Real Hypersurfaces with Killing Shape Operator in the Complex Quadric

REAL HYPERSURFACES WITH KILLING SHAPE OPERATOR IN THE COMPLEX QUADRIC JUAN DE DIOS P´ EREZ, IMSOON JEONG, JUNHYUNG KO, AND YOUNG JIN SUH Abstract. We introduce the notion of Killing shape operator for real hypersurfaces in the complex quadric Qm=SOm+2/SOmSO2. The Killing shape operator implies that the unit normal vector field Nbecomes A-principal or A-isotropic. Then according to each case, we give a complete classification of real hypersurfaces in Qm= SOm+2/SOmSO2with Killing shape operator. 1. Introduction When we consider some Hermitian symmetric spaces of rank 2, we can usually give examples of Riemannian symmetric spaces SUm+2/S(U2Um) and SU2,m/S(U2Um), which are said to be complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians respectively (see [15], [16], and [17] ). These are viewed as Hermitian symmetric spaces and quaternionic K¨ahler symmetric spaces equipped with the K¨ahler structure J and the quaternionic K¨ahler structure J. In the complex projective space CPm+1 and the quaternionic projective space QPm+1 some classifications of real hypersurfaces related to commuting Ricci tensor were investigated by Kimura [9], and P´erez and Suh [11], [12] respectively. The classification problems of real hypersurfaces of the complex 2-plane Grassmannian G2(Cm+2) = SUm+2/S(U2Um) with certain geometric conditions were mainly discussed in Jeong, Kim and Suh [2], Jeong, Machado, P´erez and Suh [3], [4], Suh [15], [16], [17], where the classification of contact hypersurfaces,parallel Ricci tensor,harmonic curvature and Jacobi operator of a real hypersurface in G2(Cm+2) were extensively studied. Moreover, in [17] we have asserted that the Reeb flow on a real hypersurface in SU2,m/S(U2Um) is isometric if and only if Mis an open part of a tube around a totally geodesic SU2,m−1/S(U2Um−1)⊂SU2,m/S(U2Um) . As another kind of Hermitian symmetric space with rank 2 of compact type different from the above ones, we can consider the example of complex quadric Qm= SOm+2/SOmSO2, which is a complex hypersurface in complex projective space CPm+1 (see Klein [5], [6], [8] and Smyth [14]). The complex quadric can also be regarded as a 2010 Mathematics Subject Classification: Primary 53C40. Secondary 53C55. Key words: Killing shape operator, A-isotropic, A-principal, K¨ahler structure, complex conjugation, complex quadric. This work was supported by grant Proj. No. NRF-2015-R1A2A1A-01002459 from National Research Foundation of Korea. The first author was supported by MCT-FEDER project MTM-2013-47828-C2-1-P, the second and third authors were supported by NRF-2017-R1A2B4005317, and the fourth by Bokhyun Research 2017. 2JUAN DE DIOS P´ EREZ, IMSOON JEONG, JUNHYUNG KO, AND YOUNG JIN SUH kind of real Grassmann manifold of compact type with rank 2 (see Kobayashi and Nomizu [10]). Accordingly, the complex quadric admits two important geometric structures, a complex conjugation structure Aand a K¨ahler structure J, which anti-commute with each other, that is, AJ =−JA. Then for m≥2 the triple (Qm, 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], [7] and Reckziegel [13]). Apart from the complex structure Jthere is another distinguished geometric structure on Qm, namely a parallel rank two vector bundle Awhich contains an S1-bundle of real structures, that is, complex conjugations Aon the tangent spaces of Qm. This geometric structure determines a maximal A-invariant subbundle Qof the tangent bundle TM of a real hypersurface Min Qm. Moreover, the derivative of the complex conjugation Aon Qmis defined by (¯ ∇XA)Y=q(X)JAY for any vector fields Xand Yon Mand qdenotes a certain 1-form defined on M. When the shape operator Sof Min Qmsatisies (∇XS)Y= (∇YS)Xfor any X, Y on Min Qm, we say that the shape operator is of Codazzi type. In [18] we gave a nonexistence property of real hypersurfaces of Codazzi type in the complex quadric Qmwith parallel shape operator as follows: Theorem A. There do not exist any real hypersurfaces in complex quadric Qm,m≥3, with shape operator of Codazzi type. Recall that a nonzero tangent vector W∈T[z]Qmis called singular if it is tangent to more than one maximal flat in Qm. There are two types of singular tangent vectors for the complex quadric Qm: 1. If there exists a conjugation A∈Asuch that W∈V(A), then Wis singular. Such a singular tangent vector is called A-principal. 2. If there exist a conjugation A∈Aand orthonormal vectors X, Y ∈V(A) such that W/||W|| = (X+JY )/√2, then Wis singular. Such a singular tangent vector is called A-isotropic. When we consider a hypersurface Min the complex quadric Qm, under the assumption of some geometric properties the unit normal vector field Nof Min Qmcan be divided into two classes if either Nis A-isotropic or A-principal (see [18] and [19]). In the first case where Nis A-isotropic, we have shown in Suh [18] that Mis locally congruent to a tube over a totally geodesic CPkin Q2k. In the second case, when the unit normal N is A-principal, we proved that a contact hypersurface Min Qmis locally congruent to a tube over a totally geodesic and totally real submanifold Smin Qm(see [19]). The shape operator Sof Min Qmis said to be Killing if the operator Ssatisfies (∇XS)Y+ (∇YS)X= 0 for any X, Y ∈TzM,z∈M. The equation is equivalent to (∇XS)X= 0 for any X∈TzM, z∈M, because of linearization. Moreover, we can give the geometric meaning of Killing Jacobi tensor as follows: KILLING SHAPE OPERATOR IN THE COMPLEX QUADRIC 3 When we consider a geodesic γwith initial conditions such that γ(0) = zand ˙γ(0) = X. Then the transformed vector field S˙γis Levi-Civita parallel along the geodesic γof the vector field X(see Blair [1] and Tachibana [23]). In the study of real hypersurfaces in the complex quadric Qmwe 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 Nis A-isotropic or A-principal. So in [19] we gave a classification with the further assumption of A-isotropic. But fortunately, when we consider Killing shape operator, first we can assert that the unit normal vector field Nbecomes either A-isotropic or A-principal as follows: Main Theorem 1. Let Mbe a Hopf real hypersurface in Qm,m≥3, with Killing shape operator. Then the unit normal vector field Nis singular, that is, Nis A-isotropic or A-principal. Then motivated by such a result, next we give a complete classification for real hypersurfaces in the complex quadric Qmwith Killing shape operator as follows: Main Theorem 2. Let Mbe a Hopf real hypersurface in the complex quadric Qm, m≥4, with Killing shape operator. Then Mhas 4distinct constant principal curvatures given by α=0, β =γ= 0, λ =(α2+ 1) + √(α2+ 1)2+ 2α2 2α,and µ=(α2+ 1) −√(α2+ 1)2+ 2α2 2α with corresponding principal curvature spaces respectively Tα= [ξ], Tβ= [AN], Tγ= [Aξ], ϕ(Tλ) = Tµ,and dim Tλ=dim Tµ=m−2. Usually, Killing shape operator is a generalization of parallel shape operator Sof Min Qm, that is, ∇XS= 0 for any tangent vector field Xon M. The parallelism of shape operator has a geometric meaning that every eigen spaces of the shape operator Sare parallel along any direction on Min Qm. Then naturally, by Theorem 2 above we give the following Corollary. There do not exist any Hopf real hypersurfaces in Qm,m≥3, with parallel shape operator. 2. The complex quadric For more background to this section we refer to [5], [10], [13], [18], [19] and [20]. The complex quadric Qmis the complex hypersurface in CPm+1 which is defined by the equation z2 0+··· +z2 m+1 = 0, where z0, . . . , zm+1 are homogeneous coordinates on CPm+1. We equip Qmwith the Riemannian metric gwhich is induced from the FubiniStudy metric ¯gon CPm+1 with constant holomorphic sectional curvature 4. The FubiniStudy metric ¯gis defined by ¯g(X, Y ) = Φ(JX, Y ) for any vector fields Xand Yon CPm+1 and a globally closed (1,1)-form Φ given by Φ = −4i∂ ¯ ∂logfjon an open set Uj={[z0, . . . , zj, . . . , zm+1]∈CPm+1|zj=0}, where the function fjdenotes fj=∑m+1 k=0 tk j¯ tk j, 4JUAN DE DIOS P´ EREZ, IMSOON JEONG, JUNHYUNG KO, AND YOUNG JIN SUH and tk j=zk zjfor j, k = 0,···, m+1. Then naturally the K¨ahler structure on CPm+1 induces canonically a K¨ahler structure (J, g) on the complex quadric Qm. The complex projective space CPm+1 is a Hermitian symmetric space of the special unitary group SUm+2, namely CPm+1 =SUm+2/S(Um+1U1). We denote by o= [0, . . . , 0,1] ∈ CPm+1 the fixed point of the action of the stabilizer S(Um+1U1). The special orthogonal group SOm+2 ⊂SUm+2 acts on CPm+1 with cohomogeneity one. The orbit containing o is a totally geodesic real projective space RPm+1 ⊂CPm+1. The second singular orbit of this action is the complex quadric Qm=SOm+2/SOmSO2. This homogeneous space model leads to the geometric interpretation of the complex quadric Qmas the Grassmann manifold G+ 2(Rm+2) of oriented 2-planes in Rm+2. It also gives a model of Qmas a Hermitian symmetric space of rank 2. The complex quadric Q1is isometric to a sphere S2 with constant curvature, and Q2is isometric to the Riemannian product of two 2-spheres with constant curvature. For this reason we will assume m≥3 from now on. In another way, the complex projective space CPm+1 is defined by using the Hopf fibration π:S2m+3→CPm+1, z→[z], which is said to be a Riemannian submersion. Then naturally we can consider the following diagram for the complex quadric Qmas follows: ˜ Q=π−1(Q)˜ i −−−→ S2m+3⊂Cm+2 π  yπ  y Q=Qmi −−−→ CPm+1 The submanifold ˜ Qof codimension 2 in S2m+3 is called the Stiefel manifold of orthonormal 2-frames in Rm+2, which is given by ˜ Q={x+iy∈Cm+2|g(x, x) = g(y, y) = 1 2and g(x, y) = 0}, where g(x, y) = ∑m+2 i=1 xiyifor any x= (x1, . . ., xm+2) and y= (y1, . . ., ym+2)∈Rm+2. Then the tangent space is decomposed as TzS2m+3 =Hz⊕Fzand Tz˜ Q=Hz(Q)⊕Fz(Q) at z=x+iy∈˜ Qrespectively, where the horizontal subspaces Hzand Hz(Q) are given by Hz= (Cz)⊥and Hz(Q) = (Cz⊕C¯z)⊥, and Fzand Fz(Q) are fibers which are isomorphic to each other. Here Hz(Q) becomes a subspace of Hzof real codimension 2 and orthogonal to the two unit normals −¯zand −J¯z. Explicitly, at the point z=x+iy∈˜ Qit can be described as Hz={u+iv∈Cm+2|g(x, u) + g(y, v) = 0, g(x, v) = g(y, u)} and Hz(Q) = {u+iv∈Hz|g(u, x) = g(u, y) = g(v, x) = g(v, y) = 0}, where Cm+2 =Rm+2⊕iRm+2, and g(u, x) = ∑m+2 i=1 uixifor any u= (u1, . . ., um+2), x= (x1, . . ., xm+2)∈Rm+2. These spaces can be naturally projected by the differential map π∗as π∗Hz=Tπ(z)CPm+1 and π∗Hz(Q) = Tπ(z)Qrespectively. This gives that at the point π(z) = [z] the tangent KILLING SHAPE OPERATOR IN THE COMPLEX QUADRIC 5 subspace T[z]Qmbecomes a complex subspace of T[z]CPm+1 with complex codimension 1 and has two unit normal vector fields −¯zand −J¯z(see Reckziegel [13]). Then let us denote by A¯zthe shape operator of Qmin CPm+1 with respect to the unit normal ¯z. It is defined by A¯zw=¯ ∇w¯z= ¯wfor a complex Euclidean connection ¯ ∇ induced from Cm+2 and all w∈T[z]Qm. That is, the shape operator A¯zis just a complex conjugation restricted to T[z]Qm. Moreover, it satisfies the following for any w∈T[z]Qm and any λ∈S1⊂C A2 λ¯zw=Aλ¯zAλ¯zw=Aλ¯zλ¯w =λA¯zλ¯w=λ¯ ∇λ¯w¯z=λ¯ λ¯ ¯w =|λ|2w=w. Accordingly, A2 λ¯z=Ifor any λ∈S1. So the shape operator A¯zbecomes an anti-commuting involution such that A2 ¯z=Iand AJ =−JA on the complex vector space T[z]Qmand T[z]Qm=V(A¯z)⊕JV (A¯z), where V(A¯z) = Rm+2 ∩T[z]Qmis the (+1)-eigenspace and JV (A¯z) = iRm+2 ∩T[z]Qmis the (−1)-eigenspace of A¯z. That is, A¯zX=Xand A¯zJX =−JX, respectively, for any X∈V(A¯z). Geometrically this means that the shape operator A¯zdefines a real structure on the complex vector space T[z]Qm, or equivalently, is a complex conjugation on T[z]Qm. Since the real codimension of Qmin CPm+1 is 2, this induces an S1-subbundle Aof the endomorphism bundle End(TQm) consisting of complex conjugations. There is a geometric interpretation of these conjugations. The complex quadric Qmcan be viewed as the complexification of the m-dimensional sphere Sm. Through each point [z]∈Qmthere exists a one-parameter family of real forms of Qmwhich are isometric to the sphere Sm. These real forms are congruent to each other under action of the center SO2of the isotropy subgroup of SOm+2 at [z]. The isometric reflection of Qmin such a real form Smis an isometry, and the differential at [z] of such a reflection is a conjugation on T[z]Qm. In this way the family Aof conjugations on T[z]Qmcorresponds to the family of real forms Smof Qmcontaining [z], and the subspaces V(A)⊂T[z]Qmcorrespond to the tangent spaces T[z]Smof the real forms Smof Qm. The Gauss equation for Qm⊂CPm+1 implies that the Riemannian curvature tensor ¯ R of Qmcan be described in terms of the complex structure Jand the complex conjugations A∈A: ¯ R(X, Y )Z=g(Y, Z)X−g(X, Z)Y+g(JY, Z)JX −g(JX, Z)JY −2g(JX, Y )JZ +g(AY, Z)AX −g(AX, Z)AY +g(JAY, Z)JAX −g(JAX, Z)JAY. Note that Jand each complex conjugation Aanti-commute, that is, AJ =−JA for each A∈A. For every unit tangent vector W∈T[z]Qmthere exist a conjugation A∈Aand orthonormal vectors X, Y ∈V(A) such that W= cos(t)X+ sin(t)JY 6JUAN DE DIOS P´ EREZ, IMSOON JEONG, JUNHYUNG KO, AND YOUNG JIN SUH for some t∈[0, π/4]. The singular tangent vectors correspond to the values t= 0 and t=π/4. When W=Xfor X∈V(A), t= 0, there exist many kinds of maximal 2-flats RX+RZfor Z∈V(A) orthogonal to X∈V(A). So the tangent vector Xis said to be singular. When W= (X+JY )/√2 for t=π 4, it becomes also a singular tangent vector, which belongs to many kinds of maximal 2-flats given by R(X+JY )+RZfor any Z∈V(A) orthogonal to X∈V(A) or R(X+JY ) + RJZ for any JZ∈JV (A). If 0 < t < π/4 then the unique maximal flat containing Wis RX⊕RJY . 3. Some general equations Let Mbe a real hypersurface in Qmand denote by (ϕ, ξ, η, g) the induced almost contact metric structure. Note that ξ=−JN, where Nis a (local) unit normal vector field of Mand ηthe corresponding 1-form defined by η(X) = g(ξ, X) for any tangent vector field Xon M. The tangent bundle TM of Msplits orthogonally into TM =C ⊕ Rξ, where C= ker(η) is the maximal complex subbundle of TM. The structure tensor field ϕ restricted to Ccoincides with the complex structure Jrestricted to C, and ϕξ = 0. At each point z∈Mwe define a maximal A-invariant subspace of TzM,z∈Mas follows: Qz={X∈TzM|AX ∈TzMfor all A∈Az}. Then we want to introduce an important lemma which will be used in the proof of our main Theorem in the introduction. Lemma 3.1. ([18]) For each z∈Mwe have (i) If Nzis A-principal, then Qz=Cz. (ii) If Nzis not A-principal, there exist a conjugation A∈Aand orthonormal vectors X, Y ∈V(A)such that Nz= cos(t)X+ sin(t)JY for some t∈(0, π/4]. Then we have Qz=Cz⊖C(JX +Y). We now assume that Mis a Hopf hypersurface. Then the Reeb vector field ξ=−JN satisfies the following Sξ =αξ, where Sdenotes the shape operator of the real hypersurface Mfor a smooth function α=g(Sξ, ξ) on M. When we consider the transformed JX by the K¨ahler structure Jon Qmfor any vector field Xon Min Qm, we may put JX =ϕX +η(X)N for a unit normal Nto M. Then we now consider the equation of Codazzi 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(JAY, Z)−g(Y, Aξ)g(JAX, Z). (3.1) Putting Z=ξin (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(JY, Aξ) + g(Y, Aξ)g(JX, Aξ). KILLING SHAPE OPERATOR IN THE COMPLEX QUADRIC 7 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=ξyields 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(JY, Aξ) + g(Y, Aξ)g(JX, 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∈Mwe can choose A∈Azsuch that N= cos(t)Z1+ sin(t)JZ2 for some orthonormal vectors Z1, Z2∈V(A) and 0 ≤t≤π 4(see Proposition 3 in [13]). Note that tis a function on M. First of all, since ξ=−JN, we have AN = cos(t)Z1−sin(t)JZ2, ξ= sin(t)Z2−cos(t)JZ1, Aξ = sin(t)Z2+ cos(t)JZ1. (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(JY, Aξ) + g(Y, Aξ)g(JX, Aξ) −2g(X, AN)g(ξ, Aξ)η(Y)+2g(Y, AN)g(ξ, Aξ)η(X). (3.4) 4. Killing shape operator and a Key Lemma By the equation of Gauss, the curvature tensor R(X, Y )Zfor a real hypersurface Min Qminduced from the curvature tensor ¯ Rof Qmcan be described in terms of the complex 8JUAN DE DIOS P´ EREZ, IMSOON JEONG, JUNHYUNG KO, AND YOUNG JIN SUH structure Jand the complex conjugation A∈Aas 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(JAY, Z)JAX −g(JAX, Z)JAY +g(SY, Z)SX −g(SX, Z)SY for any X, Y, Z∈TzM,z∈M. Now let us put AX =BX +ρ(X)N, for any vector field X∈TzQm,z∈M,ρ(X) = g(AX, N), where BX and ρ(X)Nrespectively denote the tangential and normal component of the vector field AX. Then Aξ =Bξ +ρ(ξ)Nand ρ(ξ) = g(Aξ, N) = 0. Then it follows that AN =AJξ =−JAξ =−J(Bξ +ρ(ξ)N) =−(ϕBξ +η(Bξ)N). The shape operator Sof Min Qmis said to be Killing if the operator Ssatisfies (∇XS)Y+ (∇YS)X= 0.(4.1) for any X, Y ∈TzM,z∈M. From (4.1), together with the equation of Codazzi (3.1), it follows that 2g((∇XS)Y, 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(JAY, Z)−g(Y, Aξ)g(JAX, Z). (4.2) Since we have assumed the real hypersurface Min Qmis Hopf, then Sξ =αξ. This gives (∇XS)ξ= (Xα)ξ+αϕSX −SϕSX. From this, let us put Y=ξin (4.2) and use g(Aξ, N) = 0, we see that 2g((Xα)ξ+αϕSX −SϕSX, Z) = −g(ϕX, Z) + g(X, AN)g(Aξ, Z) +g(X, Aξ)g(JAξ, Z)−g(ξ, Aξ)g(JAX, Z).(4.3) Here, let us put X=ξin (4.3) and also use g(ξ, AN) = 0, we have 2(ξα)η(Z) = g(ξ, Aξ)g(JAξ, Z)−g(ξ, Aξ)g(JAξ, Z) = 0. From this we get ξα = 0. Then the derivative Y α in section 3 becomes Y α = 2g(Y, AN)g(ξ, Aξ). From this, together with (4.3), it follows that 2g(2g(X, AN)g(ξ, Aξ)ξ+αϕSX −SϕSX, Z) = −g(ϕX, Z) + g(X, AN)g(Aξ, Z) +g(X, Aξ)g(JAξ, Z)−g(ξ, Aξ)g(JAX, Z).(4.4) Then by putting Z=ξinto (4.3), we have 4g(X, AN)g(ξ, Aξ) =g(X, AN)g(Aξ, ξ) + g(X, Aξ)g(JAξ, ξ) −g(ξ, Aξ)g(JAX, ξ) =2g(X, AN)g(Aξ, ξ). (4.5) KILLING SHAPE OPERATOR IN THE COMPLEX QUADRIC 9 Since g(Aξ, N) = 0, (4.5) gives that g(Aξ, ξ)g(AN, X) = 0. Then we have g(Aξ, ξ) = 0 or (AN)T= 0, where (AN)Tdenotes the tangential part of the vector AN. Summing up above discussions, we conclude the following Lemma 4.1. Let Mbe a Hopf real hypersurface in Qm,m≥3, with Killing shape operator. Then the unit normal vector field Nis singular, that is, Nis A-isotropic or A-principal. Proof. In above discussion, let us consider the first case g(Aξ, ξ) = 0. Then it implies that 0 = g(Aξ, ξ) = g(AJN, JN) = −g(JAN, JN) = −g(AN, N). If we insert N= cos tZ1+ sin tJZ2for Z1, Z2∈V(A) into the above equation, we have cos2t−sin2t= 0. Then by section 2, we have t=π 4, that is, N=1 √2(X+JY ) for some X, Y ∈V(A). So the unit normal Nis A-isotropic. Next we consider the case that (AN)T= 0. Then AN = (AN)T+g(AN, N)N= g(AN, N)N. So it follows that N=A2N=g(AN, N)AN =g2(AN, N)N. So g(AN, N) = ±1 gives that AN =±N. That is, the unit normal Nis A-principal.  Then we are able to consider the classification of Killing shape operator Sof Min Qm into two cases, that the unit normal Nis A-principal or Nis A-isotropic. In section 5 we will discuss a classification of real hypersurfaces in Qmwith Killing shape operator and A-isotropic unit normal and in section 6 a non-existence of Killing shape operator for hypersurfaces in Qmwhen Nis A-principal will be explained in detail. 5. Proof of Main Theorem with A-isotropic unit normal In this section let us assume that the unit normal vector field Nis A-isotropic. Then the normal vector field Ncan be written N=1 √2(Z1+JZ2) for Z1, Z2∈V(A), where V(A) denotes the (+1)-eigenspace of the complex conjugation A∈A. Then it follows that AN =1 √2(Z1−JZ2), AJN =−1 √2(JZ1+Z2),and JN =1 √2(JZ1−Z2). From this, together with (3.3) and the anti-commuting AJ =−JA, it follows that g(ξ, Aξ) = g(JN, AJN) = 0, g(ξ, AN) = 0 and g(AN, N) = 0. Then (4.3) gives the following for any X, Z∈TzM,z∈M 2g(αϕSX −SϕSX, Z) = −g(ϕX, Z) + g(X, AN)g(Aξ, Z) + g(X, Aξ)g(JAξ, Z) =−g(ϕX, Z) + g(X, AN)g(Aξ, Z)−g(X, Aξ)g(AN, Z).(5.1)