Hopf hypersurfaces in complex hyperbolic space and submanifolds in indefinite complex 2-plane Grassmannian I

Topology and its Applications 196 (2015) 594–607

Contents lists available at ScienceDirect

Topology and its Applications www.elsevier.com/locate/topol

Hopf hypersurfaces in complex hyperbolic space and submanifolds in indefinite complex 2-plane Grassmannian I Jong Taek Cho a,1 , Makoto Kimura b,∗,2 a

Department of Mathematics, Chonnam National University, CNU The Institute of Basic Sciences, Kwangju 500-757, Republic of Korea b Department of Mathematics, Faculty of Science, Ibaraki University, Mito, Ibaraki 310-8512, Japan

a r t i c l e

i n f o

Article history: Received 29 January 2014 Received in revised form 25 June 2014 Accepted 25 June 2014 Available online 27 May 2015 Keywords: Gauss map Hopf hypersurface Complex hyperbolic space Para-quaternionic Kähler structure

a b s t r a c t We define Gauss map from a real hypersurface in complex hyperbolic space to indefinite complex 2-plane Grassmannian. We show that if a real hypersurface is Hopf, then the image of the Gauss map is a half-dimensional regular submanifold and has a nice behavior under para-quaternionic Kähler structures of the Grassmannian. In particular if absolute value of the Hopf curvature of the Hopf hypersurface is greater (resp. smaller) than 2, then the Gauss image is totally complex (resp. totally para-complex) submanifold with respect to the para-quaternionic Kähler structure of indefinite complex 2-plane Grassmannian, provided that the induced metric is nondegenerate. © 2015 Elsevier B.V. All rights reserved.

1. Introduction n (c) of constant holomorphic sectional curvature The study of real hypersurfaces in complex space form M 4c has been an active field over the last forty years. A natural and important class of real hypersurfaces is the class of Hopf hypersurfaces. For a unit normal vector N of a hypersurface M the vector ξ = −JN is a . A real hypersurface M in M  is called Hopf if tangent vector of M , where J is the complex structure of M the vector ξ, called structure vector of M , is a principal direction at every point of M . Homogeneous real hypersurfaces in complex projective space CPn are classified by Takagi [22] and they are all Hopf (cf. [11,23]). Also it is known that a tube over a complex submanifold in CPn is Hopf. Structure n (1) is obtained by Cecil–Ryan [5]: Let M be a connected, theorem of Hopf hypersurfaces in CPn = M orientable Hopf hypersurface of CPn of constant holomorphic sectional curvature 4 with corresponding constant principal curvature μ = 2 cot 2r (0 < r < 2π). Suppose that the focal map, i.e., the normal exponential * Corresponding author. E-mail addresses: [email protected] (J.T. Cho), [email protected] (M. Kimura). J.T. Cho was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2014R1A1A2053665). 2 M. Kimura was supported by JSPS KAKENHI Grant Number 24540080. 1

http://dx.doi.org/10.1016/j.topol.2014.06.018 0166-8641/© 2015 Elsevier B.V. All rights reserved.

J.T. Cho, M. Kimura / Topology and its Applications 196 (2015) 594–607

595

map Φr : M → CPn of distance r has constant rank q on M . Then q is even and every point x0 has a neighborhood U such that Φr (U ) is an embedded complex q/2-dimensional submanifold of CPn . On the other hand, a result which is similar to Cecil–Ryan’s theorem for Hopf hypersurfaces in CHn = n (−1) with absolute value of the principal curvature μ corresponding to ξ is greater than 2, was obtained M by Montiel [19]. Recently Ivey and Ryan [9,10] showed that a Hopf hypersurface with |μ| < 2 in CHn may be constructed from an arbitrary pair of Legendrian submanifolds in S 2n−1 . In [13], we studied Hopf hypersurfaces in CPn from different point of view, namely by using a Gauss map. For a real hypersurface M in CPn , we defined the Gauss map ψ : M → G2 (Cn+1 ) to complex 2-plane Grassmannian: for x in M , ψ(x) is the complex 2-plane in Cn+1 which is spanned by the position vector and structure vector at x. Our main result was: Let M be a real hypersurface in CPn and let ψ : M 2n−1 → G2 (Cn+1 ) be the Gauss map. Suppose M is a Hopf hypersurface. Then the image ψ(M ) is a real (2n −2)-dimensional totally complex submanifold with respect to quaternionic Kähler structure of G2 (Cn+1 ). Note that the Gauss map is considered as a generalization of the Gauss map for hypersurfaces in sphere to complex quadric [21]. In this paper we study Hopf hypersurfaces in CHn by using Gauss map: For a real hypersurface M 2n−1 in CHn , we define the Gauss map ψ : M → G1,1 (Cn+1 ) to indefinite complex 2-plane Grassmannian as 1 follows: Let M 2n−1 be an embedded real hypersurface of CHn . Let w ∈ H12n+1 with π(w) = x for x ∈ M  be the horizontal lift of the structure vector ξ at x to Tw with respect to the fibration and let ξ  = ξw n 2n+1 π : H1 → CH , where H12n+1 is the anti de-Sitter space. Then the Gauss map ψ : M 2n−1 → G1,1 (Cn+1 ) 1 n 2n−1 in CH is defined by for a real hypersurface M ψ(x) = spanC {w, ξ  }.

(1)

As complex 2-plane Grassmannian G2 (Cn+1 ) has quaternionic Kähler structures, G1,1 (Cn+1 ) has para1 4m ˜  quaternionic Kähler structure: A 4m-dimensional manifold (M , g˜, Q) is called para-quaternionic Kähler, if   and Q ˜ is a rank 3 subbundle of End T M g˜ is a pseudo-Riemannian metric of split signature (2m, 2m) on M which satisfies the following conditions: , there is a neighborhood U of p over which there exists a local frame field {I˜1 , I˜2 , I˜3 } (i) For each p ∈ M ˜ of Q satisfying I˜12 = −1,

I˜22 = I˜32 = 1,

I˜2 I˜3 = −I˜3 I˜2 = I˜1 ,

I˜1 I˜2 = −I˜2 I˜1 = −I˜3 , I˜3 I˜1 = −I˜1 I˜3 = −I˜2 .

˜ p , g˜p is invariant by L, i.e., g˜p (LX, Y ) + g˜p (X, LY ) = 0 for X, Y ∈ Tp M . , p ∈ M (ii) For any element L ∈ Q ˜ is parallel in End T M  with respect to the Levi-Civita connection ∇  associated (iii) The vector bundle Q with g˜. In this paper we will show that if a real hypersurface M 2n−1 of CHn is Hopf, then the Gauss image ψ(M ) is a half-dimensional regular submanifold of G1,1 (Cn+1 ) and has a nice behavior under para-quaternionic 1 Kähler structures of the Grassmannian. Our main result (Theorem 5.1) is that: Let M 2n−1 be a real hypersurface in complex hyperbolic space CHn of constant holomorphic sectional curvature −4. Suppose M is a Hopf hypersurface with Hopf curvature μ with |μ| > 2 (resp. 0 ≤ |μ| < 2). Then, (i) the image ψ(M ) is a real (2n − 2)-dimensional submanifold ˜ ψ(M ) of the para-quaternionic Kähler of G1,1 (Cn+1 ), (ii) there exist sections I˜˜1 , I˜˜2 and T˜3 of the bundle Q| 1 ˜ ψ(p) for p ∈ Σ satisfying structure such that they are orthonormal with respect to natural inner product on Q 2 2 2 2 2 (I˜˜1 ) = −1 (resp. (I˜˜1 ) = 1), (I˜˜2 ) = 1 (resp. (I˜˜2 ) = −1) and (I˜˜3 ) = 1, such that dψx (Tx M ) is invariant under I˜˜1 and I˜˜2 dψx (Tx M ), I˜3 dψx (Tx M ) are orthogonal to dψx (Tx M ), (iii) the induced metric on ψ(M )

596

J.T. Cho, M. Kimura / Topology and its Applications 196 (2015) 594–607

 in G1,1 (Cn+1 ) has signature (p, q) (resp. (p, p)), where p = |λ|>1 dim{X| AX = λX, X ⊥ ξ} and  1 q = dim{X| AX = λX, X ⊥ ξ}. (iv) Furthermore if dim{X| AX = ±X, X ⊥ ξ} is 0, then |λ|<1 the induced metric of ψ(M ) is non-degenerate and ψ(M ) is a totally complex (resp. totally para-complex) submanifold of G1,1 (Cn+1 ). Similar result is obtained (Theorem 5.2) for Hopf hypersurfaces with |μ| = 2. In the forthcoming papers we will discuss converse construction from a submanifold in G1,1 (Cn+1 ) to a 1 Hopf hypersurface in CHn . The authors would like express their gratitude to the referee for his/her careful reading of the manuscript and valuable suggestions. 2. Real hypersurfaces in CHn We recall the metric of negative constant holomorphic sectional curvature on the complex hyperbolic space CHn (cf. [19,15]). The indefinite inner product  ,  of index 2 on Cn+1 is given by  z, w = Re −z0 w ¯0 +

n 

 zk w ¯k

(2)

k=1

for z = (z0 , z1 , . . . , zn ), w = (w0 , w1 , . . . , wn ) ∈ Cn+1 . We write this inner product space (Cn ,  , ) as Cn+1 . 1 The anti de Sitter space is defined by H12n+1 = {z ∈ Cn+1 | z, z = −1}. 1 H12n+1 is the principal fiber bundle over CHn with the structure group S 1 and the fibration π : H12n+1 → CHn . The tangent space of H12n+1 at a point z is Tz H12n+1 = {w ∈ Cn+1 |z, w = 0}. 1 Let Tz = {w ∈ Cn+1 |z, w = iz, w = 0}. 1 We observe that the restriction of  ,  to Tz is positive-definite. The distribution Tz defines a connection in the principal fiber bundle H12n+1 (CHn , S 1 ), because Tz is complementary to the subspace {iz} tangent to the fiber through z, and invariant under the S 1 -action. Then the metric g of constant holomorphic sectional curvature −4 is given by g(X, Y ) = X ∗ , Y ∗ , where X, Y ∈ Tx CHn , and X ∗ , Y ∗ are respectively their √ horizontal lifts at a point z with π(z) = x. The complex structure on T  defined by multiplication by −1 induces a canonical complex structure J on CHn through π∗ . n (c) be a space of constant We recall some facts about real hypersurfaces in complex space forms. Let M ˜ For a real hyholomorphic sectional curvature 4c with real dimension 2n and Levi-Civita connection ∇. 2n−1  persurface M in M , the Levi-Civita connection ∇ of the induced metric and the shape operator A are characterized by ˜ X Y = ∇X Y + AX, Y N ∇ and ˜ X N = −AX ∇  → M  be the complex structure with properties where N is a local choice of unit normal. Let J : T M 2  = 0, and JX, JY  = X, Y . We define the structure vector of M as ξ = −JN . Clearly J = −1, ∇J ξ ∈ T M and |ξ| = 1.

J.T. Cho, M. Kimura / Topology and its Applications 196 (2015) 594–607

597

Define a skew-symmetric (1, 1)-tensor φ from the tangent projection of J by JX = φX + η(X)N where η is a 1-form on M defined by η(X) = ξ, X. Then we have φ2 X = −X + η(X)ξ,

φX, φY  = X, Y  − η(X)η(Y ),

φξ = 0.

˜ = 0 implies (φ, ξ, η,  , ) determines an almost contact metric structure on M . Furthermore ∇J (∇X φ)Y = η(Y )AX − AX, Y ξ,

∇X ξ = φAX.

, M is called a Hopf If ξ is a principal vector, i.e., Aξ = μξ holds for the shape operator of M in M hypersurface and μ is called Hopf curvature. The following fact is proved by Y. Maeda [17] and Ki-Suh [16] (cf. [20], Corollary 2.3 (i)). Proposition 2.1. Let M 2n−1 (n ≥ 2) be a Hopf hypersurface in a complex space form of constant holomorphic sectional curvature 4c = 0, and let μ be the Hopf curvature. Then μ must be a constant. Furthermore if X ⊥ ξ and AX = λX, then 

μ λ− AφX = 2







 λμ + c φX. 2

(3)

Hence

AφX =

λμ + 2c 2λ − μ

φX

provided

λ−

μ = 0 . 2

(4)

We note that μ is not constant for Hopf hypersurfaces in Cn+1 in general ([20], Theorem 2.1). Corollary 2.1. Let M 2n−1 (n ≥ 2) be a Hopf hypersurface with Hopf curvature μ in a complex space form of constant holomorphic sectional curvature 4c = 0. If there exists a (locally defined) vector field X = 0 √ √ orthogonal to ξ with AX = (μ/2)X, then c < 0, |μ| = 2 −c and |λ| = −c. In particular when either √ c > 0 or c < 0 with |μ| = 2 −c, there exists no tangent vector field X = 0 orthogonal to ξ satisfying AX = (μ/2)X. In fact, in (3) if λ = μ/2, then we have μ2 + 4c = 0 and so c < 0. Corollary 2.2. Let M 2n−1 (n ≥ 2) be a Hopf hypersurface with Hopf curvature μ in a complex projective space CPn of constant holomorphic sectional curvature 4 (c = 1). If we put μ = 2 cot 2r for some r ∈ (0, π/2) and if a (locally defined) tangent vector field X orthogonal to ξ satisfies AX = cot(r + θ)X for some θ, then we have AφX = cot(r − θ)φX. Corollary 2.3. Let M 2n−1 (n ≥ 2) be a Hopf hypersurface with Hopf curvature μ in a complex projective space CHn of constant holomorphic sectional curvature −4 (c = −1). (i) In the case |μ| > 2, we put μ = 2 coth 2r for some r = 0. Let X be a principal vector orthogonal to ξ satisfying AX = λX.

J.T. Cho, M. Kimura / Topology and its Applications 196 (2015) 594–607

598

(a) When |λ| > 1, if we put λ = coth(r + θ) for some θ ∈ R, then we have λμ − 2 = coth(r − θ) 2λ − μ

and

λμ − 2 2λ − μ > 1.

(b) When |λ| < 1, if we put λ = tanh(r + θ) for some θ ∈ R, then we have λμ − 2 = tanh(r − θ) 2λ − μ

and

λμ − 2 2λ − μ < 1.

(c) If λ = ±1, then λμ − 2 = ∓1. 2λ − μ (d) Hence with respect to eigenvalues λ1 , · · · , λ2n−2 of A|{ξ}⊥ , #{i| |λi | > 1}

and

#{i| |λi | < 1}

are even. (ii) In the case |μ| < 2, we put μ = 2 tanh 2r for some r ∈ R. Let X be a principal vector orthogonal to ξ satisfying AX = λX. (a) When |λ| > 1, if we put λ = coth(r + θ) for some θ ∈ R, then we have λμ − 2 = tanh(r − θ) 2λ − μ

and

λμ − 2 2λ − μ < 1.

(b) When |λ| < 1, if we put λ = tanh(r + θ) for some θ ∈ R, then we have λμ − 2 = coth(r − θ) 2λ − μ

and

λμ − 2 2λ − μ > 1.

(c) If λ = ±1, then λμ − 2 = ∓1. 2λ − μ (d) Hence with respect to eigenvalues λ1 , · · · , λ2n−2 of A|{ξ}⊥ , we have #{i| |λi | > 1} = #{i| |λi | < 1}. (iii) In the case μ = 2 (resp. μ = −2), if 2λ − μ = 0, then λμ − 2 =1 2λ − μ

(resp.

λμ − 2 = −1). 2λ − μ

So we have {ξ}⊥ =

{X| AX = λX, X ⊥ ξ} ⊕ {X| AX = X, X ⊥ ξ}, λ=1

(resp. {ξ}⊥ =

{X| AX = λX, X ⊥ ξ} ⊕ {X| AX = −X, X ⊥ ξ}),

λ=−1

J.T. Cho, M. Kimura / Topology and its Applications 196 (2015) 594–607

599

and φ{X| AX = λX, λ = 1 and X ⊥ ξ} ⊂ {X| AX = ±X, X ⊥ ξ}. (resp. φ{X| AX = λX, λ = −1 and X ⊥ ξ} ⊂ {X| AX = −X, X ⊥ ξ}.) A fundamental result for Hopf hypersurfaces in complex projective space CPn was proved by Cecil– Ryan [5]: 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) = expx (rNx ) (x ∈ M , Nx ∈ Tx⊥ M , |Nx | = 1) has constant rank q on M . Then: (i) q is even and every point x0 ∈ M has a neighborhood U such that φr U is an embedded complex (q/2)-dimensional submanifold of CPn . (ii) For each point x in such a neighborhood U , the leaf of the foliation T0 through x intersects U in an open subset of a geodesic hypersphere in the totally geodesic CP(m−q/2) orthogonal to Tp (φr U ) at p = φr (x). Thus U lies on the tube of radius r over φr U . Borisenko [3] obtained a structure theorem for Hopf hypersurfaces in CPn without any assumption about rank of the focal map. Similar result for Hopf hypersurfaces with large Hopf curvature in complex hyperbolic space CHn was proved by Montiel [19]: Theorem 2.2. Let M be an orientable Hopf hypersurface of CHn and we assume that φr (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 neighborhood U of x0 such that φr U is a (q/2)-dimensional complex submanifold embedded in CHn . Moreover U lies in a tube of radius r over φr U . On the other hand, with respect to Hopf hypersurfaces with small Hopf curvature in CHn , Ivey [9] and Ivey–Ryan [10] proved that a Hopf hypersurface with |μ| < 2 in CHn may be constructed from an arbitrary pair of Legendrian submanifolds in S 2n−1 . 3. Totally (para-)complex submanifolds in (para-)quaternionic Kähler manifolds In this section, we will give basic definitions and facts on totally complex (resp. para-complex) submanifolds of a quaternionic (resp. para-complex) Kähler manifold (cf. [24,18]). First we recall real Clifford algebras (cf. [8,18]) and (quaternionic) para-complex structure (cf. [18]). Let (V = R(p, q),  , ) be a real symmetric inner product space of signature p, q. The Clifford algebra C(p, q) is the quotient ⊗V /I(V ), where I(V ) is the two-sided ideal in ⊗V generated by all elements: x ⊗ x + x, x with x ∈ V. Then Clifford algebras with p + q = 1 or 2 are given as: (i) C = C(0, 1), complex numbers: z = x + iy (x, y ∈ R), i2 = −1, |z|2 = x2 + y 2 , there is no zero divisors.  = C(1, 0), split-complex numbers: z = x + jy (x, y ∈ R), j 2 = 1, |z|2 = x2 − y 2 , there exists zero (ii) C divisors. (iii) H = C(0, 2), quaternions: q = q0 + iq1 + jq2 + kq3 (q0 , q1 , q2 , q3 ∈ R), i2 = j 2 = k 2 = ijk = −1, |q|2 = q q¯ = q02 + q12 + q22 + q32 , there is no zero divisors, H ∼ = C2 , q = z1 (q) + jz2 (q).

600

J.T. Cho, M. Kimura / Topology and its Applications 196 (2015) 594–607

 = C(2, 0) = C(1, 1), split-quaternions: q = q0 + iq1 + jq2 + kq3 (q0 , q1 , q2 , q3 ∈ R), i2 = −1, (iv) H ∼ j 2 = k2 = ijk = 1, |q|2 = q02 + q12 − q22 − q32 , there exists zero divisors, H = C2 , q = z1 (q) + jz2 (q). There is a natural correspondence between complex numbers C and Euclidean plane R2 , and split-complex  are naturally identified with real symmetric inner product space R(1, 1) of signature (1, 1). Also numbers C there is a natural correspondence between quaternions H and Euclidean 4-space R4 , and split-quaternions  are naturally identified with real symmetric inner product space R(2, 2) of signature (2, 2). H According to unit complex number i, i2 = −1, the following definitions are well-known: Definition 3.1. A tensor field J of type (1, 1) on a differentiable manifold M is called almost complex structure if J 2 = −1. A Riemannian manifold (M,  , ) with almost complex structure J is called almost Hermitian if JX, Y  + X, JY  = 0 for X, Y ∈ T M . Furthermore almost Hermitian manifold (M,  , , J) is Kähler) if and only if ∇J = 0 for the Levi-Civita connection ∇ of M . Similarly, according to unit para-complex number j, j 2 = 1, there are the following definitions (cf. [12]): Definition 3.2. ([4,14]) A tensor field J˜ of type (1, 1) on a differentiable manifold M is called almost product structure (resp. almost para-complex structure) if (5) (resp. (5), (6)) is valid: J˜2 = 1,

(5)

˜ = ±X} = dimR M/2. dimR {X|JX

(6)

A pseudo-Riemannian manifold (M,  , ) with almost para-complex structure J˜ is called almost paraHermitian (resp. para-Kähler) if (5)–(7) (resp. (5)–(8)) hold: ˜ Y  + X, JY ˜  = 0, JX,

(X, Y ∈ T M )

∇J˜ = 0,

(7) (8)

where ∇ denotes the Levi-Civita connection of M . Note that the metric (7) is of split signature (m, m), 2m = dim M . A para-Kähler structure can be regarded as a pair of complementary integrable Lagrangian distributions (T + , T − ) with respect to symplectic ˜ , where T ± denotes eigendistribution of J. ˜ Such a structure is often called structure Ω(X, Y ) = X, JY a bi-Lagrangian structure or a Lagrangian 2-web. Both Kähler and para-Kähler manifolds are important classes of symplectic manifolds. Next, according to quaternions and split-quaternions, we recall definitions of quaternionic and paraquaternionic Kähler manifolds: Let (M 4m , g, Q) be a quaternionic Kähler manifold with the quaternionic Kähler structure (g, Q), that is, g is the Riemannian metric on M and Q is a rank 3 subbundle of End T M which satisfies the following conditions: (i) For each p ∈ M , there is a neighborhood U of p over which there exists a local frame field {I1 , I2 , I3 } of Q satisfying I12 = I22 = I32 = −1,

I1 I2 = −I2 I1 = I3 ,

I2 I3 = −I3 I2 = I1 ,

I3 I1 = −I1 I3 = I2 .

(ii) For any element L ∈ Qp , gp is invariant by L, i.e., gp (LX, Y ) + gp (X, LY ) = 0 for X, Y ∈ Tp M , p ∈ M .

J.T. Cho, M. Kimura / Topology and its Applications 196 (2015) 594–607

601

(iii) The vector bundle Q is parallel in End T M with respect to the Levi-Civita connection ∇ associated with g. We note that Qp is naturally identified with Lie algebra su(2) and isometric to Euclidean 3-space R3 with respect to the Killing form: Qp = {aI1 + bI2 + cI3 | a, b, c ∈ R} ∼ = su(2) ∼ = R3 , and Sp := {I ∈ Qp | I 2 = −1} ∼ = S 2 : (unit 2-sphere).

(9)

A submanifold Σ2m of M is said to be almost Hermitian if there exists a section I of the bundle Q|Σ such that (1) I 2 = −1, (2) IT Σ = T Σ (cf. D.V. Alekseevsky and S. Marchiafava [2]). We denote by IΣ the almost complex structure on Σ induced from I. Evidently (Σ, IΣ ) with the induced metric gΣ is an almost Hermitian manifold. If (Σ, gΣ , IΣ ) is Kähler, we call it a Kähler submanifold of a quaternionic Kähler manifold M . An almost Hermitian submanifold Σ together with a section IΣ of Q|Σ is said to be totally complex if at each point p ∈ Σ we have Lp Tp Σ ⊥ Tp Σ, for each Lp ∈ Qp with gp (Lp , (IΣ )p ) = 0 (cf. S. Funabashi [7]), where gp is the natural inner product on Qp such that I1 , I2 and I3 are orthonormal with respect to gp . It is known that a 2m (m ≥ 2)-dimensional almost Hermitian submanifold Σ2m is Kähler if and only if it is totally complex ([2] Theorem 1.12). 4m , g˜, Q) ˜ be a para-quaternionic Kähler manifold with the para-quaternionic Kähler structure Next let (M ˜ that is, g˜ is the pseudo-Riemannian metric of split signature (2m, 2m) on M  and Q ˜ is a rank 3 (˜ g , Q),  subbundle of End T M which satisfies the following conditions: , there is a neighborhood U of p over which there exists a local frame field {I˜1 , I˜2 , I˜3 } (i) For each p ∈ M ˜ of Q satisfying I˜12 = −1,

I˜22 = I˜32 = 1,

I˜2 I˜3 = −I˜3 I˜2 = I˜1 ,

I˜1 I˜2 = −I˜2 I˜1 = −I˜3 , I˜3 I˜1 = −I˜1 I˜3 = −I˜2 .

˜ p , g˜p is invariant by L, i.e., g˜p (LX, Y ) + g˜p (X, LY ) = 0 for X, Y ∈ Tp M , p ∈ M . (ii) For any element L ∈ Q ˜ is parallel in End T M  with respect to the Levi-Civita connection ∇  associated (iii) The vector bundle Q with g˜. ˜ p is naturally identified with Lie algebra su(1, 1) and isometric to Minkowski 3-space R3 We note that Q 1 with respect to the Killing form: ˜ p = {aI1 + bI2 + cI3 | a, b, c ∈ R} ∼ Q = su(1, 1) ∼ = R31 , and ˜ p | I˜2 = 1} ∼ (S+ )p := {I˜ ∈ Q = S12 : (de-Sitter 2-space),

(10)

˜ p | I = −1} ∼ (S− )p := {I˜ ∈ Q = H : (hyperbolic 2-space),

(11)

˜ p | I˜2 = 0} ∼ (S0 )p := {I˜ ∈ Q = L2 : (light cone).

(12)

˜2

2

 is said to be almost Hermitian (resp. almost paraA pseudo-Riemannian submanifold Σ2m of M ˜ Σ such that (1) I 2 = −1 (resp. I 2 = 1), Hermitian) if there exists a section I of the bundle Q|

602

J.T. Cho, M. Kimura / Topology and its Applications 196 (2015) 594–607

(2) IT Σ = T Σ. We denote by IΣ the almost (para-)complex structure on Σ induced from I. Evidently (Σ, IΣ ) with the induced metric gΣ is an almost (para-)Hermitian manifold. If (Σ, gΣ , IΣ ) is (para-)Kähler, . An almost Hermitian we call it a (para-)Kähler submanifold of a para-quaternionic Kähler manifold M ˜ (resp. para-Hermitian) submanifold Σ together with a section IΣ of Q|Σ is said to be totally complex (resp. ˜ p with gp (Lp , Ip ) = 0 totally para-complex) if at each point p ∈ Σ we have LTp Σ ⊥ Tp Σ, for each Lp ∈ Q ˜p ∼ (cf. [25]) with respect to natural inner product gp on Q = R31 . 4. Indefinite complex 2-plane Grassmannian G1,1 (Cn+1 ) 1 We consider Grassmann manifold G1,1 (Cn+1 ) of complex 2-planes with signature (1, 1) in Cn+1 . We put 1 1 n × n matrix Jn as ⎛

⎞ −1 0 · · · 0 ⎜ ⎟ ⎜ 0 1 ··· 0⎟ ⎜ Jn = ⎜ . .. . . .⎟ ⎟. . .. ⎠ . ⎝ .. 0 0 ··· 1  (n + 1, 2; C) be the space of complex matrices Z with n + 1 rows and 2 columns such that  = M Let M t¯ ZJn+1 Z = J2 (or, equivalently, the 2 column vectors are orthonormal with respect to the inner product in  is identified with complex (indefinite) Stiefel manifold V1,1 (Cn+1 ). The group U (1, 1) acts freely Cn+1 ). M 1 1  on the right: Z → ZB, where B ∈ U (1, 1). We may consider complex (1, 1)-plane Grassmannian on M  with group U (1, 1). G1,1 (Cn+1 ) as the base space of the principal fiber bundle M 1    , the tangent space TZ (M  ) is For each Z ∈ M  ) = {W ∈ M (n + 1, 2; C)| t W Jn+1 Z + t ZJn+1 W = 0}. TZ (M  ) we have an inner product In TZ (M g(W1 , W2 ) = Re trace (J2 t W 2 Jn+1 W1 ).

(13)

If we write W1 = (w11 , w12 ) and W2 = (w21 , w22 ) by column vectors, then we have g(W1 , W2 ) = −w11 , w21  + w12 , w22 ,

(14)

 ) and let where  ,  is the inner product on Cn+1 defined by (2). Let TZ = {ZA| A ∈ u(1, 1)} ⊂ TZ (M 1      ) with respect to g. We see that π  → G1,1 (Cn+1 ) TZ be the orthogonal complement of TZ in TZ (M ˜ :M 1 n+1 induces a linear isomorphism of TZ onto Tπ˜ (Z) (G1,1 (C1 )). By transferring the inner product g on TZ by  , the kernel of π π ˜∗ we get the (indefinite) metric on G1,1 (Cn+1 ). Note that for Z ∈ M ˜∗ is TZ . Also we can 1 see that the metric g has signature (2n − 2, 2n − 2) by (14). Let U be an open subset in G1,1 (Cn+1 ) and let Z˜ : U → π ˜ −1 (U ) be a cross section with respect to the 1 n+1    → G1,1 (C fibration π ˜ :M ). Then for each p ∈ U , the subspace TZ(p) admits 3 linear endomorphisms ˜ 1 I1 , I2 , I3 :  I1 : W → W

−i 0 0 i



 ,

I2 : W → W 

I3 : W → W

0 i

−i 0

 0 1 1 0

,

 .

(15)

J.T. Cho, M. Kimura / Topology and its Applications 196 (2015) 594–607

603

 By transferring I1 , I2 , I3 on TZ(p) (p ∈ U ) by π ˜∗ we get local basis I˜1 , I˜2 , I˜3 of para-quaternionic Kähler ˜ ˜ p on U ⊂ G1,1 (Cn+1 ) (cf. [1,6]). structure Q 1

5. Gauss map of real hypersurfaces in CHn to G1,1 (Cn+1 ) 1 In this section we will investigate the Gauss map from a Hopf hypersurface M 2n−1 in CHn to G1,1 (Cn+1 ). 1 n 2n−1 We assume that constant holomorphic sectional curvature c of CH is equal to −4, and M is an  embedded real hypersurface of CHn . Let w ∈ H12n+1 with π(w) = x for x ∈ M and let ξ  = ξw be the horizontal lift of the structure vector ξ at x to Tw with respect to the fibration π : H12n+1 → CHn . Then the Gauss map ψ : M 2n−1 → G1,1 (Cn+1 ) for a real hypersurface M 2n−1 in CHn is defined by 1 ψ(x) = π ˜ (w, ξ  ),

(16)

 = V1,1 (Cn+1 ) → G1,1 (Cn+1 ) given in Section 4. This definition is independent of the choice of where π ˜:M 1 1 w ∈ π −1 (x). Now we suppose that M is a Hopf hypersurface in CHn with Hopf curvature μ. Without loss of generality,  = we may assume μ ≥ 0. With respect to a section CHn ⊃ M 2n−1 → H12n+1 , x → w, let ψ˜ : M 2n−1 → M n+1 V1,1 (C1 ) be the map defined by ˜ ψ(x) = (w, ξ  ).

(17)

˜ Differential of ψ˜ at x with respect to the direction of the structure vector ξ is Then we have ψ = π ˜ ◦ ψ. ˜ ξ ξ) ) dψ˜x (ξ) = (ξ  , Dξ ξ  ) = (ξ  , −Dξ ξ  , ww + ∇ξ ξ  ) = (ξ  , w + ∇ξ ξ  ) = (ξ  , w + (∇ = (ξ  , w + μN  ) = (ξ  , w) + μ(0, N  ). Hence we obtain dψx (ξ) = 0. dψx (ξ) = d˜ πψ(x) (dψ˜x (ξ)) = d˜ πψ(x) ((ξ  , −w) + μ(0, N  )) ˜ ˜  μ = d˜ πψ(x) (ξ  , −w) + ((iw, iξ  ) + (−iw, iξ  )) = 0. ˜ 2

(18)

This is also deduced from the fact that: Each integral curve γ of ξ on a Hopf hypersurface lies in a totally geodesic CH1 in CHn , hence ψ is constant along γ. First we consider the case: |μ| = 2. Let X be a tangent vector of M such that X ⊥ ξ and AX = λX. Then by Corollary 2.1, we have 2λ − μ = 0. Hence Proposition 2.1 implies AφX =

λμ − 2 φX. 2λ − μ

Differentials of ψ with respect to X and φX are given by dψx (X) = d˜ πψ(x) (dψ˜x (X)) = d˜ πψ(x) (X  , DX  ξ  ) = d˜ πψ(x) (X  , (φAX) ) ˜ ˜ ˜ = d˜ πψ(x) (X  , λ(φX) ), ˜ dψx (φX) = d˜ πψ(x) (dψ˜x (φX)) = d˜ πψ(x) ((φX) , D(φX) ξ  ) = d˜ πψ(x) ((φX) , (φAφX) ) ˜ ˜ ˜

 λμ − 2  (φX) , − X . = d˜ πψ(x) ˜ 2λ − μ

(19)

(20)

J.T. Cho, M. Kimura / Topology and its Applications 196 (2015) 594–607

604

Hence (18), (19) and (20) yield that ψ(M ) is a real (2n − 2)-dimensional submanifold in G1,1 (Cn+1 ) by 1 implicit function theorem. Suppose μ > 2 (resp. 0 ≤ μ < 2). We put μ = 2 coth 2r (resp. μ = 2 tanh 2r) for some r > 0 (resp. ˜ ˜ r ≥ 0). According to local basis I˜1 , I˜2 , I˜3 at (15) with respect to Z(ψ(x)) = ψ(x) of para-quaternionic ˜ ψ(x) , we put Kähler structure Q 1 I˜˜1 =  (μI˜1 + 2I˜2 ) 2 |μ − 4| = cosh 2rI˜1 + sinh 2rI˜2 .



 resp. sinh 2rI˜1 + cosh 2rI˜2 .

(21)

Note that (I˜˜1 )2 = −1 (resp. (I˜˜1 )2 = 1).

(22)

Then we have 

|μ2 − 4|(I˜˜1 dψx (X)) = (μI˜1 + 2I˜2 )d˜ πψ(x) (X  , λ(φX) ) ˜ = μd˜ πψ(x) (−(φX) , −λX  ) + 2d˜ πψ(x) (λ(φX) , X  ) ˜ ˜ = d˜ πψ(x) ((2λ − μ)(φX) , −(λμ − 2)X  ) ˜ = (2λ − μ)dψx (φX),

and 2λ − μ I˜˜1 dψx (X) =  dψx (φX), |μ2 − 4|

 I˜˜1 dψx (φX) = −

|μ2 − 4| dψx (X). 2λ − μ

Hence we obtain I˜˜1 dψx (Tx M ) ⊂ dψx (Tx M ).

(23)

1 I˜˜2 =  (2I˜1 + μI˜2 ). 2 |μ − 4|

(24)

Let

Then (I˜˜2 )2 = 1 (resp. (I˜˜2 )2 = −1). We get 

|μ2 − 4|(I˜˜2 dψx (X)) = (2I˜1 + μI˜2 )d˜ πψ(x) (X  , λ(φX) ) ˜ = 2d˜ πψ(x) (−(φX) , −λX  ) + μd˜ πψ(x) (λ(φX) , X  ) ˜ ˜ = d˜ πψ(x) ((λμ − 2)(φX) , (−2λ + μ)X  ), ˜

(25)

and  |μ2 − 4|(I˜˜2 dψx (φX)) = d˜ πψ(x) (−(λμ − 2)X  , (−2λ + μ)(φX) ). ˜

(26)

J.T. Cho, M. Kimura / Topology and its Applications 196 (2015) 594–607

605

Also I˜3 dψx (X) = I˜3 d˜ πψ(x) (X  , λ(φX) ) = d˜ πψ(x) (−λX  , −(φX) ), ˜ ˜

 λμ − 2  (φX) , − X πψ(x) I˜3 dψx (φX) = I˜3 d˜ ˜ 2λ − μ

 λμ − 2   = −d˜ πψ(x) (φX) , X . ˜ 2λ − μ

(27)

(28)

Hence (23), (25), (26), (27) and (28) yield that with respect to the para-Hermitian metric g given by (13), I˜˜2 dψx (Tx M ), I˜3 dψx (Tx M ) ⊥ dψx (Tx M ).

(29)

Let λ1 , · · · , λ2n−2 be the eigenvalues of A|{ξ}⊥ . We put p = #{i| |λi | > 1},

q = #{i| |λi | < 1}.

(30)

Then by Corollary 2.3, p and q are even (resp. p = q).

(31)

Using (19) and (20), we see that for X ∈ Tx M with X ⊥ ξ and |X| = 1,

g(dψx (X), dψx (X)) = λ2 − 1 and g(dψx (φX), dψx (φX)) =

λμ − 2 2λ − μ

2 − 1,

and the induced metric on ψ(M )(⊂ G1,1 (Cn+1 )) has signature 1 (p, q) (resp. (p, p)).

(32)

Now we obtain the following results: Theorem 5.1. Let M 2n−1 be a real hypersurface in complex hyperbolic space CHn of constant holomorphic sectional curvature −4 and let ψ : M → G1,1 (Cn+1 ) be the Gauss map defined by (16). Suppose M is a 1 Hopf hypersurface with Hopf curvature μ with |μ| > 2 (resp. 0 ≤ |μ| < 2) Then: (i) The image ψ(M ) is a real (2n − 2)-dimensional submanifold of G1,1 (Cn+1 ). 1 ˜ ψ(M ) of the para-quaternionic Kähler structure such (ii) There exist sections I˜˜1 , I˜˜2 and T˜3 of the bundle Q| ˜ ψ(p) for p ∈ Σ satisfying that they are orthonormal with respect to natural inner product on Q (I˜˜1 )2 = −1 (I˜˜2 )2 = 1

(resp.(I˜˜1 )2 = 1),

(resp.(I˜˜2 )2 = −1)

and

(I˜˜3 )2 = 1,

such that dψx (Tx M ) is invariant under I˜˜1 and I˜˜2 dψx (Tx M ), I˜3 dψx (Tx M ) are orthogonal to dψx (Tx M ). (iii) The induced metric on ψ(M ) in G1,1 (Cn+1 ) has signature (p, q) (resp. (p, p)), where 1 p=

 |λ|>1

dim{X| AX = λX, X ⊥ ξ},

q=



dim{X| AX = λX, X ⊥ ξ}.

|λ|<1

(iv) Furthermore if dim{X| AX = ±X, X ⊥ ξ} is 0, then the induced metric of ψ(M ) is non-degenerate and ψ(M ) is a totally complex (resp. totally para-complex) submanifold of G1,1 (Cn+1 ).

J.T. Cho, M. Kimura / Topology and its Applications 196 (2015) 594–607

606

Finally we study the case |μ| = 2. Without loss of generality, we may assume μ = 2. For a pair X, φX of eigenvectors of A|{ξ}⊥ , we may put AX = λX

and AφX = φX,

by Corollary 2.3(iii). Then (19) and (20) are written as dψx (X) = d˜ πψ(x) (X  , λ(φX) ), ˜

dψx (φX) = d˜ πψ(x) ((φX) , −X  ). ˜

By (15), we have I˜3 dψx (X) = d˜ πψ(x) (−λX  , −(φX) ), ˜

I˜3 dψx (φX) = d˜ πψ(x) (−(φX) , X  ), ˜

(I˜1 + I˜2 )dψx (X) = d˜ πψ(x) ((λ − 1)(φX) , (1 − λ)X  ), ˜

(I˜1 + I˜2 )dψx (φX) = 0.

Note that (I˜3 )2 = 1 and (I˜1 + I˜2 )2 = 0 hold. Hence if we denote I˜˜1 = I˜3 and I˜˜2 = I˜1 + I˜2 , then we obtain Theorem 5.2. Let M 2n−1 be a real hypersurface in complex hyperbolic space CHn of constant holomorphic sectional curvature −4 and let ψ : M → G1,1 (Cn+1 ) be the Gauss map defined by (16). Suppose M is a 1 Hopf hypersurface with Hopf curvature μ with |μ| = 2. Then: (i) The image ψ(M ) is a real (2n − 2)-dimensional submanifold of G1,1 (Cn+1 ). 1 ˜ ˜ ˜ (ii) There exist sections I˜1 , and T2 of the bundle Q|ψ(M ) of the para-quaternionic Kähler structure such ˜ ψ(p) for p ∈ Σ satisfying that they are orthonormal with respect to natural inner product on Q (I˜˜1 )2 = 1

and

(I˜˜2 )2 = 0,

such that I˜˜1 dψx (Tx M ), I˜˜2 dψx (Tx M ) are orthogonal to dψx (Tx M ). (iii) The induced metric on ψ(M ) in G1,1 (Cn+1 ) has signature (p, q) (resp. (p, p)), where 1 p=



dim{X| AX = λX, X ⊥ ξ},

|λ|>1

q=



dim{X| AX = λX, X ⊥ ξ},

|λ|<1

and satisfies p + q ≤ n − 1. References [1] D.V. Alekseevsky, V. Cortes, Classification of pseudo-Riemannian symmetric spaces of quaternionic Kähler type, Transl. Am. Math. Soc. 213 (2) (2005) 33–62. [2] D.V. Alekseevsky, S. Marchiafava, Hermitian and Kähler submanifolds of a quaternionic Kähler manifold, Osaka J. Math. 38 (4) (2001) 869–904. [3] A.A. Borisenko, On the global structure of Hopf hypersurfaces in a complex space form, Ill. J. Math. 45 (1) (2001) 265–277. [4] V. Cruceanu, P. Fortuny, P.M. Gadea, A survey on paracomplex geometry, Rocky Mt. J. Math. 26 (1996) 83–115. [5] T.E. Cecil, P.J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Am. Math. Soc. 269 (2) (1982) 481–499. [6] A.S. Dancer, H.R. Joergensen, A.F. Swann, Metric geometries over the split quaternions, Rend. Semin. Mat. (Torino) 63 (2) (2005) 119–139. [7] S. Funabashi, Totally complex submanifolds of a quaternionic Kaehlerian manifold, Kodai Math. J. 2 (3) (1979) 314–336. [8] F.R. Harvey, Spinors and Calibrations, Perspectives in Mathematics, vol. 9, Academic Press, Inc., Boston, MA, ISBN 0-12-329650-1, 1990, pp. xiv+323. [9] T.E. Ivey, A d’Alembert formula for Hopf hypersurfaces, Results Math. 60 (1–4) (2011) 293–309. [10] T.E. Ivey, P.J. Ryan, Hopf hypersurfaces of small Hopf principal curvature in CH 2 , Geom. Dedic. 141 (2009) 147–161. [11] M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Am. Math. Soc. 296 (1986) 137–149.

J.T. Cho, M. Kimura / Topology and its Applications 196 (2015) 594–607

607

[12] M. Kimura, Space of geodesics in hyperbolic spaces and Lorentz numbers, Mem. Fac. Sci. Eng. Shimane Univ. Ser. B Math. Sci. 36 (2003) 61–67. [13] M. Kimura, Hopf hypersurfaces in complex projective space and half-dimensional totally complex submanifolds in complex 2-plane Grassmannians I, Differ. Geom. Appl. (2015), in press. [14] S. Kaneyuki, M. Kozai, Paracomplex structures and affine symmetric spaces, Tokyo J. Math. 8 (1985) 81–98. [15] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. II, Interscience Tracts in Pure and Applied Mathematics, vol. 15, Interscience Publishers, John Wiley & Sons, Inc., New York–London–Sydney, 1969, pp. xv+470. [16] U.-H. Ki, Y.J. Suh, On real hypersurfaces of a complex space form, Math. J. Okayama Univ. 32 (1990) 207–221. [17] Y. Maeda, On real hypersurfaces of a complex projective space, J. Math. Soc. Jpn. 28 (1976) 529–540. [18] S. Marchiafava, Submanifolds of (para-)quaternionic Kähler manifolds, Note Mat. 28 (2009), [2008 on verso], suppl. 1, 295–316. [19] S. Montiel, Real hypersurfaces of a complex hyperbolic space, J. Math. Soc. Jpn. 37 (3) (1985) 515–535. [20] R. Niebergall, P.J. Ryan, Real hypersurfaces in complex space forms, in: Tight and Taut Submanifolds, Berkeley, CA, 1994, in: Math. Sci. Res. Inst. Publ., vol. 32, Cambridge Univ. Press, Cambridge, 1997, pp. 233–305. [21] B. Palmer, Hamiltonian minimality and Hamiltonian stability of Gauss maps, Differ. Geom. Appl. 7 (1) (1997) 51–58. [22] R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973) 495–506. [23] R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures, J. Math. Soc. Jpn. 27 (1) (1975) 43–53. [24] K. Tsukada, Fundamental theorem for totally complex submanifolds (English summary) Hokkaido Math. J. 36 (3) (2007) 641–668. [25] M. Vaccaro, (Para-)Hermitian and (para-)Kähler submanifolds of a para-quaternionic Kähler manifolds, Differ. Geom. Appl. 30 (2012) 347–364.