Ruled Real Hypersurfaces in he Complex Quadric

RULED REAL HYPERSURFACES IN THE COMPLEX QUADRIC MAKOTO KIMURA, HYUNJIN LEE, JUAN DE DIOS P´ EREZ AND YOUNG JIN SUH* Abstract. First we introduce the notions of η-parallel and η-commuting shape operator for real hypersurfaces in the complex quadric Qm=SOm+2/SOmSO2. Next we give a complete classification of real hypersurfaces in the complex quadric Qmwith such kind of shape operators. By virtue of this classification we give a new characterization of ruled real hypersurface foliated by complex totally geodesic hyperplanes Qm−1in Qmwhose unit normal vector field in Qmis A-principal. 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 some classifications of real hypersurfaces related to η-parallel shape operator were investigated by Kimura [4], Kimura and Maeda [6] 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 P´erez and Suh [10], and Suh [15], [16], [17], where the classification of contact hypersurfaces,parallel Ricci tensor,harmonic curvature and structure 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 Kobayashi and Nomizu [8] and Smyth [12], [13] and [14]). The complex quadric can 2010 Mathematics Subject Classification: Primary 53C40. Secondary 53C55. *: Corresponding author Key words:η-parallel shape operator, A-isotropic, A-principal, ruled real hypersurface, complex conjugation, complex quadric. The first author was supported by JSPS KAKENHI Grant Number JP16K05119, the second by NRF2016-R1A6A3A-11931947, the third by MCT-FEDER project MTM-2016-78807-C2-1-P, and the fourth by grant Proj. No. NRF-2018-R1D1A1B-05040381 from National Research Foundation of Korea 2 M. KIMURA, H. LEE, J. D. P´ EREZ & Y. J. SUH also be regarded as a kind of real Grassmann manifold of compact type with rank 2. 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 [7] and Reckziegel [11]). 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 given by (¯ ∇XA)Y=q(X)JAY for any vector fields Xand Yon M, where qdenotes a certain 1-form defined on M. 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) := Eig(A, 1), 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 considered of 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 Nis Aprincipal, 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 η-parallel if it satisfies g((∇XS)Y, Z)=0 for any X, Y, Z∈Cz,z∈M, where Czdenotes the orthogonal complement of the Reeb vector field ξz=JNzof Min TzM. Moreover, if the shape operator Sof Min Qmsatisfies g((Sφ −φS)X, Y ) = 0 for any X, Y ∈C, we say that Mis η-commuting. When the Reeb vector field ξis a principal vector field of the shape operator of M in Qm, a real hypersurface Mis said to be Hopf. Now let us introduce another kind of real hypersurfaces which is said to be ruled real hypersurfaces in the complex quadric Qm which are not Hopf as follows: Let γ:I→Qmbe an integral curve of the Reeb vector field ξsuch that γ0(0) = ξp. The distribution C={X∈TM|X⊥ξ}is said to be integrable if [X, Y ]∈C for any vector RULED REAL HYPERSURFACES 3 fields X, Y ∈C. When Mis foliated by the integrable totally geodesic complex hyperplane Qm−1in Qm, then M={x∈Qm−1(t)|t∈I}. In such a case we say that Mis a ruled real hypersurface in Qm. In such a case, the expression of the shape operator Sof the ruled real hypersurface Min Qmbecomes Sξ =αξ +βU SU =βξ SX = 0 for any vector field X⊥ξ, U, where Uis a unit vector field in C,αand βare functions on Mand βdoes not vanish. Then the above expression holds if and only if g(SX, Y ) = 0 for any vector fields Xand Yin C. By the totally geodesic property of the complex hyperplane Qm−1in Qmin the construction of the ruled real hypersurface in Qm, it naturally satisfies the above expression of the shape operator, and conversely if the shape operator satisfies the above formula, we can construct the ruled real hypersurface in Qm. So as a characterization of ruled real hypersurfaces in Qm, we summarize this one as follows: Theorem A. Let Mbe a real hypersurface in Qm,m≥3. Then Mis locally congruent to a ruled real hypersurface foliated by complex totally geodesic Qm−1in Qmif and only if the shape operator Ssatisfies g(SX, Y )=0for any X, Y ∈C. This Theorem A implies that the shape operator Sis η-parallel, that is, g((∇XS)Y, Z) = 0 for any X, Y, Z∈C. By linearization, g((∇XS)X, X) = 0 for any X∈C. Then this is equivalent to the constancy of g(Sγ0, γ0) = ¯g(¯ ∇γ0γ0,¯ ∇γ0γ0), where ¯gand ¯ ∇denote respectively the Riemannian metric and the Riemannian connection of the complex quadric Qm. This means that every geodesic γ:I→Min Qmwhich is orthogonal to the Reeb vector field ξ, that is γ0(0)⊥ξp, and γ(0) = p, has constant first curvature. When the stucture tensor φcommutes with the shape operator S, that is, Sφ =φS, we say that Mhas commuting shape operator. Motivated by this one, Berndt and Suh [2] have proved the following Theorem B. Let Mbe a complete real hypersurface in Qm,m≥3, with commuting shape operator. Then Mis locally congruent to a tube over CPkin Q2k,m= 2k. Motivated by Theorems A and B, and Theorems 5.3 and 5.4 in section 5, we can assert the following Main Theorem. Let Mbe a real hypersurface in the complex quadric Qm,m≥4, with η-parallel and η-commuting shape operator. Then Mis locally congruent to a ruled hypersurface foliated by totally geodesic complex hypersurfaces Qm−1in Qmwith A-principal unit normal vector field. If Mis Hopf and η-commuting, the shape operator of Mcommutes with the structure tensor φ. Then by a result due to Berndt and Suh [3] Mis locally congruent to a tube over a totally geodesic CPkin Q2k. In such a case the unit normal vector field Nis A-isotropic. In section 5 we prove that the unit normal vector field Nof a ruled real hypersurface is A-principal. But in this case Mis non-Hopf. 4 M. KIMURA, H. LEE, J. D. P´ EREZ & Y. J. SUH Remark 1.1. In Remark 4.4, we have mentioned that the unit normal vector field N of a ruled real hypersurface in Qmis either A-principal or A-isotropic. Remark 1.2. In section 6, we construct an example of minimal ruled real hypersurface which is foliated by totally geodesics Qm−1in the complex quadric Qmfrom curves in real projective space RPm+1. 2. The complex quadric For more background to this section we refer to [7], [8], [11], [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|zj6=0}, where the function fjdenotes fj=Pm+1 k=0 tk j¯ tk j, 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}, RULED REAL HYPERSURFACES 5 where g(x, y) = Pm+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) = Pm+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 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 [11]). 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 6 M. KIMURA, H. LEE, J. D. P´ EREZ & Y. J. SUH 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 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, RULED REAL HYPERSURFACES 7 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). 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 [11]). 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) 8 M. KIMURA, H. LEE, J. D. P´ EREZ & Y. J. SUH 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. η-parallel 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 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). Then we assert the following: Lemma 4.1. Let Mbe a real hypersurface in Qm,m≥3, with η-parallel and η-commuting shape operator. Then for any X, Y, Z∈C we have 0 =g(X, AN)g(AY, Z) + g(Y, Aξ)g(AX, φZ)−g(φZ, Aξ)g(AX, Y ) −η(SφZ)g(Y, SX) + g(X, V )g(Y, SZ) + g(Y, V )g(X, SZ). where Cdenotes the orthogonal complement of the Reeb vector field ξand Vis given by φSξ. Proof. The notion of η-commuting shape operator gives g((Sφ −φS)X, Y ) = 0 for any X, Y ∈C. By differentiating this, we have g((∇XS)Y,φZ) + g((∇XS)Z, φY ) = η(SY )g(X, SZ) + η(SZ)g(Y, SX) +g(X, SφY )g(Z, V ) + g(X, SφZ)g(Y, V ).(4.1) Then let us consider cyclic formulas with respect X,Yand Zas follows: g((∇YS)Z,φX) + g((∇YS)X, φZ) = η(SZ)g(Y, SX) + η(SX)g(Z, SY ) +g(Y, SφZ)g(X, V ) + g(Y, SφX)g(Z, V )(4.2) RULED REAL HYPERSURFACES 9 and g((∇ZS)X,φY ) + g((∇ZS)Y, φX) = η(SX)g(Z, SY ) + η(SY )g(X, SZ) +g(Z, SφX)g(Y, V ) + g(Z, SφY )g(X, V )(4.3) Then substract the third one (4.3) from summing up (4.1) and (4.2). From such an obtained equation, and using the equation of Codazzi, it follows that g((∇XS)Y, φZ) + g((∇YS)X, φZ) + g((∇XS)Z−(∇ZS)X, φY ) +g((∇YS)Z−(∇ZS)Y, φX) =2η(SZ)g(Y, SX)+2g(X, V )g(Y, SφZ)+2g(Y, V )g(X, SφZ) =2g((∇XS)Y, φZ)−{g(X, AN)g(AY, φZ)−g(Y, AN)g(AX, φZ) +g(X, Aξ)g(JAY, φZ)−g(Y, Aξ)g(JAX, φZ)} +{g(X, AN)g(AZ, φY )−g(Z, AN)g(AX, φY ) +g(X, Aξ)g(JAZ, φY )−g(Z, Aξ)g(JAX, φY )} +{g(Y, AN)g(AZ, φX)−g(Z, AN)g(AY, φX) +g(X, Aξ)g(JAZ, φX)−g(Z, Aξ)g(JAY, φX)}. (4.4) From this, together with η-commuting property, and using g(JAY, φZ) = −g(AY, JφZ) = g(AY, Z) for any Y, Z∈C, we have g((∇XS)Y, φZ)−g(X, AN)g(AY, φZ)−g(Y, Aξ)g(AX, Z)−g(Z, Aξ)g(AX, Y ) =η(SZ)g(Y, SX) + g(X, V )g(Y, SφZ) + g(Y, V )g(X, SφZ)(4.5) for any X, Y, Z∈C. Then by replacing Zby φZ in (4.5), we have g((∇XS)Y, Z) = g(X, AN)g(AY, Z) + g(Y, Aξ)g(AX, φZ)−g(φZ, Aξ)g(AX, Y ) −η(SφZ)g(Y, SX) + g(X, V )g(Y, SZ) + g(Y, V )g(X, SZ).(4.6) This gives a complete proof of our Lemma.  Remark 4.2. Let Mbe a tube over a totally complex geodesic k-dimensional complex projective space CPkin Q2k. Then the unit normal vector field Nis A-isotropic and the shape operator Scommutes with the structure tensor φ. So the Reeb vector field ξ is principal and the vector field V=φSξ = 0. It can be easily seen that the vectors Aξ and AN belong to the distribution C. Then by (4.6) we have g((∇XS)Y, Z) = 0 for any X, Y, Z∈C orthogonal to the vectors Aξ and AN. Moreover, (4.6) gives the following formulas g((∇AξS)Aξ, Aξ) = −g(Aξ, Aξ)g(A2ξ, φZ) + g(Aξ, Aξ)g(ξ, φZ)=0, g((∇AN S)AN, Aξ) = g(AN, AN)g(A2N, Aξ)−g(AN, AN)g(A2N, Aξ) = 0, g((∇AξS)AN, AN) = −g(Aξ, Aξ)g(A2N, φAN) + g(AN, Aξ)g(A2ξ, φAN) = 0, and g((∇AξS)Aξ, AN) = −g(Aξ, Aξ)g(A2ξ, φAN) + g(Aξ, Aξ)g(A2ξ, φAN)=0. 16 M. KIMURA, H. LEE, J. D. P´ EREZ & Y. J. SUH So we have obtained that AN =g(AN, N)Nand N=g(AN, N)AN =g(AN, N)2N. This gives that g(AN, N)2= 1, which means cos2(2t) = 1. As 0≤t≤π 4, the unique possibility is 2t= 0, that is, t= 0 and Nis A-principal. Case 2) Suppose g(AX, φX) = 0 for any X∈CU. This yields g(AX, Y ) = 0 for any X, Y ∈CU. Take X, Y =φX∈CU,Z=Uin (5.1). We have 0 = 2g(X, AN)g(X, AφU). Therefore we assert g(X, AN)g(X, AφU) = 0 (5.25) for any X∈CU. And taking X, Y =φX∈CU,Z=φU in (5.1) we obtain 2g(X, AN)g(AX, U) = βg(SX, φX) (5.26) for any X∈CU. Taking X∈CU,Y=U,Z=Uin (5.1) we get g(X, AN)g(AU, U)−g(U, Aξ)g(X, JAN)−g(U, AN)g(AX, U) = 0 (5.27) for any X∈CU, and taking X∈CU,Y=φU,Z=φU in (5.1) we obtain 0 = −g(X, AN)g(AU, U)−g(U, AN)g(AX, U) −g(U, Aξ)g(X, JAU)+2βg(SφU, X).(5.28) From (5.7) and (5.8) we have g(X, AN)g(AU, U) = 2βg(SφU, X) (5.29) for any X∈CU. Let us suppose that g(X, AN) = 0 for any X∈CU. Then 0 = g(φX, AN) = −g(X, JAN) = g(X, AJN) = −g(X, Aξ). Therefore g(X, Aξ) = 0 for any X∈CU. As we suppose g(AX, Y ) = 0 for any X, Y ∈CU, we know AX =g(AX, U)U+g(AX, φU)φU, for any X∈CU, that is, X=g(AX, U)AU +g(AX, φU)AφU for any X∈CU. Accordingly, it follows that CU= Span{AU, AφU}and dimCU≤2. Therefore dimM≤5 or m≤3, which is impossible. If g(AX, φU) = 0 for any X∈CUwe have g(φX, AφU) = −g(φX, JAU) = −g(X, AU) = 0, and in this case AX =g(AX, ξ)ξ+g(AX, N)Nwhich yields X=g(AX, ξ)Aξ +g(AX, N)AN for any X∈CU. We arrive at the same contradiction. Therefore we must suppose that there exists X∈CUsuch that g(X, AN)6=0 and from (5.25) g(X, AφU) = 0. Taking X, Y ∈CU,Z=Uin (5.1) we obtain g(X, AN)g(AY, U) + g(Y, Aξ)g(AX, φU) = 0 (5.30) for any X, Y ∈CU. Taking, in particular, our previous X∈CUin (5.30) we obtain g(AY, U) = 0 for any Y∈CU. As above, this also yields g(AY, φU) = 0 for any Y∈CU. Then for any Y∈CUwe have AY =g(AY, ξ)ξ+g(AY, N)N and, as above, this gives a contradiction. Accordingly, the Case 2) can not appear. So only the Case 1) remains valid. From this, together with Theorem A, we have proved. RULED REAL HYPERSURFACES 17 Theorem 5.4. Let Mbe a non-Hopf real hypersurface in the complex quadric Qm, m≥4, with η-parallel and η-commuting shape operator. Then the unit normal vector field Nof Mis A-principal and Mis locally congruent to a ruled real hypersurface foliated by complex totally geodesic Qm−1in Qm. Summing up above two Theorems 5.3 and 5.4 we give a complete proof of our Main Theorem in the introduction. 6. Examples of ruled real hypersurfaces in complex quadric In this section, let us construct ruled real hypersurfaces M2m−1in complex quadric Qm, i.e., real hypersurfaces which are foliated by totally geodesic complex hyperquadric Qm−1, from curves in real projective space RPm+1. First we recall Stiefel manifold (cf. [5]). Let V2(Rm+2) = {(v1, v2)|v1, v2∈Rm+2,kv1k=kv2k= 1,hv1, v2i= 0} be the Stiefel manifold of orthonormal 2-frames in Rm+2. Then the tangent space T(v1,v2)V2(Rm+2) is given as R(−v2, v1)⊕{(x1, x2)∈Rm+2 ×Rm+2|x1, x2⊥span{v1, v2}}. Let e G2(Rm+2) be the Grassmannian manifolds of oriented 2-planes in Rm+2 and let πG: V2(Rm+2)→e G2(Rm+2) be the projection defined by πG(v1, v2) = span(v1, v2). Then with respect to the metric on V2(Rm+2) induced from Euclidean space Rm+2 ×Rm+2 =Cm+2 as a submanifold, we can define a Riemannian metric on e G2(Rm+2) such that πGis a Riemannian submersion. We consider an embedding: ˜ i:V2(Rm+2)→S2m+3 ⊂Cm+2,˜ i(v1, v2) = (v1+iv2)/√2. The tangent space T˜ i(v1,v2)˜ i(V2(Rm+1)) is given as R(−v2+iv1)⊕{x1+ix2∈Cm+2|x1, x2⊥span{v1, v2}}. Then we have a commutative diagram V2(Rm+2)−−−→ ˜ i S2m+3 πG  y  yπ, G2(Rm+2)−−−→ i CPm+1 (6.1) where πis the Hopf fibration and iis the embedding induced from ˜ i. Then i(G2(Rm+2)) is identified with the complex quadric Qn. Let Ibe an interval and let γ:I→RPm+1 be a real 1-dimensional regular curve in real projective space. We denote Γ : I→SO(n) a horizontal lift of γwith respect to the natural projection SO(m+ 2) →RPm+1 =SO(m+ 2)/S(O(m+ 1) ×O(1)). For an expression of the matrix Γ(t) = (e1(t),··· , em+1(t), em+2(t)) by column vectors, we may assume e0 j(t) = λj(t)em+2(t) (j= 1,··· , m + 1), e0 m+2(t) = − m+1 X j=1 λj(t)ej(t).(6.2) 18 M. KIMURA, H. LEE, J. D. P´ EREZ & Y. J. SUH Let e Φ : I×V2(Rm+1)→S2m+3 ⊂Cm+2 be a map defined by e Φ(t, (v1, v2)) = Γ(t)(v1+iv2)/√2 0.(6.3) Then we have the induced map Φ : I×G2(Rm+1)→CPm+1 defined by Φ(t, πG((v1, v2))) = π(e Φ(t, (v1, v2))) (6.4) such that the following diagram is commutative: I×V2(Rm+1)−−−→ ˜ Φ S2m+3 id×πG  y  yπ, I×G2(Rm+1)−−−→ Φ CPm+1 (6.5) and the image Φ(I×G2(Rm+1)) lies in the complex quadric Qmin CPm+1 and for each t∈I, Φ({t}×G2(Rm+1))) is a totally geodesic complex hypersurface Qm−1in Qm. We compute the differential of e Φ. Using (6.2) we have de Φ((∂/∂t),0) = Γ0(t)(v1+iv2)/√2 0 = Γ(t)O−λ(t) t λ(t) 0 (v1+iv2)/√2 0 = Γ(t)0 (hλ(t), v1i+ihλ(t), v2i)/√2,(6.6) where we put λ(t) =t(λ1(t),··· , λm+1(t)). Also we obtain V:= de Φ(0,(−v2, v1)) = Γ(t) √2−v2+iv1 0,(6.7) and de Φ(0,(x1, x2)) = Γ(t) √2x1+ix2 0,(6.8) where x1, x2⊥v1, v2. Here Vis a vertical vector with respect to the fibration id ×πG: I×V2(Rm+1)→I×G2(Cm+1). The metric on I×V2(Rm+1) induced by e Φ is written as: kde Φ((∂/∂t),0)k2=hλ(t), v1i2+hλ(t), v2i2 2, kVk2= 1,kde Φ(0,(x1, x2))k2=kx1k2+kx2k2 2, hde Φ((∂/∂t),0), V i=hde Φ((∂/∂t),0), de Φ(0,(x1, x2))i=hV, de Φ(0,(x1, x2))i= 0. Hence e Φ is regular at (t, (v1, v2)) ⇔ hλ(t), v1i2+hλ(t), v2i26= 0,(6.9) Proposition 6.1. Let γ:I→RPm+1 be a real 1-dimensional regular curve in real projective space and let Γ : I→SO(n)be a horizontal lift of γwith respect to the natural projection SO(m+ 2) →RPm+1 =SO(m+ 2)/S(O(m+ 1) ×O(1)). Then the map RULED REAL HYPERSURFACES 19 Φ : I×G2(Rm+1)→CPm+1 defined by (6.4) is regular at (t, (v1, v2)) if and only if (6.9) holds. A unit normal vector of e Φ at (t, (v1, v2)) is given by e N=Γ(t) phλ(t), v1i2+hλ(t), v2i20 −hλ(t), v2i+ihλ(t), v1i.(6.10) Now we compute the condition for which e Φ (and Φ) is minimal. By (6.8) and (6.10), we see that hA e Φ(0,(x1, x2)),(0,(y1, y2))i= 0,(x1, x2, y1, y2⊥v1, v2) where A e Φis the shape operator of e Φ. Hence e Φ (and Φ) is a minimal immersion at the regular points of e Φ if and only if hA e Φ(∂/∂t, 0),(∂/∂t, 0)i= 0. Using (6.2) and (6.6), we obtain Dd e Φ((∂/∂t),0)de Φ((∂/∂t),0) = Γ0(t) √20 (hλ(t), v1i+ihλ(t), v2i) +Γ(t) √20 (hλ0(t), v1i+ihλ0(t), v2i) =Γ(t) √2O−λ(t) t λ(t) 0  0 (hλ(t), v1i+ihλ(t), v2i) +Γ(t) √20 (hλ0(t), v1i+ihλ0(t), v2i) =Γ(t) √2−(hλ(t), v1i+ihλ(t), v2i)λ(t) (hλ0(t), v1i+ihλ0(t), v2i).(6.11) Hence e Φ (and Φ) is minimal if and only if −hλ0(t), v1ihλ(t), v2i+hλ0(t), v2ihλ(t), v1i= 0 (6.12) for any (v1, v2)∈V2(Rm+2). We may assume that kλ(t)k2= 1, by changing parameter tif necessarily, so we have λ(t)⊥λ0(t). 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Suh, Real hypersurfaces in the complex quadric with parallel Ricci tensor, Advances in Math. 281 (2015), 886-905. [20] Y.J. Suh, Real hypersurfaces in the complex quadric with harmonic curvature, J. Math. Pures Appl. 106 (2016), 393-410. Makoto Kimura Ibaraki University, Department of Mathematics, Mito, Ibaraki, 310-8512, JAPAN E-mail address:[email protected] Hyunjin Lee Kyungpook National University, Research Institute of, Real and Complex Manifolds, Daegu 41566, Republic of Korea E-mail address:[email protected] Juan de Dios P´ erez University of Granada, Department of Geometry and Topology, Granada 18071, Spain E-mail address:[email protected] RULED REAL HYPERSURFACES 21 Young Jin Suh Kyungpook National University, College of Natural Sciences, Department of Mathematics, and Research Institute of Real & Complex Manifolds, Daegu 41566, Republic of Korea E-mail address:[email protected]