, AN)g(Y, Aξ)−g(Y, AN)g(X, Aξ)

−g(X, Aξ)g(J Y, Aξ) +g(Y, Aξ)g(J X, Aξ).

On the other hand, we have

g((∇XS)Y −(∇YS)X, ξ)

= g((∇XS)ξ, Y)−g((∇YS)ξ, X)

= (Xα)η(Y)−(Y α)η(X) +αg((Sϕ+ϕS)X, Y)−2g(SϕSX, Y).

Comparing the previous two equations and putting X =ξ yields

Y α= (ξα)η(Y)−2g(ξ, AN)g(Y, Aξ) + 2g(Y, AN)g(ξ, Aξ).

Reinserting this into the previous equation yields g((∇XS)Y −(∇YS)X, ξ)

= −2g(ξ, AN)g(X, Aξ)η(Y) + 2g(X, AN)g(ξ, Aξ)η(Y) +2g(ξ, AN)g(Y, Aξ)η(X)−2g(Y, AN)g(ξ, Aξ)η(X) +αg((ϕS +Sϕ)X, Y)−2g(SϕSX, Y).

Altogether this implies

0 =2g(SϕSX, Y)−αg((ϕS+Sϕ)X, Y)−2g(ϕX, Y) +g(X, AN)g(Y, Aξ)−g(Y, AN)g(X, Aξ)

−g(X, Aξ)g(J Y, Aξ) +g(Y, Aξ)g(J X, Aξ)

+ 2g(ξ, AN)g(X, Aξ)η(Y)−2g(X, AN)g(ξ, Aξ)η(Y)

−2g(ξ, AN)g(Y, Aξ)η(X) + 2g(Y, AN)g(ξ, Aξ)η(X).

(3.2)

At each point z ∈M we can choose A∈Az such that N = cos(t)Z₁+ sin(t)J Z₂

for some orthonormal vectors Z₁, Z₂ ∈ V(A) and 0 ≤ t ≤ ^π₄ (see Proposition 3 in [10]).

Note thatt is a function onM. First of all, since ξ=−J N, we have N = cos(t)Z₁+ sin(t)J Z₂,

AN = cos(t)Z₁−sin(t)J Z₂, ξ= sin(t)Z₂−cos(t)J Z₁, Aξ = sin(t)Z₂+ cos(t)J Z₁.

(3.3)

This implies g(ξ, AN) = 0 and hence

0 =2g(SϕSX, Y)−αg((ϕS+Sϕ)X, Y)−2g(ϕX, Y) +g(X, AN)g(Y, Aξ)−g(Y, AN)g(X, Aξ)

−g(X, Aξ)g(J Y, Aξ) +g(Y, Aξ)g(J X, Aξ)

−2g(X, AN)g(ξ, Aξ)η(Y) + 2g(Y, AN)g(ξ, Aξ)η(X).

(3.4)

4. Proof of Theorem 1

By the equation of Gauss, the curvature tensorR(X, Y)Z for a real hypersurfaceM in Q^m induced from the curvature tensor ¯R of Q^m can be described in terms of the complex structure J and the complex conjugation A∈A as follows:

R(X, Y)Z = g(Y, Z)X−g(X, Z)Y +g(ϕY, Z)ϕX−g(ϕX, Z)ϕY −2g(ϕX, Y)ϕZ +g(AY, Z)AX−g(AX, Z)AY +g(J AY, Z)J AX −g(J AX, Z)J AY +g(SY, Z)SX −g(SX, Z)SY

for any X, Y, Z∈T_zM, z∈M.

From this, contractingY and Z onM inQ^m, we have Ric(X) =(2m−1)X−X−ϕ²X−2ϕ²X

−g(AN, N)AX−X+g(AX, N)AN −g(J AN, N)J AX

−X+g(J AX, N)J AN+ (trS)SX −S²X

=(2m−1)X−3η(X)ξ−g(AN, N)AX+g(AX, N)AN

−g(J AN, N)J AX +g(J AX, N)J AN + (trS)SX −S²X,

(4.1)

where we have used the following

∑2m−1

i=1 g(Ae_i, e_i) = T rA−g(AN, N) = −g(AN, N),

∑2m−1

i=1 g(AX, e_i)Ae_i = ∑2m

i=1g(AX, e_i)Ae_i−g(AX, N)AN =X−g(AX, N)AN,

∑2m−1

i=1 g(J Ae_i, e_i)J AX = ∑2m

i=1g(J Ae_i.e_i)J AX−g(J AN, N)J AX

= −g(J AN, N)J AX, and

∑2m−1

i=1 g(J AX, e_i)J Ae_i = ∑2m

i=1g(J AX, e_i)J Ae_i−g(J AX, N)J AN

= J AJ AX−g(J AX, N)J AN

= X−g(J AX, N)J AN.

Now in this section we consider only an A-principal normal vector field N, that is, AN =N, for a real hypersurfaceM inQ^m with parallel Ricci tensor. Then (4.1) becomes Ric(X) = (2m−1)X−2η(X)ξ−AX+hSX−S²X, (4.2)

YOUNG JIN SUH*

where h = trS denotes the mean curvature and is defined by the trace of the shape operator S of M in Q^m. Then from this, by the parallel Ricci tensor, we have

0 =−2g(∇Xξ, Y)ξ−2η(Y)∇Xξ−(∇XA)Y + (Xh)SY +h(∇XS)Y −(∇XS²)Y

=−2g(ϕSX, Y)ξ−2η(Y)ϕSX−(∇XA)Y + (Xh)SY +h(∇XS)Y −(∇XS²)Y,

(4.3)

where (∇XA)Y = ∇X(AY)−A∇XY. Here, AY belongs to T_zM, z∈M, from the fact that g(AY, N) = g(Y, AN) = g(Y, N) = 0 for any tangent vector Y on M. Then by puttingY =ξ in (4.3), we know that

2ϕSX =−(∇XA)ξ+ (Xh)Sξ +h(∇XS)ξ−(∇XS²)ξ

=−q(X)J Aξ−αη(X)AN +α(Xh)ξ +h(∇XS)ξ−(∇XS²)ξ.

(4.4)

In order to get the equation (4.4) we have used the following (∇XA)ξ = ∇X(Aξ)−A∇Xξ

= ( ¯∇X(Aξ))^T −A∇Xξ

= {

( ¯∇XA)ξ+A∇¯Xξ }T

−AϕSX

= q(X)J Aξ+AϕSX+g(SX, ξ)AN −AϕSX

= q(X)J Aξ+αη(X)AN,

where (· · ·)^T denotes the tangential component of the vector (· · ·) in Q^m. Moreover, we get

(∇XS)ξ = ∇X(Sξ)−S∇Xξ = (Xα)ξ+αϕSX −SϕSX, and

(∇XS²)ξ = ∇X(S²ξ)−S²∇Xξ = (Xα²)ξ+α²ϕSX−S²ϕSX.

Then (4.4) can be written as follows:

2ϕSX = −q(X)J Aξ−αη(X)AN +α(Xh)ξ+h(Xα)ξ+hαϕSX−hSϕSX

−(Xα²)ξ−α²ϕSX+S²ϕSX.

Then it can be reduced as follows:

2ϕSX = −q(X)J Aξ−αη(X)AN +hαϕSX−hSϕSX

−α²ϕSX+S²ϕSX.

From this, if we take a tangential part, we have the following:

(2 +α² −hα)ϕSX =−hSϕSX +S²ϕSX (4.5) for any tangent vectorX∈T_zM,z∈M, because we have assumed that the unit vector field N isA-principal, that is, AN =N, and J Aξ =−AJ ξ =−AN.

On the other hand, by (3.4) in section 3 for hypersurface in complex quadricQ^m with A-principal normal vector field N, we have

2SϕSX =α(ϕS+Sϕ)X+ 2ϕX,

where we used g(X, AN) = 0 and g(J X, Aξ) = g(J X, J AN) = 0 from the facts that A-principal normalAN =N and anti-commuting AJ =−J A. From this, it follows that

2S²ϕSX =α(SϕS+S²ϕ)X+ 2SϕX

=α({α

2(Sϕ+ϕS)X+ϕX}

+S²ϕ)

X+ 2SϕX

= α²

2 (Sϕ+ϕS)X+αϕX +αS²ϕX + 2SϕX

(4.6)

Then summing up (4.5) and (4.6), we have (2 +α²−hα)ϕSX

=−h {α

2(Sϕ+ϕS)X+ϕX }

+α²

4 (Sϕ+ϕS)X +α

2ϕX+α

2S²ϕX +SϕX.

(4.7)

Remark 4.1. It is known that in Berndt and Suh [4] a real hypersurface M is a tube over S^m in Q^m if and only if the shape operator S of M satisfies Sϕ+ϕS = kϕ for an non-zero constant k. Then let us check whether a tube overS^m could satisfy (4.7) or not.

Then (4.7) gives

(2 +α²−hα)ϕSX = −h{αk 2 + 1}

ϕX+ α² 4 kϕX +α

2ϕX +α

2S²ϕX +SϕX.

If we consider an eigen vector such that SX = λX, then (Sϕ+ϕS)X =kϕX gives that SϕX = (k−λ)ϕX. From this, together with (4.7) usingαk =−2, the principal curvatures satisfy a quadratic equation such that

αx² −2(α²−hα+ 1)x= 0.

Then λ = 0 or µ = √

2 tan√

2r (see [4]). Moreover, the trace h of the shape operator becomes h=α+ (m−1)k (see also [4]). But for a tube over a sphere S^m we know that

√2 tan√

2r = 2

α(α²−hα+ 1)

= 2(α−h) + 2 α

= 4(m−1) α + 2

α

= 2(2m−1) α

= −(2m−1)√

2 tan√ 2r,

YOUNG JIN SUH*

where in the third equality we used α−h =−(m−1)k = ^2(m−1)_α and in the fifth equality α=−√

2 cot(√

2r)respectively (see Prop. 3.4 in[4]). This gives that2m√

2 tan√

2r = 0, which gives us a contradiction. So we conclude that a real hypersurface in Q^m which is a tube over a m-dimensional sphere S^m does not admit a parallel Ricci tensor. Of course, in this case the unit normal N is A-principal.

If we put SX =λX, then (4.5) gives

(2 +α²−hα)λϕX =−hλSϕX +λS²ϕX.

Moreover, (4.6) gives that

SϕX = αλ+ 2 2λ−αϕX.

From this, together with the above formula, we have (2 +α² −hα)λϕX =−hλ

(αλ+ 2 2λ−α

)

ϕX +λ

(αλ+ 2 2λ−α

)2

ϕX. (4.8)

First we consider the case that λ̸=0 and µ = ^αλ+2_2λ−α are both non-vanishing. Then the function µ= ^αλ+2_2λ−α satisfies the following equation

µ²−hµ+hα−α²−2 = 0.

Then it is equivalent to the following for the function λ λ²−hλ+hα−α²−2 = 0.

Combining these two equations, we have

(λ−µ)(λ+µ−h) = 0.

Then we know that the functionsλ andµare distinct. So it implies thath=λ+µ. Then it follows that

h=λ+µ=α+ (m−1)(λ+µ) =α+ (m−1)h, this implies that h=−_m^α₋₂, which gives us a contradiction.

In fact, in (4.7) with h=−_m^α₋₂ it becomes {

2 + (m−1)α² m−2

}

ϕSX = α²

2(m−2)(ϕS+Sϕ)X−hϕX+ α²

4 (Sϕ+ϕS)X + α

2ϕX+α

2S²ϕX+SϕX.

From this, by taking a symmetric part, we get the following {

3 + (m−1)α² m−2

}

g((ϕS −Sϕ)X, Y) = α

2g((S²ϕ−ϕS²)X, Y).

Then for any principal vector X such that SX = λX and SϕX = µϕX with distinct principal curvatures λ and µwe know

(λ−µ) {

3 + (m−1)α² m−2 − α

2(λ+µ) }

ϕX = 0.

This implies 3 + ^(2(m_2(m−2)^−1)+1)α2 = 0, which gives a contradiction.

Next we consider for the case that λ = 0 and µ= −_α²̸=0. In this case, we can easily verify that h=−_α². Then the equation (4.7), together with hα=−2, gives

(α²+ 4)ϕSX =(Sϕ+ϕS)X−hϕX+ α²

4 (Sϕ+ϕS)X +α

2ϕX+α

2S²ϕX +SϕX.

(4.9) Then by taking a skew-symmetric part of (4.9)we have

(α²+ 5)(ϕS −Sϕ) = α

2(S²ϕ−ϕS²). (4.10)

From this, using SX =λX and SϕX =µϕX, whereµ= ^αλ+2_2λ−α, then we have (λ−µ){

(α²+ 5)−α

2(λ+µ)}

ϕX = 0 (4.11)

Since λ̸=µand hα =−2, (4.10) implies the following 0 =α²+ 5− α

2(λ+µ) = α²+ 5− α

2h=α²+ 6,

which gives also a contradiction even in this case. So we conclude that there do not exist any real hypersurfaces in complex quadric Q^m with parallel Ricci tensor andA-principal normal vector field.

5. Proof of Theorem 2

In this section we want to prove Theorem 2 for real hypersurfaces with parallel Ricci tensor and A-isotropic unit normal vector field. Then in this case we want to make the derivative of the Ricci tensor as follows:

(∇YRic)X =∇Y(Ric(X))−Ric(∇YX)

=−3(∇Yη)(X)ξ−3η(X)∇Yξ

+g(X,∇Y(AN))AN −g(AX, N)∇Y(AN) +g((∇Y(Aξ), X)Aξ+η(AX)∇Y(Aξ) + (Y h)SX +h(∇YS)X−(∇YS²)X.

(5.1)

Since we assumed that the unit normal N isA-isotropic, by the defintion in section 3 we know that t= ^π₄. Then from the expression of the A-isotropic unit normal vector field it becomes N = √¹

2Z₁+√¹

2Z₂ . This implies that

g(ξ, Aξ) = 0, g(ξ, AN) = 0, g(AN, N) = 0, g(Aξ, N) = 0, and

g(J AN, ξ) =−g(AN, N) = 0.

Then it follows that

∇Y(BN) =∇Y(AN) ={( ¯∇YA)N +A∇¯YN}^T ={q(Y)J AN −ASY}^T, and

∇Y(Aξ) ={( ¯∇YA)ξ+A∇¯Yξ}^T

={q(Y)J Aξ+AϕSY}^T,

YOUNG JIN SUH*

where (· · ·)^T denotes the tangential component of the vector (· · ·) inQ^m. By our assump- tion of Ricci parallel, the above formula becomes

0 =−3g(ϕSY, X)ξ−3η(X)ϕSY +{q(Y)g(J AN, X)−g(ASY, X)}AN

−g(AX, N){q(Y)J AN −ASY}^T +{q(Y)g(J Aξ, X) +g(AϕSY, X)}Aξ+η(AX){q(Y)J Aξ+AϕSY}^T + (Y h)SX +h(∇YS)X−(∇YS²)X.

(5.2)

By taking an inner product (5.2) with the Reeb vector field ξ, we have 0 =−3g(ϕSY, X) +g(AX, N)η(ASY) +η(AX)η(AϕSY)

+ (Y h)αη(X) +hg((∇YS)X, ξ)

−g((∇YS²)X, ξ).

(5.3) On the other hand, let us use the following calculation

(∇XS)ξ = (Xα)ξ+αϕSX −SϕSX, (∇XS²)ξ = (Xα²)ξ+α²ϕSX−S²ϕSX.

From this, together with puttingX =ξ in (5.2) and usingg(ξ, AN) = g(Aξ, ξ) = 0, we have

3ϕSY =(Y h)Sξ+h(∇YS)ξ−(∇YS²)ξ

−g(ASY, ξ)AN +g(AϕSY, ξ)Aξ

=(Y h)αξ+h{(Y α)ξ+αϕSY −SϕSY}

− {(Y α²)ξ+α²ϕSY −S²ϕSY}

=αhϕSY −hSϕSY −α²ϕSY +S²ϕSY.

(5.4)

From this, it follows that the functionhα−α²is constant onM. Then it can be rearranged as follows:

(3 +α²−αh)ϕSY =−hSϕSY +S²ϕSY −g(ASY, ξ)AN +g(AϕSY, ξ)Aξ. (5.5) Since the unit normal N isA-isotropic, we know thatg(ξ, Aξ) = 0. Moreover, by Lemma 4.2 due to [14], we have the following

2SϕSX =α(Sϕ+ϕS)X+ 2ϕX−2g(X, AN)Aξ+ 2g(X, Aξ)AN. (5.6) Now let us consider the distribution Q^⊥, which is an orthogonal complement of the maximalA-invariant subspaceQin the complex subbundleC of T_zM,z∈M inQ^m. Then by Lemma 3.1 in section 3, the orthogonal complement Q^⊥ = C⊖Q becomes C⊖Q = Span [AN, Aξ]. From the assumption ofSϕ =ϕS on the distributionQ^⊥ it can be easily checked that the distribution Q^⊥ is invariant by the shape operator S. Then (5.6) gives the following for SAN =λAN

(2λ−α)SϕAN = (αλ+ 2)ϕAN −2Aξ

= (αλ+ 2)ϕAN −2ϕAN

= αλϕAN.

Then Aξ =ϕAN gives the following

SAξ = αλ

2λ−αAξ. (5.7)

Then from the assumptionSϕ =ϕS on Q^⊥ =C⊖Q it follows that λ= _2λ^αλ_−α gives

λ= 0 or λ=α. (5.8)

On the other hand, on a distributionQwe know thatAX∈T_zM,z∈M, becauseAN∈Q.

So (5.6), together with the fact thatg(X, Aξ) = 0 andg(X, AN) = 0 for anyX∈Q, imply that

2SϕSX =α(Sϕ+ϕS)X+ 2ϕX. (5.9)

Then we can take an orthonormal basis X₁,· · ·, X_2(m−2)∈Q such that AX_i = λ_iX_i for i= 1,· · ·, m−2. Then by (5.6) we know that

SϕXi = αλ_i + 2 2λ_i−αϕXi.

Accordingly, by (5.8) the shape operator S can be expressed in such a way that

S =

















α 0 0 0 · · · 0 0 · · · 0 0 0(α) 0 0 · · · 0 0 · · · 0 0 0 0(α) 0 · · · 0 0 · · · 0 0 0 0 λ₁ · · · 0 0 · · · 0 ... ... ... ... . .. ... ... · · · ... 0 0 0 0 · · · λm−2 0 · · · 0 0 0 0 0 · · · 0 µ1 · · · 0 ... ... ... ... ... ... ... . .. ... 0 0 0 0 · · · 0 0 · · · µ_m−2

















So on the distribution Q, for any X, ϕX∈Q such that SX = λX and SϕX = µϕX, µ= ^αλ+2_2λ−α, (5.5) becomes the following

(3 +α² −αh)λϕX =−hλSϕX+λS²ϕX

+λg(X, Aξ)AN +λg(AϕX, ξ)Aξ

=−hλSϕX+λµ²ϕX

=−hλµϕX+λµ²ϕX,

(5.10)

where we used C⊖Q= Span [Aξ, AN] in the second equality.

From this, if there exists a non-vanishing principal curvature λ̸=0, then any principal curvature of the shape operator on the distribution Q satisfies the following quadratic equation

x²−hx+ (αh−α²−3) = 0. (5.11)

Then the discreminant always D = (h−2α)² + 12 > 0, we conclude that there exist two principal curvatures λ and µ satisfy h = λ+µ. Moreover, by the assumption of parallel Ricci tensor ∇Ric = 0, and (5.4) the function hα−α² should be constant. Then summing up these properties and the expressions above, let us consider the first case for non-vanishing λ̸=µas follows:

YOUNG JIN SUH*

h=λ+µ=α+ (m−2)(λ+µ)

=α+ (m−2)h. (5.12)

From this, the constant hα−α² becomes

hα−α² =α{α+ (m−2)(λ+µ} −α²

=(m−2)(λ+µ)α

=(m−2)hα.

Then the constanthαimplies thatα must be constant. Thus the trace of the shape oper- atorhshould be constant. So the above quadratic equation has with constant coefficients.

On the other hand, for SAξ=αAξ and SAN =αAN the constant h becomes h=λ+µ

=3α+ (m−2)(λ+µ)

=3α+ (m−2)h.

(5.13) So if we compare the above two equations (5.12) and (5.13) for the constant mean curvature h, we know that the Reeb functionα should vanish, that is, α= 0. From this, together with the assumptionm≥4, it follows that the trace h=λ+µ= 0, that is,

λ=−µ. (5.14)

Moreover, from the equation (5.11) we may put the principal curvature λ=−√

3 tanr and µ=√

3 cotr. (5.15)

Since the equation (5.11) has only distinct two principal curvatures λ and µ and the princicipal curvature µrelated to the almost complex structure ϕ such that SϕX =µϕX for SX =λX on the distribution Q, that is, Q=V(λ)⊕V(µ), where dimQ= 2(m−2), V(λ) = {X∈Q|SX = λX} and V(µ) = {X∈Q|SX = µX}. Then, we know that dimV(λ) = dimV(µ) =m−2, because λ and µ are different roots of the equation (5.11) and ϕV(λ) = V(µ). Moreover, we know that the functionρ= 0 or ρ=α by the property of the vector AN and Aξ.

Summing up these results, together with (5.14) and (5.15), we may put α = 0, ρ= 0, λ=−√

3 and µ=√

3 respectively with multiplicities 1, 2, (m−2) and (m−2).

Next for the case whereλ= 0 we conclude that on the distributionξ⊕[C⊖Q]⊕Q, where C⊖Q= Span[AN, Aξ], the shape operator becomes

S =

















α 0 0 0 · · · 0 0 · · · 0 0 0(α) 0 0 · · · 0 0 · · · 0 0 0 0(α) 0 · · · 0 0 · · · 0 0 0 0 −_α² · · · 0 0 · · · 0 ... ... ... ... . .. ... ... · · · 0 0 0 0 0 · · · −_α² 0 · · · 0 0 0 0 0 · · · 0 0 · · · 0 ... ... ... ... ... ... ... . .. ...

0 0 0 0 · · · 0 0 · · · 0

















When SAN = 0 and SAξ= 0, the expression for the constant h becomes h =λ+µ=α+ (m−2)(λ+µ).

This gives that α+ (m−3)h= 0.

On the other hand, for λ = 0 we know µ = −_α². So it follows that h =λ+µ = −_α². Then these two equations give α² = 2(m−3). So let us putα=√

2(m−3). Then from the expressions above the trace of the shape operator becomes respectively

h=√

2(m−3)− 2(m−2)

√2(m−3) =− 2

√2(m−3) or

h=√

2(m−3) + 2√

2(m−3)− 2(m−2)

√2(m−3) =− 2

√2(m−3)

according to the eigenvalue 0 or α on the distribution C⊖Q = Span[Aξ, AN]. The first equation is satisfied for all of m. So in this case we may put

α=√

2(m−3), γ = 0, λ= 0, and µ=− 2

√2(m−3) with corresponding principal curvature spaces

T_α = [ξ], T_γ = [Aξ, AN], ϕ(T_λ) = T_µ,dim T_λ = dim T_µ=m−2.

But if we compare the latter equation, we have m= 3. This gives us a contradiction. So the latter case can not be appeared.

Summing up the above discussions, we give a complete proof of our Main Theorem 2 in the introduction.

Remark 5.1. In this remark let us check that whether the Ricci tensor of the tube M over a totally geodesic P_kC in Q^m, m = 2k, mentioned in Berndt and Suh [3] is parallel or not. Then by a theorem due to [3], the shape operator S commutes with the structure tensor ϕ, that is, Sϕ=ϕS. In this case we know that the normal vector field N of M in Q^2k isA-isotropic. So let us suppose that the Ricci tensor of M is parallel. Then for any Y∈Q such that SY =λY the equation (5.5) gives that

(3 +α²−αh)λϕY =−hλ²ϕY +λ³ϕY. (5.16) On the other hand, by (5.6), together with the commuting property Sϕ = ϕS, we know that

(2λ−α)λϕY = (αλ+ 2)ϕY (5.17) From this, we see that the function λ̸=0. Then by (5.16) with λ̸=0, we get

λ²−hλ+ (αh−α²−3) = 0.

Moreover, (5.17) gives that

λ²−αλ−1 = 0.

From these two equations we know that h =α and finally gives us a contradiction. So a tube over P_kC never has a parallel Ricci tensor.

YOUNG JIN SUH*

Acknowledgements The present author would like to express his deep gratitude to Professor J¨urgen Berndt for his great contributions to our works due to [3] and [4].

Motivated by these works, the author has considered such a notion of Ricci parallel for real hypersurfaces in the complex quadricQ^m. The author also wish his thanks to the referee for many kinds of valuable comments to develop the first version of this manuscript.

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Kyungpook National University, College of Natural Sciences, Department of Math- ematics, Daegu 702-701, Republic of Korea