Hopf hypersurfaces in nonflat complex space forms

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Hopf hypersurfaces in nonflat complex space forms

Makoto Kimura

Department of Mathematics, Ibaraki University, Mito, Ibaraki 310-8512, Japan

e-mail : [email protected] (2010 Mathematics Subject Classification : 53C40.)

Abstract. We study Hopf hypersurfaces in non-flat complex space forms by using some Gauss maps from a real hypersurface to complex (either positive definite or indefinite) 2-plane Grassmannian and quaternionic (para-)K¨ahler structures.

1 Introduction

Among real hypersurfaces in complex space forms, Hopf hypersurfaces have good geometric properties and studied widely. A real hypersurfaceM in a complex space form Mf(c) (of constant holomorphic sectional curvature 4c) is called Hopf if the structure vector ξ=−J N is an eigenvector of the shape operatorA of M. We say that the (constant) eigenvalue µ with Aξ =µξ a Hopf curvature of Hopf hypersurface M. Whenc >0, Cecil and Ryan [4] proved that a Hopf hypersurface M with Hopf curvature µ = 2 cot 2r (0 < r < π/2) lies on the tube of radius r over a complex submanifold in CPⁿ, provided that the focal mapφr :M →CPⁿ, φr(x) = exp_x(rNx) (x ∈ M, Nx ∈ T_x^⊥M, |Nx| = 1) has constant rank on M. After that Montiel [15] proved similar result for Hopf hypersurfaces withlarge Hopf curvature, (i.e., |µ|>2) in complex hyperbolic spaceCHⁿ (with c=−1). On the other hand, Ivey [7] and Ivey-Ryan [8] proved that a Hopf hypersurface with|µ|<2 in CHⁿ may be constructed from an arbitrary pair of Legendrian submanifolds in S²ⁿ⁻¹.

Here we will study (Hopf) hypersurfacesM inCPⁿ (resp. CHⁿ) by using some Gauss map G(resp. G) frome M to complex 2-plane GrassmannianG₂(Cⁿ⁺¹) (resp.

Grassmannian G_1,1(Cⁿ⁺¹1 ) of 2-planes with signature (1,1) in Cⁿ⁺¹1 ), which maps each point of M to the 2-plane spanned by position vector and structure vector

Key words and phrases:

* This work was supported by grant Proj. No. 24540080 from Grants-in-Aid for Scientific Research, the Ministry of Education, Science, Sports and Culture, Japan.

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at the point in M. We see that: If M is a Hopf hypersurface in CPⁿ, then the Gauss imageG(M) is a real (2n−2)-dimensionaltotally complex submanifold with respect to quaternionic K¨ahler structure in G₂(Cⁿ⁺¹) (Theorem 4.1). If M is a Hopf hypersurface inCHⁿ with (constant) Hopf curvatureµ, then the Gauss image G(Me ) is a real (2n−2)-dimensional submanifold and invariant under a quaternionic (para-)K¨ahler structureI ofG1,1(Cⁿ⁺¹1 ) satisfying

1. I²=−1 provided|µ|>2, 2. I²= 1 provided|µ|<2, 3. I²= 0 provided|µ|= 2

(Theorem 6.1). Detailed descriptions will appear elsewhere.

The author would like to thank Professor Young Jin Suh for inviting me to this international workshop.

2 Real hypersurfaces in CPⁿ and CHⁿ

First we recall the Fubini-Study metric on the complex projective space CPⁿ (cf. [4, 11]). The Euclidean metric h, ion Cⁿ⁺¹ is given byhz, wi= Re(^tzw) for¯ z, w∈Cⁿ⁺¹. The unit sphereS²ⁿ⁺¹ inCⁿ⁺¹ is the principal fiber bundle overCPⁿ with the structure groupS¹ and the Hopf fibrationπ:S²ⁿ⁺¹→CPⁿ. The tangent space ofS²ⁿ⁺¹ at a pointz is

TzS²ⁿ⁺¹={w∈Cⁿ⁺¹|hz, wi= 0}.

Let

T_z⁰ ={w∈Cⁿ⁺¹|hz, wi=hiz, wi= 0}.

With respect to the principal fiber bundle S²ⁿ⁺¹(CPⁿ, S¹), there is a connection such that T_z⁰ is the horizontal subspace at z. Then the Fubini-Study metric g of constant holomorphic sectional curvature 4 is given byg(X, Y) =hX^∗, Y^∗i, where X, Y ∈T_xCPⁿ, andX^∗, Y^∗ are respectively their horizontal lifts at a pointz with π(z) =x. The complex structure onT⁰ defined by multiplication by√

−1 induces a canonical complex structureJ onCPⁿ throughπ_∗.

Next we recall the metric of negative constant holomorphic sectional curvature on the complex hyperbolic space CHⁿ (cf. [15, 11]). The indefinite metrich, iof index 2 onCⁿ⁺¹ is given by

hz, wi= Re −z0w¯0+

n

X

k=1

zkw¯k

!

for z = (z0, z1, . . . , zn), w= (w0, w1, . . . , wn)∈ Cⁿ⁺¹. The anti de Sitter space is defined by

H₁²ⁿ⁺¹={z∈Cⁿ⁺¹| hz, zi=−1}.

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H₁²ⁿ⁺¹ is the principal fiber bundle overCHⁿ with the structure group S¹ and the fibrationπ:H₁²ⁿ⁺¹ →CHⁿ. The tangent space ofH₁²ⁿ⁺¹ at a pointzis

TzH₁²ⁿ⁺¹={w∈Cⁿ⁺¹|hz, wi= 0}.

Let

T_z⁰ ={w∈Cⁿ⁺¹|hz, wi=hiz, wi= 0}.

We observe that the restriction ofh, ito T_z⁰ is positive-definite. The distribution T_z⁰ defines a connection in the principal fiber bundleH₁²ⁿ⁺¹(CHⁿ, S¹), becauseT_z⁰ is complementary to the subspace {iz} tangent to the fibre throughz, and invariant under theS¹-action. Then the metricgof constant holomorphic sectional curvature

−4 is given byg(X, Y) =hX^∗, Y^∗i, whereX, Y ∈TxCHⁿ, andX^∗, Y^∗ are respectively their horizontal lifts at a point z withπ(z) =x. The complex structure on T⁰ defined by multiplication by √

−1 induces a canonical complex structureJ on CHⁿ throughπ∗.

We recall some facts about real hypersurfaces in complex space forms. Let Mfⁿ(c) be a space of constant holomorphic sectional curvature 4cwith real dimension 2n. LetJ:TMf→Mfbe the complex structure with properties J²=−1,∇Je = 0, and hJ X, J Yi=hX, Yi, where∇e is the Levi-Civita connection of Mf. LetM²ⁿ⁻¹ be a real hypersurface inMf(c) and letN be a local unit normal vector field ofM in Mf. We definestructure vector ofM asξ=−J N.

Ifξis a principal vector, i.e.,Aξ=µξholds for the shape operator ofM inMf, M is called a Hopf hypersurface and µ is called Hopf curvature. It is well known ([13], [12]) that Hopf curvature µof Hopf hypersurface M is constant for non flat complex space forms. We note that µ is not constant for Hopf hypersurfaces in Cⁿ⁺¹in general ([16], Theorem 2.1).

Fundamental result for Hopf hypersurfaces in complex projective spaceCPⁿwas proved by Cecil-Ryan [4]:

Theorem 2.1. Let M be a connected, orientable Hopf hypersurface of CPⁿ with Hopf curvature µ = 2 cot 2r (0 < r < π/2). Suppose the focal map φr(x) = exp_x(rN_x)(x∈M,N_x∈T_x^⊥M,|Nx|= 1) has constant rank qon M. Then:

1. q is even and every point x₀ ∈ M has a neighborhood U such that φ_rU is an embedded complex(q/2)-dimensional submanifold ofCPⁿ.

2. For each pointxin such a neighborhoodU, the leaf of the foliationT₀ throughx intersects U in an open subset of a geodesic hypersphere in the totally geodesic CP^(m−q/2)orthogonal toTp(φrU)atp=φr(x). ThusU lies on the tube of radius r overφrU.

Similar result for Hopf hypersurfaces with large Hopf curvature in complex hyperbolic spaceCHⁿ was proved by Montiel [15]:

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Theorem 2.2. Let M be an orientable Hopf hypersurface ofCHⁿ and we assume that φ_r (r >0) has constant rank q on M. Then, if µ = 2 coth 2r (hence µ > 2), for every x₀ ∈M there exists an open neighbourhood W of x₀ such that φ_rW is a (q/2)-dimensional complex submanifold embedded in CHⁿ. Moreover W lies in a tube of radius roverφrW.

On the other hand, with respect to Hopf hypersurfaces with small Hopf curvature inCHⁿ, Ivey [7] and Ivey-Ryan [8] proved that a Hopf hypersurface with|µ|<2 in CHⁿ may be constructed from an arbitrary pair of Legendrian submanifolds in S²ⁿ⁻¹.

3 Totally complex submanifolds in Quaternionic K¨ahler manifolds

We recall the basic definitions and facts on totally complex submanifolds of a quaternionic K¨ahler manifold (cf. [18]) Let (Mf^4m,g, Q) be a quaternionic K¨˜ ahler manifold with the quaternionic K¨ahler structure (˜g, Q), that is, ˜gis the Riemannian metric on MfandQis a rank 3 subbundle of EndTMfwhich satisfies the following conditions:

1. For each p∈Mf, there is a neighborhood U of pover which there exists a local frame field{I˜₁,I˜₂,I˜₃} ofQsatisfying

I˜₁²= ˜I₂²= ˜I₃²=−1, I˜1I˜2=−I˜2I˜1= ˜I3, I˜2I˜3=−I˜3I˜2= ˜I1, I˜3I˜1=−I˜1I˜3= ˜I2.

2. For any elementL∈Qp, ˜gpis invariant byL, i.e., ˜gp(LX, Y) + ˜gp(X, LY) = 0 forX, Y ∈T_pMf,p∈Mf.

3. The vector bundleQis parallel in EndTMfwith respect to the Riemannian connection∇e associated with ˜g.

A submanifoldM^2mofMfis said to bealmost Hermitian if there exists a section ˜I of the bundleQ|_M such that (1) ˜I²=−1, (2) ˜IT M =T M (cf. D.V. Alekseevsky and S. Marchiafava [1]). We denote by I the almost complex structure on M induced from ˜I. Evidently (M, I) with the induced metricgis an almost Hermitian manifold. If (M, g, I) is Kähler, we call it aKähler submanifold of a quaternionic Kähler manifoldMf. An almost Hermitian submanifoldM together with a section ˜I ofQ|_M is said to betotally complex if at each pointp∈M we haveLT_pM ⊥T_pM, for each L ∈ Qp with ˜g(L,I˜p) = 0 (cf. S. Funabashi [5]). It is known that a 2m (m≥2)-dimensional almost Hermitian submanifoldM^2mis Kähler if and only if it is totally complex ([1] Theorem 1.12).

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4 Gauss map of real hypersurfaces inCPⁿ

We recall complex Grassmann manifoldG₂(Cⁿ⁺¹) according to [11], Example 10.8. Let M⁰ = M⁰(n+ 1,2;C) be the space of complex matrices Z with n+ 1 rows and 2 columns such that^tZZ¯ =E₂ (or, equivalently, the 2 column vectors are orthonormal with respect to the standard inner product inCⁿ⁺¹). M⁰ is identified with complex Stiefel manifoldV2(Cⁿ⁺¹). The groupU(2) acts freely onM⁰ on the right: Z7→ZB, whereB∈U(2). We may consider complex 2-plane Grassmannian G2(Cⁿ⁺¹) as the base space of the principal fiber bundleM⁰ with groupU(2).

For eachZ ∈M⁰, the tangent spaceTZ(M⁰) is

TZ(M⁰) ={W ∈M(n+ 1,2;C)|^tW Z+^tZW = 0}.

In T_Z(M⁰) we have an inner product g(W₁, W₂) = Re trace ^tW₂W₁. Let T_Z⁰⁰ = {ZA| A ∈ u(2)} ⊂ T_Z(M⁰) and let T_Z⁰ be the orthogonal complement of T_Z⁰⁰ in T_Z(M⁰) with respect tog. The subspaceT_Z⁰ admits a complex structureW 7→iW. We see that ˜π : M⁰ → G₂(Cⁿ⁺¹) induces a linear isomorphism of T_Z⁰ onto T_π(Z)_˜ (G₂(Cⁿ⁺¹)). By transferring the complex structure i and the inner product g on T_Z⁰ by ˜π_∗ we get the usual complex structure J on G2(Cⁿ⁺¹) and the Hermitian metric on G2(Cⁿ⁺¹), respectively. Note that G2(Cⁿ⁺¹) is a complex 2n−2-dimensional Hermitian symmetric space.

Let U be an open subset in G2(Cⁿ⁺¹) and let ˜Z : U → π˜⁻¹(U) be a cross section with respect to the fibration ˜π:M⁰→G2(Cⁿ⁺¹). Then for eachp∈U, the subspace T⁰_˜

Z(p) admits 3 linear endomorphismsI1, I2, I3: I1:W 7→W

−i 0

0 i

, I2:W 7→W

0 −1

1 0

, I3:W 7→W

0 i i 0

. By transferring I₁, I₂, I₃ on T⁰_˜

Z(p) (p ∈ U) by ˜π_∗ we get local basis ˜I₁,I˜₂,I˜₃ of quaternionic K¨ahler structureQp onU ⊂G2(Cⁿ⁺¹) (cf. [2, 19]).

Let M²ⁿ⁻¹ be a real hypersurface in complex projective space CPⁿ and let w ∈S²ⁿ⁺¹ with π(w) = xforx ∈M. We denote ξ_w⁰ as the horizontal lift of the structure vector ξatxtoT_w⁰ with respect to the Hopf fibrationπ:S²ⁿ⁺¹→CPⁿ. Then theGauss map G:M²ⁿ⁻¹→G2(Cⁿ⁺¹) ofM is defined by

(4.1) G(x) = ˜π(w, ξ_w⁰ ),

where ˜π: M⁰ =V₂(Cⁿ⁺¹) → G₂(Cⁿ⁺¹) given as above. It is seen that G is well defined.

Theorem 4.1. Let M be a real hypersurface in CPⁿ and let G : M²ⁿ⁻¹ → G₂(Cⁿ⁺¹)be the Gauss map defined by (4.1). SupposeM is a Hopf hypersurface.

Then the image G(M) is a real (2n−2)-dimensional totally complex submanifold of G₂(Cⁿ⁺¹).

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5 Totally (para-)complex submanifolds in Quaternionic Para-K¨ahler manifolds

We recall real Clifford algebras (cf. [6, 14]) and (quaternionic) para-complex structure (cf. [14]). Let (V =R(p, q),h, i) be a real symmetric inner product space of signaturep, q. The Clifford algebraC(p, q) is the quotient⊗V /I(V), whereI(V) is the two-sided ideal in⊗V generated by all elements:

x⊗x+hx, xi withx∈V.

Examples of Clifford algebras are:

1. C=C(0,1), Complex numbers: z=x+iy,i²=−1,|z|²=x²+y², there is no zero divisors.

2. Ce =C(1,0), Para-complex numbers: z=x+jy, j²= 1, |z|²=x²−y², there exists zero divisors.

3. H=C(0,2),Quaternions: q=q0+iq1+jq2+kq3,i²=j²=k²=−1,|q|²= q¯q=q²₀+q₁²+q₂²+q₃², there is no zero divisors,H∼=C²,q=z1(q) +jz2(q).

4. He =C(2,0) =C(1,1),Para-quaternions: q=q0+iq1+jq2+kq3,i²=−1, j² = k² = 1, |q|² = q²₀+q₁²−q₂²−q²₃, there exists zero divisors, He ∼=C², q=z1(q) +jz2(q).

There is a natural correspondence between complex numbersCand Euclidean plane R², and para-complex numbersCe are naturally identified with real symmetric inner product space R(1,1) of signature (1,1). Also there is a natural correspondence between quaternionsHand Euclidean 4-spaceR⁴, and para-quaternionsHe are naturally identified with real symmetric inner product spaceR(2,2) of signature (2,2).

Let M be a differentiable manifold. We consider the following structures on a real vector space V =T_xM, x ∈ M, where one assumes that I, J, K are given endomorphisms:

1. Complex: J²=−1,V ⊗C=V_J⁺⊕V_J⁻,V_J^±={u∈V ⊗C| J u=±iu}, 2. Para-complex: J²= 1, V ⊗C=V_J⁺⊕V_J⁻,V_J^±={u∈V ⊗C|J u=±iu},

dimV_J⁺= dimV_J⁻,

3. Hypercomplex: H = (I, J, K), I² = J² = K² = −1, IJ = −J I = K (J K=−KJ=I,KI =−IK =J),

4. Para-hypercomplex: He = (I, J, K),I²=−1,J²=K²= 1,IJ =−J I = K (J K =−KJ =−I,KI =−IK =J).

It is known that the space of (oriented) geodesics (i.e., great circles S¹) in round sphere Sⁿ is naturally identified with (oriented) real 2-plane Grassmannian G₂(Rⁿ⁺¹), and it has complex structure. On the other hand, the space of (oriented) geodesics (i.e., hyperbolic linesH¹) in hyperbolic spaceHⁿ is naturally identified with (oriented) GrassmannianG1,1(Rⁿ⁺¹1 ) of 2-plane with signature (1,1) in Minkowski spaceRⁿ⁺¹1 , and it has para-complex structure [3, 9, 10].

A (para-)hypercomplex structure on V generates respectively a structure:

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1. Quaternionic, generated by H = (I, J, K): Q = hHi = RI+RJ +RK, (I, J, K) determined up toA∈SO(3),S(Q) ={L∈Q| kLk²= 1},

2. Para-quaternionic, generated byHe = (I, J, K): Qe=hHei=RI+RJ+RK, (I, J, K) determined up toA∈SO(2,1), S(Q) =e S⁺(Q)e ∪S⁻(Q),e S⁺(Q) = {L∈Q|e L²= 1},S⁻(Q) ={L∈Q|e L²=−1}.

To treat the quaternionic and para-quaternionic case simultaneously, we setη= 1 orη=−1 and (1, 2, 3) = (−1,−1,−1) or (1, 2, 3) = (−1,1,1) respectively for hypercomplex or para-hypercomplex structure (I1, I2, I3): then the multiplication table looks

I_α² =α1, IαIβ=−IβIα=ηγIγ (α= 1,2,3) where (α, β, γ) is any circular permutation of (1,2,3).

Analmost (para-)quaternionic structure on a differentiable manifold M⁴ⁿ is a rank 3 subbundleQ⊂End(T M), which is locally spanned by a fieldH = (I₁, I₂, I₃) of (para-)hypercomplex structures. Such a locally defined triple H = (I_α), α = 1,2,3, will be called a(local) admissible basis ofQ.

Analmost (para-)quaternionic connection is a linear connection ∇ which preserves Q; equivalently, for any local admissible basisH = (I_α) ofQ, one has

∇Iα=−βω_γ⊗I_β+_γω_β⊗I_γ (P.C.)

where (P.C.) means that (α, β, γ) is any circular permutation of (1,2,3).

An almost (para-)quaternionic structureQis called a(para-)quaternionic structure if M admits a(para-)quaternionic connection, i.e. a torsion-free almost (para- )quaternionic connection. An(almost) (para-)quaternionic manifold (M⁴ⁿ, Q) is a manifoldM⁴ⁿ endowed with an (almost) (para-)quaternionic structureQ.

Analmost (para-)quaternionic Hermitian manifold(M, Q, g) is an almost (para- )quaternionic manifold (M, Q) endowed with a pseudo Riemannian metricgwhich is Q-Hermitian, i.e. any endomorphism ofQisg-skew-symmetric. (M, Q, g), (n >1), is called a (para-)quaternionic K¨ahler manifold if the Levi-Civita connection of g preserves Q, i.e. ∇^g is a (para-)quaternionic connection.

Let (fM^4m,˜g, Q) be a para-quaternionic K¨ahler manifold with the para- quaternionic K¨ahler structure (˜g, Q). A submanifoldM^2mofMfis said to bealmost Hermitian (resp. almost para-Hermitian) if there exists a section ˜I of the bundle Q|M such that (1) ˜I² =−1 (resp. ˜I² = 1), (2) ˜IT M =T M. An almost (para- )Hermitian submanifold M together with a section ˜I of Q|M is said to be totally (para-)complex if at each pointp ∈ M we haveLTpM ⊥ TpM, for each L ∈ Qp

with ˜g(L,I˜p) = 0.

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6 Gauss map of real hypersurfaces inCHⁿ

We consider Grassmann manifold G_1,1(Cⁿ⁺¹1 ) of 2-planes with signature (1,1) inCⁿ⁺¹1 . We putn×nmatrixJ_n as

Jn =







−1 0 · · · 0 0 1 · · · 0 ... ... . .. ... 0 0 · · · 1





 .

Let Mf⁰ = Mf⁰(n+ 1,2;C) be the space of complex matrices Z with n+ 1 rows and 2 columns such that ^tZJ¯ _n+1Z = J₂ (or, equivalently, the 2 column vectors are orthonormal with respect to the inner product inCⁿ⁺¹1 ). Mf⁰ is identified with complex (indefinite) Stiefel manifold V_1,1(Cⁿ⁺¹1 ). The groupU(1,1) acts freely on Mf⁰ on the right: Z 7→ZB, where B ∈U(1,1). We may consider complex (1,1)- plane GrassmannianG_1,1(Cⁿ⁺¹1 ) as the base space of the principal fiber bundle Mf⁰ with groupU(1,1).

For each Z∈Mf⁰, the tangent spaceTZ(Mf⁰) is

TZ(Mf⁰) ={W ∈M(n+ 1,2;C)|^tW Jn+1Z+^tZJn+1W = 0}.

InTZ(M⁰) we have an inner producteg(W1, W2) = Re trace^tW2Jn+1W1. LetT_Z⁰⁰= {ZA| A ∈ u(1,1)} ⊂TZ(M⁰) and let T_Z⁰ be the orthogonal complement of T_Z⁰⁰ in TZ(M⁰) with respect to eg. The subspace T_Z⁰ admits a complex structure W 7→

iW. We see that ˜π : Mf⁰ →G_1,1(Cⁿ⁺¹1 ) induces a linear isomorphism of T_Z⁰ onto T_˜_π(Z)(G_1,1(Cⁿ⁺¹1 )). By transferring the complex structureiand the inner product g onT_Z⁰ by ˜π_∗ we get the complex structureJ onG_1,1(Cⁿ⁺¹1 ) and the (indefinite) Hermitian metric onG_1,1(Cⁿ⁺¹1 ), respectively.

Let U be an open subset in G1,1(Cⁿ⁺¹1 ) and let ˜Z : U → π˜⁻¹(U) be a cross section with respect to the fibration ˜π :Mf⁰ → G1,1(Cⁿ⁺¹1 ). Then for each p∈U, the subspaceT_Z(p)⁰_˜ admits 3 linear endomorphismsI1, I2, I3:

I1:W 7→W

−i 0 0 i

, I2:W 7→W 0 1

1 0

, I3:W 7→W

0 −i i 0

.

By transferring I1, I2, I3 on T_Z(p)⁰_˜ (p ∈ U) by ˜π∗ we get local basis ˜I1,I˜2,I˜3 of quaternionic para-K¨ahler structureQp onU ⊂G1,1(Cⁿ⁺¹1 ).

Let M²ⁿ⁻¹ be a real hypersurface in complex hyperbolic space CHⁿ and let w∈ H₁²ⁿ⁺¹ with π(w) =xforx∈M. We denote ξ⁰_w as the horizontal lift of the structure vectorξatxtoT_w⁰ with respect to the fibrationπ:H₁²ⁿ⁺¹→CHⁿ. Then theGauss map Ge:M²ⁿ⁻¹→G_1,1(Cⁿ⁺¹1 ) ofM is defined by

(6.1) G(x) = ˜e π(w, ξ⁰_w),

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where ˜π:Mf⁰ =V1,1(Cⁿ⁺¹1 )→G1,1(Cⁿ⁺¹1 ) given as above. It is seen thatGe is well defined.

Theorem 6.1. Let M²ⁿ⁻¹ be a real hypersurface in CHⁿ and let Ge : M → G1,1(Cⁿ⁺¹1 ) be the Gauss map defined by (6.1). Suppose M is a Hopf hypersurface with Hopf curvature µ. Then the imageG(M) is a real (2n−2)-dimensional submanifold of G1,1(Cⁿ⁺¹1 )such that

1. G(M)is totally complex provided |µ|>2, 2. G(M)is totally para-complex provided|µ|<2,

3. There exists a sectionI˜of the bundleQ|_G(M₎ satisfyingI˜²= 0andIT G(M˜ ) = T G(M)provided|µ|= 2.

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