Real hypersurfaces with constant ϕ-sectional curvature in complex projective space

Differential Geometry and its Applications 68 (2020) 101573

Contents lists available at ScienceDirect

Differential Geometry and its Applications www.elsevier.com/locate/difgeo

Real hypersurfaces with constant φ-sectional curvature in complex projective space ✩ Jong Taek Cho a , Makoto Kimura b,∗ a b

Department of Mathematics, Chonnam National University, Gwangju 61186, Republic of Korea 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 23 July 2019 Accepted 21 September 2019 Available online xxxx Communicated by J. Berndt MSC: primary 53C40

a b s t r a c t We will give a geometric description of real hypersurfaces with constant φ-sectional curvature in complex projective space. Besides geodesic hypersurfaces, such real hypersurfaces are obtained as the image of either a curve or a surface in complex projective space under the polar map. As a consequence, we obtain a classification of real hypersurface M 3 in complex projective plane such that the structure vector ξ is an eigenvector of the Ricci tensor. © 2019 Elsevier B.V. All rights reserved.

Keywords: Real hypersurface Sectional curvature Tube Focal map

1. Introduction In Riemannian geometry, sectional curvature K plays central role and real space forms of constant sectional curvature, i.e., Euclidean space Rn (K = 0), sphere Sn (K > 0) and (real) hyperbolic space Hn (K < 0) are most interesting objects in geometry. Also, holomorphic sectional curvature H plays important role for Hermitian and Kähler manifolds and complex space forms of constant sectional curvature, i.e., complex Euclidean space C n (H = 0), complex projective space CP n (H > 0) and complex hyperbolic space CHn (H < 0) are fundamental objects in complex geometry. In this paper we investigate real hypersurfaces M 2n−1 in complex projective space CP n with complex structure J and Fubini-Study metric of constant holomorphic sectional curvature 4. It is well-known that M has induced almost contact metric structure (φ, η, ξ, g), and φ-sectional curvature H(X) with respect to ✩ 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 Grant Number 2019R1F1A1040829. M. Kimura was supported by JSPS KAKENHI Grant Number JP16K05119. * Corresponding author. E-mail addresses: [email protected] (J.T. Cho), [email protected] (M. Kimura).

https://doi.org/10.1016/j.difgeo.2019.101573 0926-2245/© 2019 Elsevier B.V. All rights reserved.

2

J.T. Cho, M. Kimura / Differential Geometry and its Applications 68 (2020) 101573

a unit tangent vector X orthogonal to ξ is defined by H(X) = g(R(X, φX)φX, X). In [6], we classified real hypersurfaces M 2n−1 in CP n with which H is constant in the sense that H depends neither a unit tangent vector X(⊥ ξ) nor a point in M as follows (cf. [4]): Theorem 1. [6] Let M be a real hypersurface in CP n (n ≥ 3) on which H is constant. Then M is one of the following: 1. an open subset of a geodesic hypersphere (H > 4), which we call of type A0 , 2. a ruled hypersurface (H = 4). More precisely, let T0 be the distribution defined by T0 (x) = {X ∈ Tx (M )| X ⊥ ξ} for x ∈ M , then T0 is integrable, and its integral manifolds are a totally geodesic CP n−1 (H = 4), which we call of type F1 , 3. a real hypersurface on which there is a foliation of codimension 2 such that each leaf of the foliation is contained in come complex hyperplane CP n−1 as a ruled hypersurface (H = 4), which we call of type F2 . Real hypersurface of type A0 and F1 are well-studied but it seems that neither explicit examples nor method of construction is known for real hypersurfaces of type F2 . We will give a new description for real hypersurfaces of types F1 and F2 . Among real hypersurfaces in CP n , most interesting and important class is Hopf hypersurfaces, i.e., the structure vector ξ is an eigenvector of the shape operator A satisfying Aξ = μξ. Typical example of Hopf hypersurface is a geodesic hypersphere (type A0 ) in CP n . It is known that μ (which is called Hopf curvature) is locally constant on M . Fundamental results about Hopf hypersurfaces are obtained by Cecil-Ryan [1]. If M 2n−1 lies on a tube over a complex submanifold Σ in CP n , then M is Hopf. Conversely let M 2n−1 be a Hopf hypersurface with μ = 2 cot 2r for r ∈ (0, π/2). If the rank of the focal map (§2) ϕr is constant, then M lies on a tube of radius r over a complex submanifold Σ in CP n . Recently we investigated Hopf hypersurfaces in non-flat complex space forms by using Gauss map to (indefinite) complex 2-plane Grassmannian and its twistor space [4], [5], [7], [8]. In §2, we recall focal map, tube and parallel hypersurface in CP n . Let M be an embedded real n-dimensional submanifold of CP n and let BM denote the unit normal bundle of M with projection P onto M . For N ∈ BM and r ∈ (0, π/2), let ϕr (N ) be the point in CP n reached by traversing a distance r along the geodesic γ(s) = π(cos sw + sin sN  ),

(1.1)

in CP n with γ(0) = x = π(w) and γ  (0) = N . The tube of radius r over M is the map ϕr : BM → CP n . For sufficiently small values of r at least, ϕr gives a real hypersurface in CP n . When M is a real hypersurface, we may consider ϕr : M → CP n . For values of r such that ϕr is an immersion, ϕr (M ) is called the parallel hypersurface at oriented distance r from M . When r = π/2, the image ϕπ/2 (BM ) is considered as a set of unit normal vector of M by (1.1), and also regarded as a generalization of polar surface given by Lawson [11]. So we call ϕπ/2 : BM → CP n a polar map. In §3, we show that the image of polar map ϕπ/2 of a ruled real hypersurface (resp. a real hypersurface of type F2 ) M 2n−1 is a real 1-dimensional curve Σ1 (resp. a real 2-dimensional surface Σ2 which is not J-invariant) in CP n . Hence a ruled real hypersurface (resp. a real hypersurface of type F2) lies on a tube of radius π/2 over a curve Σ1 (a surface Σ2 ) in CP n (Proposition 2 and 3). In §4 and §5, we calculate the shape operator of the image of the polar map ϕπ/2 for a curve Σ1 and a surface Σ2 which is not J-invariant, respectively. Then we show that ϕπ/2 (Σ1 ) (resp. ϕπ/2 (Σ1 )) is indeed a ruled real hypersurface (type F1 ) (resp. a real hypersurface of type F2 ) (Theorem 4 and 5). In §6, we study real hypersurfaces M 3 in complex projective plane CP 2 such that the structure vector ξ is an eigenvector of the Ricci tensor S. As easily seen that if M 3 is Hopf, then also ξ is an eigenvector of

J.T. Cho, M. Kimura / Differential Geometry and its Applications 68 (2020) 101573

3

S. But converse does not hold, for a ruled real hypersurface is not Hopf but ξ is an eigenvector of S. Then we show that on M 3 in CP 2 ξ is an eigenvector of S is and only if M 3 is either (1) a geodesic hypersphere, (2) a ruled real hypersurface or (3) the image of the polar map of a Lagrangian surface Σ2 in CP 2 . 2. Focal map and parallel hypersurfaces in complex projective space We recall the arguments about focal map for submanifolds in CP n due to Cecil-Ryan [1]. Let (z, w) =

n 

zk w ¯k

k=0

be the natural Hermitian inner product on C n+1 . Then the (real) Euclidean metric of C n+! is given by  ,  = Re(z, w). The unit sphere S 2n+1 in C n+1 is the principal fiber bundle over CP n with the structure group S 1 and the Hopf fibration π : S 2n+1 → CP n . The tangent space of S 2n+1 at a point z is Tz S 2n+1 = {w ∈ C n+1 |z, w = 0}. Let Tz = {w ∈ C n+1 |z, w = iz, w = 0}. With respect to the principal fiber bundle S 2n+1 (CP n , S 1 ), there is a connection such that Tz is the horizontal subspace at z. Then the Fubini-Study metric g of constant holomorphic sectional curvature c is given by g(X, Y ) = (c/4)X ∗ , Y ∗ , where X, Y ∈ Tx CP n , and X ∗ , Y ∗ are respectively their horizontal lifts at a point z with π(z) = x. For convenience, we will take c = 4. The complex structure on T  defined by √ multiplication by −1 induces a canonical complex structure J on CP n through π∗ . Given a vector field X on CP n , there is a corresponding basic vector field X  on S 2n+1 such that at z ∈ S 2n+1 , Xz ∈ Tz and (π∗ )z Xz = Xπ(z) . If X, Y are vector fields on CP n , the Kähler covariant derivative takes the form ˜ X Y = (π∗ )∇X  Y  ∇ where X  , Y  are the basic vector fields corresponding to X, Y and ∇ is the Levi-Civita connection on S 2n+1 . The (oriented) geodesic γ in CP n with γ(0) = x and γ  (0) = X ∈ Tw CP n is given by γ(s) = π(cos sw + sin sX  ), where w ∈ S 2n+1 with π(w) = x and X  is the horizontal lift of X to Tw . Let M be an embedded real n-dimensional submanifold of CP n and let BM denote the unit normal bundle of M with projection P onto M . For N ∈ BM and r ∈ (0, π/2), let ϕr (N ) be the point in CP n reached by traversing a distance r along the geodesic γ(t) in CP n with γ(0) = x = π(w) and γ  (0) = N . The tube of radius r over M is the map ϕr : BM → CP n . For sufficiently small values of r at least, ϕr gives a real hypersurface in CP n . When M is an oriented real hypersurface, BM is identified with M and we may consider ϕr : M → CP n . For values of r such that ϕr is an immersion, ϕr (M ) is called the parallel hypersurface at oriented distance r from M . A point p ∈ CP n is called a focal point of multiplicity ν > 0 of (M, x) if p = ϕr (N ) and dim ker(ϕr )∗ = ν at N ∈ BM . For a unit normal vector Nx ∈ BM at x ∈ M , we identify the tangent space of BM as TNx (BM ) = Tx M × (Tx⊥ M ∩ {Nx }⊥ ).

4

J.T. Cho, M. Kimura / Differential Geometry and its Applications 68 (2020) 101573

Cecil and Ryan (Proposition 2.1 in [1]) computed the differential map of ϕr : BM → CP n (0 < r < π/2) at N ∈ BM are as follows: (ϕr )∗ (0, N1 ) = (π∗ )z sin r(N1 − sin rN1 , iN  )iz, (ϕr )∗ (X, 0) = (π∗ )z (cos rXw − sin r((AN X)w ) − Xw , iN  iw),

(2.1) (2.2)

where z = cos rw + sin rNw and the vector on the right is in Tz in case (2.1), but not necessarily in (2.2). Now according to Lawson’s polar surface in [11], we define the polar map as ϕ = ϕπ/2 : BM → CP n ,

ϕ(Nx ) = π(Nw ).

(2.3)

3. Image under the Polar map of real hypersurfaces with constant φ-sectional curvature in CP n Let M 2n−1 be an oriented real hypersurface in CP n . Then M is of type F1 , i.e., ruled real hypersurface in CP n if and only if the shape operator A of M is given by Aξ = μξ + νU,

AU = νξ (ν ≡ 0),

AX = 0 (X ⊥ ξ, U ),

(3.1)

where ξ is the structure vector and U is a unit tangent vector orthogonal to ξ, of M , respectively (cf. [6]). Let ϕ : M → CP n be the polar map (cf. (2.3)) defined by ϕ(x) = π(Nw ),

(3.2)

where x ∈ M , w ∈ S 2n+1 with π(w) = x and Nw is a horizontal lift at w of unit normal vector Nx of M at x, respectively. By (2.2) with r = π/2, we have (ϕ∗ )(X) = (π∗ )N  (−(AX) + X  , iN  iw),

(3.3)

where we identify Tx M = TNx BM . Hence we obtain (ϕ∗ )(ξ) = (π∗ )N  (−(Aξ) + ξ  , iN  iw) = (π∗ )N  (−μξ  − νU  − iw) = −(π∗ )N  (νU  + iw), (ϕ∗ )(U ) = (π∗ )N  (−(AU ) + U  , iN  iw) = (π∗ )N  (−νξ  ) = 0, (ϕ∗ )(X) = 0 (X ⊥ ξ, U ). So at each point of M , the rank of ϕ∗ is equal to 1. Using the constant rank theorem (cf. [2]), we obtain: Proposition 2. Let M 2n−1 be a real hypersurface of type F1 , i.e., ruled real hypersurface in CP n , and let ϕ : M → CP n be the polar map defined by ϕ(x) = π(Nw ), where x ∈ M , w ∈ S 2n+1 with π(w) = x and Nw is a horizontal lift at w of unit normal vector Nx of M at x, respectively. Then the image ϕ(M ) is a (real) 1-dimensional curve Σ1 in CP n . Hence M lies on a tube of radius π/2 over a curve Σ1 in CP n . Let M 2n−1 be an oriented real hypersurface in CP n . Then M is of type F2 , i.e., a real hypersurface on which there is a foliation of codimension 2 such that each leaf of the foliation is contained in come complex hyperplane CP n−1 as a ruled hypersurface, if and only if the shape operator A of M is given by

J.T. Cho, M. Kimura / Differential Geometry and its Applications 68 (2020) 101573

5

Aξ = μξ + ν1 U + ν¯1 φU + ν2 V (ν12 + ν¯12 + ν22 ≡ 0), AU = ν1 ξ + λU (λ = 0), AV = ν2 ξ, AX = 0 (X ⊥ ξ, U, φU, V ),

AφU = ν¯1 ξ,

(3.4)

where ξ is the structure vector and ξ, U, φU, V are mutually unit tangent orthogonal vectors of M , [6]. Let ϕ : M → CP n be the polar map defined by (3.2). Then similar computations as above imply (ϕ∗ )(ξ) = −(π∗ )N  (ν1 U  + ν¯1 iU  + ν2 V  + iw), (ϕ∗ )(U ) = −(π∗ )N  (λU  ), (ϕ∗ )(φU ) = (ϕ∗ )(V ) = (ϕ∗ )(X) = 0 (X ⊥ ξ, U, φU, V ). So at each point of M , the rank of ϕ∗ is equal to 2. Using the constant rank theorem, we obtain: Proposition 3. Let M 2n−1 be a real hypersurface of type F2 , and let ϕ : M → CP n be the polar map defined by (3.2). Then the image ϕ(M ) is a (real) 2-dimensional surface Σ2 in CP n , which is not invariant under the complex structure J of CP n . Hence M lies on a tube of radius π/2 over a real 2-dimensional surface Σ2 in CP n , which is not J-invariant. Note that Σ2 is totally real, i.e., JT Σ ⊂ T ⊥ Σ in CP n if and only if ν¯1 = 0. 4. Polar hypersurfaces of curves in CP n Let γ : I → CP n be a parametrization of real 1-dimensional curve in CP n and let Σ1 = γ(I) be the image of γ. We denote BΣ as the unit normal bundle of Σ in CP n . Then the polar map ϕ : BΣ → CP n is given by ϕ(Nx ) = π(Nw ), where Nx ∈ BΣ and Nw is the horizontal lift of Nx at w ∈ S 2n+1 with π(w) = x. We compute the differential and locate focal points of ϕ. The unit normal bundle BΣ of Σ1 in CP n is decomposed into a disjoint union as BΣ = B 0 Σ ∪ B 1 Σ ∪ B 2 Σ, where B 0 Σ = {N ∈ BΣ| JN ∈ T ⊥ Σ},

(4.1)

/ T Σ ∪ T ⊥ Σ}, B 1 Σ = {N ∈ BΣ| JN ∈

(4.2)

B Σ = {N ∈ BΣ| JN ∈ T Σ}.

(4.3)

2

For a unit normal vector Nx ∈ BΣ at x ∈ Σ, we identify the tangent space of BΣ as TNx (BΣ) = Tx Σ × (Tx⊥ Σ ∩ {Nx }⊥ ).

(4.4)

(ϕ∗ )((0, JNx )) = 0 for Nx ∈ B 0 Σ.

(4.5)

By (2.1), we have

Hence, for each Nx ∈ B 0 Σ, ϕ(Nx ) = π(Nw ) is a focal point of (Σ, x). For Nx ∈ B 1 Σ, there exists θ with 0 < θ < π/2 such that JNx = sin θe + cos θN1 ,

iNw = sin θe + cos θN1 ,

where e ∈ Tx Σ is a unit tangent vector at x ∈ Σ and N1 ∈ Bx Σ is a unit normal vector at x ∈ Σ with N1 ⊥ Nx . We have

J.T. Cho, M. Kimura / Differential Geometry and its Applications 68 (2020) 101573

6

(ϕ∗ )((0, N1 )) = (π∗ )N  (N1 − N1 , iN  iN  ) = (π∗ )N  (N1 − cos θiN  ) = sin θ(π∗ )N  (− cos θe + sin θN1 ).

(4.6)

Also for N2 ∈ BΣ with N2 ⊥ Nx , JNx , one has (ϕ∗ )((0, N2 )) = (π∗ )N  (N2 ).

(4.7)

On the other hand, (2.2) with r = π/2 implies (ϕ∗ )(e, 0) = (π∗ )N  (sin θiw − kN e + kN e , iN  iN  ) = (π∗ )N  (sin θiw + kN cos θ(− cos θe + sin θN1 )),

(4.8)

where kN = γ  , N  is the normal curvature of γ : I → CP n with respect to a unit normal vector N . Hence ϕ(B 1 Σ) = has no focal point of (Σ, x). The vector (π∗ )N  (−w) is a unit normal to ϕ at ϕ(Nx ) = π(Nw ) (page 487 in [1]). Since the structure vector ξ with respect to ϕ is (ϕ∗ )ξ = (π∗ )N  (iw), using (4.6) and (4.8), we obtain ξ=

1 (e, −kN cot θN1 ). sin θ

The shape operator Aϕ for ϕ is given by (cf. page 487 in [1]) ˜ ϕ (X,V ) w. ϕ∗ (Aϕ (X, V )) = ∇ ∗

(4.9)

Then using (4.6) and similar computations as §3 of [1], we obtain ˜ ϕ (0,N ) w = − cos θ(π∗ )N  (iw) = − cos θ(ϕ∗ )ξ. ϕ∗ (Aϕ (0, N1 )) = ∇ ∗ 1 If we put U =−

1 (0, N1 ), sin θ

then U is a unit tangent vector with respect to the metric induced by ϕ on BΣ and Aϕ U = cot θξ. Also for N2 ∈ BΣ with N2 ⊥ Nx , JNx , one has ˜ ϕ (0,N ) w = 0, (ϕ∗ )(Aϕ (0, N2 )) = ∇ ∗ 2 and

(4.10)

J.T. Cho, M. Kimura / Differential Geometry and its Applications 68 (2020) 101573

7

Aϕ (0, N2 )) = 0. Finally for unit tangent vector e ∈ Tx Σ, ˜ ϕ (e,0) w = (π∗ )N  (e + kN sin θiw) ϕ∗ (Aϕ (e, 0)) = ∇ ∗ = (π∗ )N  (kN sin θiw − cos θ(− cos θe + sin θN1 )) = kN sin θ(ϕ∗ )(ξ) − cot θ(ϕ∗ )(N1 ),

(4.11)

implies Aϕ (e, 0) = kN sin θξ − cot θ(0, N1 ), and Aϕ ξ =

kN ξ + cot θU. sin2 θ

When Nx ∈ B 2 Σ, quite similar computations yield (ϕ∗ )((0, N2 )) = (π∗ )N  (N2 )

and (ϕ∗ )((e, 0)) = (π∗ )N  (iw) = (ϕ∗ )(ξ),

so no focal point occurs, and the shape operator Aϕ is given by above with θ = π/2. Using (3.1), we obtain: Theorem 4. Let Σ1 be a real 1-dimensional curve and let ϕ : BΣ → CP n be the polar map. Then ϕ(B 1 Σ ∪ B 2 Σ) is a real hypersurface of type F1 , i.e., ruled real hypersurface in CP n . 5. Polar hypersurfaces of surfaces in CP n Let Σ2 be a (real) 2-dimensional oriented surface in CP n . We denote BΣ as the unit normal bundle of Σ in CP n . Then as in the previous section, the polar map ϕ : BΣ → CP n is given by ϕ(Nx ) = π(Nw ), where Nx ∈ BΣ and Nw is the horizontal lift of Nx at w ∈ S 2n+1 with π(w) = x. We compute the differential and locate focal points of ϕ. The unit normal bundle BΣ of Σ1 in CP n is decomposed into a disjoint union of B 0 Σ ∪ B 1 Σ ∪ B 2 Σ, as (4.1), (4.2) and (4.3). Under the identification (4.4), (4.5) valid. Hence, for each Nx ∈ B 0 Σ, ϕ(Nx ) = π(Nw ) is a focal point of (Σ, x). For Nx ∈ B 1 Σ, there exists θ with 0 < θ < π/2 such that JNx = sin θe1 + cos θN1 ,

iNw = sin θe1 + cos θN1 ,

(5.1)

where e1 ∈ Tx Σ is a unit tangent vector at x ∈ Σ and N1 ∈ Bx Σ is a unit normal vector at x ∈ Σ with N1 ⊥ Nx . Then quite the same argument implies (ϕ∗ )((0, N1 )) = (π∗ )N  (N1 − N1 , iN  iN  ) = (π∗ )N  (N1 − cos θiN  ) = sin θ(π∗ )N  (− cos θe1 + sin θN1 ), and (4.7) hold. On the other hand, (2.2) with r = π/2 implies    (ϕ∗ )(X, 0) = (π∗ )N  (−(AΣ N X) + sin θe1 , X iw), ⊥ where X ∈ Tx Σ and AΣ N is the shape operator with respect to N ∈ Tx Σ.

(5.2)

J.T. Cho, M. Kimura / Differential Geometry and its Applications 68 (2020) 101573

8

Let {v1 , v2 } be an orthonormal basis, compatible with the orientation, of Tx Σ satisfying N AΣ N vj = kj vj

(j = 1, 2).

(5.3)

Then for another oriented orthonormal basis {e1 , e2 } of Tx Σ, we have v1 = cos αe1 + sin αe2 ,

v2 = − sin αe1 + cos αe2 ,

(5.4)

for some α ∈ [0, π/2)]. Substituting v1 and v2 into (5.2), we have (ϕ∗ )((v1 , 0)) = (π∗ )N  (−k1N v1 + sin θ cos αiw), (ϕ∗ )((v2 , 0)) = (π∗ )N  (−k2N v2 − sin θ sin αiw). By taking the orthogonal components to iN  , we obtain (ϕ∗ )((v1 , 0)) = (π∗ )N  (sin θ cos αiw − k1N v1 + k1N v1 , iN  iN  ) = (π∗ )N  (sin θ cos αiw − k1N v1 + k1N sin θ cos αiN  ) = sin θ cos α(π∗ )N  (iw) + k1N cos θ cos α(π∗ )N  (− cos θe1 + sin θN1 ) −k1N sin α(π∗ )N  (e2 ),

(5.5)

and (ϕ∗ )((v2 , 0)) = (π∗ )N  (− sin θ sin αiw − k2N v2 + k2N v2 , iN  iN  ) = (π∗ )N  (− sin θ sin αiw − k2N v2 − k2N sin θ sin αiN  ) = − sin θ sin α(π∗ )N  (iw) − k2N cos θ sin α(π∗ )N  (− cos θe1 + sin θN1 ) −k2N cos α(π∗ )N  (e2 ).

(5.6)

Now we have that ϕ(Nx ) is a focal point of (Σ, x) if and only if dimR {(ϕ∗ )((0, N1 )), (ϕ∗ )((v1 , 0)), (ϕ∗ )((v2 , 0))} < 3 ⇔ k1N sin2 α + k2N cos2 α = 0. Now we assume that with respect to Nx ∈ B 1 Σ, k1N sin2 α + k2N cos2 α = 0

(5.7)

holds. Then ϕ is an immersion at Nx ∈ B 1 Σ. As in the previous section, the vector (π∗ )N  (−w) is a unit normal to ϕ vector of real hypersurface ϕ(BΣ) at ϕ(Nx ) = π(Nw ), and the structure vector ξ with respect to ϕ is (ϕ∗ )ξ = (π∗ )N  (iw). We define unit tangent vectors U and V in TNx (BΣ) as (ϕ∗ )U = (π∗ )N  (e2 ),

V =

1 (0, N1 ). sin θ

Then (5.5) and (5.6) are written as (v1 , 0) = sin θ cos αξ − k1N sin αU + k1N cos α cos θV, (v2 , 0) = − sin θ sin αξ − k2N cos αU − k2N sin α cos θV,

J.T. Cho, M. Kimura / Differential Geometry and its Applications 68 (2020) 101573

9

and we have ξ= U=

(k2N cos αv1 − k1N sin αv2 , −k1N k2N cot θN1 ) , (k1N sin2 α + k2N cos2 α) sin θ

(− sin αv1 − cos αv2 , (k1N − k2N ) cos α sin α cot θN1 ) . k1N sin2 α + k2N cos2 α

We compute the shape operator Aϕ for ϕ, which is given by (4.9). Quite the same calculation as (4.10) yields Aϕ ((0, N1 )) = − cos θξ,

AV = − cot θξ.

(5.8)

Also for N2 ∈ BΣ with N2 ⊥ Nx , JNx , we have Aϕ (0, N2 )) = 0.

(5.9)

By the computation similar to (4.11), (ϕ∗ )(Aϕ (v1 , 0)) = (π∗ )N  (k1N sin θ cos αiw + v1 ) = (π∗ )N  (k1N sin θ cos αiw + v1 − v1 , iN  iN  ) = (π∗ )N  (k1N sin θ cos αiw + v1 − sin θ cos αiN  ) = k1N sin θ cos α(ϕ∗ )(ξ) + sin α(ϕ∗ )(U ) − cos α cot θ(ϕ∗ )(0, N1 ) implies Aϕ v1 = k1N sin θ cos αξ + sin αU − cos α cos θV. Similarly we obtain Aϕ v2 = −k2N sin θ sin αξ + cos αU + sin α cos θV. Hence we get Aϕ ξ =

k1N k2N ξ + (k2N − k1N ) cos α sin α sin θU − cot θV, (k1N sin2 α + k2N cos2 α) sin2 θ

(5.10)

(k2N − k1N ) cos α sin αξ − sin θU . (k1N sin2 α + k2N cos2 α) sin θ

(5.11)

Aϕ U =

Finally when Nx ∈ B 2 Σ, the above computations hold with θ = π/2. Since the equations (5.8)-(5.11) are equivalent to the shape operator (3.4) of real hypersurface of type F2 , we obtain: Theorem 5. Let Σ1 be a real 2-dimensional surface which is not J-invariant and let ϕ : BΣ → CP n be the polar map. Then ϕ({Nx ∈ B 1 Σ ∪ B 2 Σ| k1N sin2 α + k2N cos2 α = 0}) is a real hypersurface of type F2 in CP n .

10

J.T. Cho, M. Kimura / Differential Geometry and its Applications 68 (2020) 101573

Remark. With respect to the real hypersurface M 2n−1 of type F2 in CP n , λ = AU, U  in (3.4) is given by λ=−

k1N

1 sin α + k2N cos2 α 2

in (5.11). For Nx ∈ B 1 Σ ∪ B 2 Σ with k1N sin2 α + k2N cos2 α = 0, if we change Nx to −Nx , then k1N sin2 α + k2N cos2 α is changed to −(k1N sin2 α + k2N cos2 α). Hence the function λ on M is not constant. 6. Polar hypersurfaces of Lagrangian surfaces in CP 2 Let Σ2 be a (real) 2-dimensional oriented Lagrangian surface in CP 2 , i.e., JT Σ = T ⊥ Σ. Then the unit normal bundle BΣ over Σ is equal to B 2 Σ defined by (4.3). Hence θ defined by (5.1) is equal to π/2 and we denote JN = e1 . Let {e1 , e2 } be an orthonormal basis of Tx Σ compatible with the orientation of Σ. Also let {v1 , v2 } be an oriented orthonormal basis satisfying AΣ N vj = kj vj (j = 1, 2) and we define α ∈ [0, π/2) by (5.4). Then by (5.7), polar map ϕ is regular at Nx ∈ BΣ if and only if k1N sin2 α + k2N cos2 α = 0.

(6.1)

Using (5.8), (5.10) and (5.11), we obtain that the shape operator Aϕ of ϕ(BΣ) with respect to on orthonormal basis {ξ, U, V = φU } is written by Aϕ ξ = Aϕ U =

k1N k2N ξ + (k2N − k1N ) cos α sin αU , k1N sin2 α + k2N cos2 α

(k2N − k1N ) cos α sin αξ − U , k1N sin2 α + k2N cos2 α

Aϕ V = 0.

Let M 3 be a non-Hopf real hypersurface (i.e., the structure vector ξ is not an eigenvector of the shape operator A) in CP 2 = CP 2 (4) of constant holomorphic sectional curvature 4, and suppose that ξ is an eigenvector of the Ricci tensor S of M . By the Gauss equation, we have SX = 5X −3η(X)ξ +(trace A)AX − A2 X for a tangent vector X of M 3 and Sξ = 2ξ + (trace A)Aξ − A2 ξ. Then we can put an orthonormal basis {ξ, U, V = φU } of M such that the shape operator A satisfies Aξ = μξ + νU,

AU = νξ + λU + βV,

AV = βU + ζV,

where μ, ν, λ, β and ζ are functions on M and ν = 0, for M is non-Hopf. Also trace A = μ + λ + ζ. By direct computations, we see that the component of A2 ξ − (trace A)ξ orthogonal to ξ is ν(ζU + βV ) and we have β = ζ = 0. Hence Aξ = μξ + νU,

AU = νξ + λU,

AV = 0.

(6.2)

Furthermore the φ-sectional curvature of M 3 is constant (cf. [6]) and λ is identically equal to 0 if and only if M is of type F1 , i.e., a ruled real hypersurface and λ = 0 if and only if M is of type F2 . Theorem 6. Let M 3 be a real hypersurface in CP 2 and suppose the structure vector ξ is an eigenvector of the Ricci tensor S. Then M is locally congruent to one of the following: 1. A Hopf hypersurface, 2. A ruled real hypersurface, 3. The image of the polar map ϕ of a Lagrangian surface Σ2 in CP 2 .

J.T. Cho, M. Kimura / Differential Geometry and its Applications 68 (2020) 101573

11

Let Σ2 be a non-totally geodesic Lagrangian surface in CP 2 . Then the corresponding polar hypersurface M is of type F2 on the open subset of regular points satisfying (6.2). Typical example of such Lagrangian surface is the minimal Clifford Torus T 2 in CP 2 (cf. [3] and [12]). Note that M. Kon ([9] and [10]) gave characterizations of Hopf hypersurfaces and ruled real hypersurfaces among real hypersurfaces in non-flat complex space forms such that ξ is an eigenvector of the Ricci tensor S. 3

References [1] T.E. Cecil, P.J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Am. Math. Soc. 269 (1982) 481–499. [2] T.E. Cecil, P.J. Ryan, Geometry of Hypersurfaces, Springer Monogr. Math., Springer-Verlag, New-York, 2015. [3] B-Y. Chen, K. Ogiue, On totally real submanifolds, Trans. Am. Math. Soc. 193 (1974) 257–266. [4] J.T. Cho, M. Kimura, Pseudo-holomorphic sectional curvatures of real hypersurfaces in a complex space form, Kyushu J. Math. 62 (1) (2008) 75–87. [5] J.T. Cho, M. Kimura, Hopf hypersurfaces in complex hyperbolic space and submanifolds in indefinite complex 2-plane Grassmannian I, Topol. Appl. 196 (2015) 594–607. [6] M. Kimura, Sectional curvatures of holomorphic planes on a real hypersurface in P n (C), Math. Ann. 276 (3) (1987) 487–497. [7] M. Kimura, Hopf hypersurfaces in complex projective space and half dimensional totally complex submanifolds in complex 2-plane Grassmannian I, Differ. Geom. Appl. 35 (suppl.) (2014) 156–163. [8] M. Kimura, Hopf hypersurfaces in complex projective space and half dimensional totally complex submanifolds in complex 2-plane Grassmannian II, Differ. Geom. Appl. 54 (part A) (2017) 44–52. [9] M. Kon, 3-dimensional real hypersurfaces and Ricci operator, Differ. Geom. Dyn. Syst. 16 (2014) 189–202. [10] M. Kon, Ricci tensor of real hypersurfaces, Pac. J. Math. 281 (1) (2016) 103–123. [11] H.B. Lawson, Complete minimal surfaces in S 3 , Ann. Math. 92 (1970) 335–374. [12] G.D. Ludden, M. Okumura, K. Yano, A totally real surface in CP 2 that is not totally geodesic, Proc. Am. Math. Soc. Math. 53 (1) (1975) 186–190.