CR Structures on Real Hypersurfaces of a Complex Projective Space

Jong Taek Cho and U-Hang Ki

Abstract

We define the generalized Tanaka connection for real hypersurfaces in Kählerian manifolds, and further classify a real hypersurface of a complex projective space whose shape operator or Ricci tensor is parallel with respect to the generalized Tanaka connection, respectively.

1991 Mathematics Subject Classification: 53C15, 53C40 Key words: real hypersurfaces, associated CR-structures, generalized Tanaka connection

Introduction

Let P_{n}(C) be an n-dimensional complex projective space with Fubini-Study metric of constant holomorphic sectional curvature 4, and let M be an orientable real hypersurface of P_{n}(C). Then M has an almost contact metric structure (η, ϕ, g) induced from the Kählerian structure of P_{n}(C) (see section 2). We denote by ξ the structure vector field dual to η. One of the typical examples of M is a geodesic hypersphere. We denote by ∇, A, R and S, the Levi-Civita connection with respect to g, the shape operator, the curvature tensor and the Ricci tensor on M, respectively. It is well-known that there does not exist a real hypersurface M in P_{n}(C) with the parallel second fundamental tensor(∇A = 0). Also the second author ([4]) proved that there does not exist a real hypersurface M with the parallel Ricci tensor(∇S = 0) in P_{n}(C), n ≥ 3. As an immediate consequence of this result, P_{n}(C)(n ≥ 3) does not admit a locally symmetric(∇R = 0) real hypersurface M. For real hypersurfaces in Kählerian manifolds, the CR structure associated with the almost contact metric structure is integrable, but is not in general non-degenerate. On the other hand, N.Tanaka ([13]) defined tha canonical affine connection on a non-degenerate integrable CR manifold. And S.Tanno ([14]) defined the generalized Tanaka connection for contact metric manifolds by relaxing the integrability condition of their associated CR structures. In the present paper, for a non-zero real number k we define the generalized Tanaka connection ˇ∇^{(k)} for real hypersurfaces in Kählerian manifolds by the naturally extended one of S.Tanno’s generalized Tanaka connection (k = 1) for contact metric manifolds. The generalized Tanaka connection ˇ∇^{(k)} coincides with the Tanaka connection if real hypersurfaces satisfy ϕA + Aϕ = 2kϕ (see section 2). In section 3 we show that there are real hypersurfaces M of P_{n}(C) such that its almost contact metric structures is not contact metric structure but, ˇ∇^{(k)} defined on M for some k ≠ 0 coincides with the Tanaka connection (see Remark 1).

Balkan Journal of Geometry and Its Applications, Vol.2, No.2, 1997, pp. 7-18 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

R. Takagi ([11]) classified homogeneous real hypersurfaces of P_{n}(C) by means of six model spaces of type A_{1}, A_{2}, B, C, D, and E, further he explicitely write down their principal curvatures and multiplicities in a table in [12]. T.E.Cecil and P.J.Ryan([3]) extensively investigated a real hypersurface which is realized as a tube of constant radius r over a complex submanifold of P_{n}(C) on which ξ is a principal curvature vector field with principal curvature α = 2 cot 2r and the corresponding focal map φ_{r} has constant rank. By making use of these two works, M. Kimura ([5]) proved the following Theorem 0. Let M be a connected real hypersurface of P_{n}(C). Then M has constant principal curvatures and the structure vector field ξ is principal if and only if M is locally congruent to one of the following spaces:

(A_{1}) a geodesic hypersphere (that is, a tube of radius r over a hyperplane P_{n−1}(C)), where 0 < r < \frac{π}{2} ;

(A_{2}) a tube of radius r over a totally geodesic P_{k}(C)(1 ≤ k ≤ n − 2), where 0 < r < \frac{π}{2} ;

(B) a tube of radius r over a complex quadric Q_{n−1}, where 0 < r < \frac{π}{4} ; (C) a tube of radius r over P_{1}(C) × P_{\frac{n−1}{2}} (C), where 0 < r < \frac{π}{4} ; (D) a tube of radius r over a complex Grassmann G_{2,5}(C), where 0 < r < \frac{π}{4}

and n = 9;

(E) a tube of radius r over a Hermitian symmetric space SO(10)/U(5), where

0 < r < \frac{π}{4} and n = 15.

In these circumstaces, we will investigate real hypersurfaces of P_{n}(C) whose shape operator is parallel with respect to the generalized Tanaka connection, and further the Ricci tensor is parallel with respect to the generalized Tanaka connection. More specifically, in section 4, we prove: Theorem 1. Let M be a real hypersurface of P_{n}(C). If the shape operator A is ˇ∇^{(k)}-parallel ( ˇ∇^{(k)}A = 0), then ξ is a principal curvature vector field and further M is locally congruent to one of the homogeneous real hypersurfaces of type A_{1}, A_{2} or B, and vice versa. Proposition 1. Let M be a real hypersurface of P_{n}(C) whose structure vector ξ is a principal curvature vector field. If the Ricci tensor S of M is ˇ∇^{(k)}-parallel ( ˇ∇^{(k)}S = 0), then M is locally congruent to one of homogeneous real hypersurfaces of type A_{1}, A_{2}, or B, and vice versa.

Recently, a ruled real hypersurface is defined by a foliated one by complex hyperplanes P_{n−1}(C) and investigated in [6]. We see that the ruled real hypersurface satisfies that ϕA + Aϕ = 0 resticted on the distribution D which is determined by the kernel of η (or that D is integrable). In section 5, we prove that there does not exist the ruled real hypersurface with the ˇ∇^{(k)}-parallel Ricci tensor.

In this paper, all manifolds are assumed to be connected and of class C^{∞} and the real hypersurfaces are supposed to be oriented.

1 Almost contact metric structures and the associated CR structures

First, we give a brief review of several fundamental notions and formulas which we will need later on. An odd–dimensional Riemannian manifold M with metric tensor g is said to have an almost contact metric structure if it admits a (1,1)- tensor field ϕ, a vector field ξ and a 1-form η satisfying

(1.1) ϕ^{2}X = −X + η(X)ξ, η(ξ) = 1, g(ϕX, ϕY) = g(X, Y) − η(X)η(Y).

From (1.1) we get

(1.2)  ϕξ = 0, η ∘ ϕ = 0, η(X) = g(X, ξ).

We call (η, ϕ, g) an almost contact metric structure of M and M = (M; η, ϕ, g) an almost contact metric manifold. The tangent space T_{p}M of M at each point p ∈ M is decomposed as T_{p}M = D_{p} ⊕ {ξ}_{p} (direct sum), where we denote D_{p} = {v ∈ T_{p}M|η(v) = 0}. Then D : p → D_{p} defines a distribution orthogonal to ξ. For an almost contact metric manifold M = (M; η, ϕ, g), one may define naturally an almost complex structure on the product manifold M × R, where R denotes real line. If the almost complex structure is integrable, M is said to be normal. The integrablity condition for the almost complex structure is the vanishing of the tensor [ϕ, ϕ]+2dη⊗ξ, where [ϕ, ϕ] denotes the Nijenhuis torsion of ϕ. Also, for an almost contact metric manifold M we define its fundamental 2-form Φ by Φ(X, Y) = g(ϕX, Y). If

(1.3)  Φ = dη,

M is called a contact metric manifold. A normal contact metric manifold is called a Sasakian manifold. For more details about the general theory of almost contact metric manifolds, we refer to [1], [9].

On the other hand, for an almost contact metric manifold M = (M; η, ϕ, g), the restriction ϕ̄ = ϕ|D of ϕ to D defines an almost complex structure to D. If

[ϕ̄X, ϕ̄Y] − [X, Y] ∈ D

and

[ϕ̄, ϕ̄](X, Y) = 0

for all X, Y ∈ D, where [ϕ̄, ϕ̄] is the Nijenhuis torsion of ϕ̄, then the pair (η, ϕ̄) is called the integrable CR structure associated with the almost contact metric structure (η, ϕ, g), and the associated Levi form defined by L(X, Y) = dη(X, ϕ̄Y), X, Y ∈ D. If the associated Levi form is nondegenerate (positive or negative definite, resp.) and hermitian, then (η, ϕ̄) is called a non-degenerate (strongly pseudo-convex, resp.) pseudo-hermitian CR structure. For further details about CR structures, we refer to [2], [14].

2 The generalized Tanaka connection on real hypersurfaces

Let M be a real hypersurface of a Kählerian manifold M̃ = (M̃; J, g̃) and N (or N′ = −N) a unit normal vector on M. By ˜∇ we denote the Levi-Civita connection in M̃. Then the Gauss and Weingarten formulas are given respectively by

˜∇_{X}Y = ∇_{X}Y + g(AX, Y)N, ˜∇_{X}N = −AX,

for any vector fields X and Y on M, where g denotes the Riemannian metric of M induced from g̃. An eigenvector(resp. eigenvalue) of the shape operator A is called a principal curvature vector (resp. principal curvature). We denote by V_{λ} the eigenspace associated with an eigenvalue λ. For any vector field X tangent to M, we put

(2.1)  JX = ϕX + η(X)N, JN = −ξ.

Then we easily see that the structure (η, ϕ, g) is an almost contact metric structure on M, and further it is known that the associated CR structure is integrable (cf. [2]). From ˜∇J = 0 and (2.1), making use of the Gauss and Weingarten formulas, we have

(2.2)  (∇_{X}ϕ)Y = η(Y)AX − g(AX, Y)ξ,

(2.3)  ∇_{X}ξ = ϕAX.

In general, the notion of contact metric structure is equivalent to the notion of strongly pseudo-convex CR structure, not necessarily integrable (see Proposition 2.1 in [14]). From (1.3) and (2.3) we have Proposition 2. Let M = (M; η, ϕ, g) be a real hypersurface of a Kählerian manifold. The almost contact metric structure of M is contact metric if and only if ϕA + Aϕ = ±2ϕ, where ± is determined by the orientation.

The Tanaka connection ([13]) is the canonical affine connection defined on non-degenerate integrable CR manifold. S.Tanno ([14]) defined the generalized Tanaka connection for contact metric manifolds by the unique linear connection which coincides with the Tanaka connection if the associated CR structure is integrable. We define the generalized Tanaka connection for real hypersurfaces of Kählerian manifolds by the naturally extended one of S.Tanno’s generalized Tanaka connection for contact metric manifolds.

Now we recall the generalized Tanaka connection ˇ∇ for contact metric manifolds,

ˇ∇_{X}Y = ∇_{X}Y + (∇_{X}η)(Y)ξ − η(Y)∇_{X}ξ − η(X)ϕY,

for all vector fields X and Y.

Thus, taking account of (2.3) the generalized Tanaka connection ˇ∇^{(k)} for real hypersurfaces of Kählerian manifolds is naturally defined by

(2.4)  ˇ∇_{X}^{(k)}Y = ∇_{X}Y + g(ϕAX, Y)ξ − η(Y)ϕAX − kη(X)ϕY,

where k is a non-zero real number. We put F_{X}Y = g(ϕAX, Y)ξ − η(Y)ϕAX − kη(X)ϕY. Then the torsion tensor is Ť^{(k)}(X, Y) = F_{X}Y − F_{Y} X. Also, by using (1.2), (1.3), (2.2), (2.3) and (2.4) we can see that

(2.5)  ˇ∇^{(k)}η = 0, ˇ∇^{(k)}ξ = 0, ˇ∇^{(k)}g = 0, ˇ∇^{(k)}ϕ = 0.

and

Ť^{(k)}(X, Y) = 2dη(X, Y)ξ, ∀X, Y ∈ D.

We note that the associated Levi form L(X, Y) = \frac{1}{2}g((ϕ̄Ā + Āϕ̄)X, ϕ̄Y), where we denote by Ā the restriction A to D. We denote the extended one of L to T M by the same letter L. If M satisfies ϕA + Aϕ = 2kϕ, then we see that the associated CR structure is strongly pseudo-convex and further satisfy Ť^{(k)}(ξ, ϕY) = −ϕŤ^{(k)}(ξ, Y), and hence the generalized Tanaka connection ˇ∇ coincides with the Tanaka connection (see [14]). That is, we have Proposition 3.Let M = (M; η, ϕ, g) be a real hypersurface of a Kählerian manifold. If M satisfies ϕA + Aϕ = 2kϕ, then the generalized Tanaka connection ˇ∇^{(k)} coincides with the Tanaka connection.

3 Real hypersurfaces of P_{n}(C)

Let M̃ be a complex projective space P_{n}(C) of constant holomorphic sectional curvature 4. Then we have the following Gauss and Codazzi equations

(3.1) 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,
(3.2)  (∇_{X}A)Y − (∇_{Y} A)X = η(X)ϕY − η(Y)ϕX − 2g(ϕX, Y)ξ,

for any tangent vector fields X, Y, Z on M. Using (1.2), (1,3), (2.2) and (2.3) we get

(3.3)  SX = (2n + 1)X − 3η(X)ξ + hAX − A^{2}X,

(3.4) (∇_{X}S)Y = −3{g(ϕAX, Y)ξ + η(Y)ϕAX} + (Xh)AY + h(∇_{X}A)Y
−A(∇_{X}A)Y − (∇_{X}A)AY,

for any tangent vector fields X, Y on M, where h = traceA.

We recall the the following : Proposition 4 ([7]). If ξ is a principal curvature vector, then the corresponding principal curvature α is constant. Proposition 5 ([7]). Assume that ξ is a principal curvature vector field with corresponding principal curvature α. If AX = λX for X orthogonal to ξ, then

we have AϕX = \frac{αλ + 2}{2λ − α}ϕX.

Under the same hypothesis as in Proposition 5, by Proposition 4 and the table in [12] we see that a real hypersurface M is locally congruent to one of type B if and only if M satisfies αλ^{2} + 4λ − α = 0, where AX = λX for X orthogonal to ξ. If M satisfies ϕA + Aϕ = cϕ for some non-zero constant c, then by using (1.1) we easily see that ξ is a principal curvature vector field, and thus taking account of Proposition 5 we have (see also [15]) Theorem 2. Let M be a real hypersurface of P_{n}(C). Then M satisfies ϕA+Aϕ = cϕ for some non-zero constant c if and only if M is locally congruent to one of

homogeneous real hypersurfaces of type A_{1} with α = \frac{c^{2} − 4}{2c} or B with α = −\frac{4}{c}.

Here we prove Proposition 6. Let M be a real hypersurface of P_{n}(C). The almost contact metric structure of M is contact metric if and only if M is locally congruent to a geodesic hypersphere of radius \frac{π}{4} or a tube of radius \frac{π}{8} over a complex quadric

Q^{n−1}. Proof. If the almost contact metric structure is contact metric, then from Proposition 2 we get

(3.5)  ϕAX + AϕX = ±2ϕX,

for all tangent vector field X on M, where ± is determined by the orientation in such a way that the almost contact metric structure is a contact metric structure. We put X = ξ in (3.5), then together with (1.2) we get ϕAξ = 0, from which we see that ξ is a principal curvature vector field. Assume that X ∈ V_{λ}. Then from (3.5) and Proposition 5, we have λ^{2} −2λ+1±α = 0. Taking account of the table given in [12], we can see that M is locally congruent to a geodesic hypersphere of radius \frac{π}{4} or a tube of radius \frac{π}{8} over a complex quadric Q_{n−1}. (Q.E.D.) Remark 1. From Theorem 2, Propositions 3 and 6 we see that the almost contact metric structure on geodesic hyperspheres of radius r ≠ \frac{π}{4} and tubes

over complex quadric Q_{n−1} of radius r ≠ \frac{π}{8} are not contact metric, but for a constant c ≠ ±2 appeared in theorem 2 the generalized Tanaka connection ˇ∇^{(c/2)} defined on them coincides with the Tanaka connection. Theorem 3 ([8]). Let M be a real hypersurface of P_{n}(C). Then M is locally congruent to one of the homogeneous real hypersurfaces of type A_{1}, A_{2} if and only if M satisfies ϕA = Aϕ.

The following Theorems 4 and 5 are very useful for the proof of Theorem 1 and Proposition 1 in section 4. Theorem 4 ([6]). Let M be a real hypersurface of P_{n}(C). Then the shape operator satisfies g((∇_{X}A)Y, Z) = 0 for any X, Y and Z which are orthogonal to ξ and ξ is a principal curvature vector field if and only if M is locally congruent to one of homogeneous real hypersurfaces of type A_{1}, A_{2} or B. Theorem 5 ([10]). Let M be a real hypersurface of P_{n}(C). Then the Ricci tensor of M satisfies g((∇_{X}S)Y, Z) = 0 for any X, Y and Z which are orthogonal to ξ and ξ is a principal curvature vector field if and only if M is locally congruent to one of homogeneous real hypersurfaces of type A_{1}, A_{2} or B.

4 Proof of Theorem 1 and Proposition 1

We define a vector field U on M by U = ∇_{ξ}ξ. Then from (1.2) and (2.3) we easily observe that

(4.1)  g(U, ξ) = 0, g(U, Aξ) = 0, ∥ U ∥^{2}= g(U, U) = β − α^{2},

where β = g(A^{2}ξ, ξ). From (1.2),(2.3) and (4.1) we have at once: Lemma 1. Aξ = αξ if and only if β − α^{2} = 0.

Taking account of (2.4), we have

(4.2)  ( ˇ∇_{X}^{(k)}A)Y = ˇ∇_{X}^{(k)}AY − A ˇ∇_{X}^{(k)}Y = (∇_{X}A)Y + F_{X}AY − AF_{X}Y
= (∇_{X}A)Y + g(ϕAX, AY)ξ − η(AY)ϕAX − kη(X)ϕAY
−g(ϕAX, Y)Aξ + η(Y)AϕAX + kη(X)AϕY,

for any vector fields X and Y on M. First, we prove that ξ is a principal curvature vector, i.e., Aξ = αξ. From (4.2) we see that ˇ∇_{ξ}^{(k)}A = 0 implies (4.3) (∇_{ξ}A)X = k(ϕAX − AϕX) + η(AX)U − g(AX, U)ξ + g(X, U)Aξ − η(X)AU,

for any vector field X on M. From (4.1) and (4.3) we easily see that dα(ξ) = 0, where d denotes the exterior derivative. The above equation (4.3), together with (3.2), yields

(4.4)  (∇_{X}A)ξ = k(ϕAX − AϕX) − ϕX − g(U, AX)ξ + η(AX)U
−η(X)AU + g(U, X)Aξ.

With U = ϕAξ and from (2.2),(2.3) and (4.4) we have

(4.5) ∇_{X}U = X +(α−k)AX −kϕAϕX +ϕAϕAX −η(A^{2}X)ξ +(α+k)η(AX)ξ
−η(X)ξ − η(AX)Aξ + g(U, X)U − η(X)ϕAU.

Also, it follows from (4.2) and ( ˇ∇_{X}^{(k)}A)ξ = 0 that

(4.6)  (∇_{X}A)ξ = g(AX, U)ξ + αϕAX + kη(X)U − AϕAX

for any vector field X on M. From (4.4) and (4.6) we get

(4.7) kg(ϕAX, Y) − kg(AϕX, Y) − g(ϕX, Y) − g(U, AX)η(Y) + η(AX)g(U, Y)
−η(X)g(AU, Y) + g(U, X)η(AY)
= g(AX, U)η(Y) + αg(ϕAX, Y) + kη(X)g(U, Y) − g(AϕAX, Y)

for any vector fields X and Y on M. We put Y = ξ in (4.7) and taking account of (1.2) and (2.1), we have

(4.8)  (α + k)g(X, U) = 3g(AX, U)

for any vector field X on M. The equation (4.8) yields that

(4.9)  3AU = (α + k)U.

Further, from (3.2) and (4.6) we get

(4.10)  (∇_{ξ}A)X = ϕX + g(AX, U)ξ + αϕAX + kη(X)U − AϕAX

for any vector field X on M. From (4.10) it follows that

(4.11) 2g(ϕX, Y)+g(AX, U)η(Y)−g(AY, U)η(X)+αg(ϕAX, Y)−αg(ϕAY, X)
+kη(X)g(U, Y) − kη(Y)g(U, X) − g(AϕAX, Y) + g(AϕAY, X) = 0

for all vector fields X and Y on M. Putting X = Aξ and Y = U in (4.11) and taking account of (2.1) and (4.9), we obtain

(4.12)  (α − 2k)g(A^{2}ξ, ϕU) = 3(2 + kα)(β − α^{2}).

Now, let W be the subset of M such that β − α^{2} ≠ 0. Then W is an open subset of M. Suppose that W is non-empty, and from now on we discuss our arguments on W. If there exists a point p ∈ W such that α(p) = 2k, then from (4.12) we see that β(p) − α(p)^{2} = 0, which is impossible. Thus we see that α ≠ 2k on W, and we get from (4.12)

(4.13)  g(A^{2}ξ, ϕU) = 3\frac{2 + kα}{α − 2k}(β − α^{2}).

From (4.13), taking account of ϕU = −Aξ + αξ, we get

(4.14)  g(AϕU, ϕU) = −\frac{α^{2} + kα + 6}{α − 2k}(β − α^{2}).

Further from (4.9) we have

(4.15)  3ϕAU = (α + k)ϕU.

Differentiating covariantly (4.15) with respect to ξ and taking the component of U, then yields with (2.2), (4.5), (4.10), (4.13) and (4.14) :

(α + k)g(AϕU, ϕU) + 3g(A^{2}ξ, ϕU) = (3β − 2α^{2} − 3α + 2)g(U, U).

Thus from (4.14) and (4.15), we obtain

(α − 2k)(3β − α^{2} − kα + 9 − k^{2}) = 0.

Since α ≠ 2k on W, we have

(4.16)  3β = α^{2} + kα − 9 + k^{2}.

Then 3∥U∥^{2} = 3(β − α^{2}) = k^{2} + αk − 2α^{2} − 9 ≥ 0 independent with k. Thus for any (fixed) point we have α^{2} ≤ −4, which is impossible. After all, we conclude that W is empty, and hence Aξ = αξ on M.

Now the equation (4.2), together with Aξ = αξ, shows that

(4.17)  g((∇_{X}A)Y, Z) = 0

for any vector fields X, Y and Z orthogonal to ξ. Thus by Theorem 4, we see that M is locally congruent to one of homogeneous real hypersurfaces of type A_{1}, A_{2} or B. By using (2.3) and Proposition 4 we can see that a homogeneous real hypersurface M of type A_{1}, A_{2} or B satisfies ( ˇ∇^{(k)}A)ξ = 0. Further M satisfies ( ˇ∇_{ξ}^{(k)}A)X = 0 for any vector field X orthogonal to ξ. In fact, from (4.2) taking account of (3.2) and Aξ = αξ, we get

( ˇ∇_{ξ}^{(k)}A)X = αϕAX − AϕAX + ϕX − k(ϕAX − AϕX),

for any vector field X orthogonal to ξ. Assume X ∈ V_{λ}. Then from Proposition 5 we have

(4.18)  ( ˇ∇_{ξ}^{(k)}A)X = (α − 2k)(λ^{2} − αλ − 1)ϕX.

From Theorem 3, we see that a real hypersurface of type A_{1} or A_{2} satisfies λ^{2} − αλ − 1 = 0. Also, from Theorem 0 we see that for a real hypersurface of type B α(= 2 cot 2r) is non-zero constant. So, from (4.18) we see that M of type A_{1}, A_{2} or B satisfies ( ˇ∇_{ξ}^{(k)}A)X = 0 for any vector field X orthogonal to ξ. Therefore we have proved Theorem 1. (Q.E.D.)

Next, we prove Proposition 1. Taking account of (2.4), we have

(4.19)  ( ˇ∇_{X}^{(k)}S)Y = ˇ∇_{X}^{(k)}SY − S ˇ∇_{X}^{(k)}Y = (∇_{X}S)Y + F_{X}SY − SF_{X}Y

= (∇_{X}S)Y + g(ϕAX, SY)ξ − g(ϕAX, Y)Sξ − η(SY)ϕAX + η(Y)SϕAX

−kη(X)ϕSY + kη(X)SϕY

for any vector fields X and Y on M. From (4.19), the hypotheses ˇ∇^{(k)}S = 0 and Aξ = αξ yield

(4.20)  g((∇_{X}S)Y, Z) = 0

for any vector fields X, Y and Z orthogonal to ξ. From (4.20) and Theorem 5 we see that M is locally congruent to one of the real hypersurfaces of type A_{1}, A_{2} or B.

For a real hypersurface M of type A_{1}, A_{2} or B, since Aξ = αξ, by making use of (2.3), (3.3) and Proposition 4 we easily see that M satisfies ( ˇ∇^{(k)}S)ξ = 0. Now from (5.1), (3,3) and (3.4) we get

(4.21) ( ˇ∇_{ξ}^{(k)}S)X = h(αϕAX − AϕAX + ϕX) − (αAϕAX − A^{2}ϕAX + AϕX)
−(αϕA^{2}X − AϕA^{2}X + ϕAX) + kh(ϕA − Aϕ)X + k(ϕA^{2} − A^{2}ϕ)X

for any vector field X orthogonal to ξ. Assume that X ∈ V_{λ}. Then from (4.21) and Proposition 5 we have

( ˇ∇_{ξ}^{(k)}S)X = (α − 2k){2λ^{4} − 2(α + h)λ^{3} + 3αhλ^{2} − (2α − 2h + α^{2}h)λ −
− 2 − αh} = (α − 2k)(λ^{2} − αλ − 1){2λ^{2} − 2hλ + (2 + αh)}.

Thus, by the similar arguments in the last part in the proof of Theorem 1 we see that a real hypersurface M of type A_{1}, A_{2} or B also satisfies ( ˇ∇_{ξ}^{(k)}S)X = 0 for any vector field X orthogonal to ξ. Therefore we have proved Proposition 1. (Q.E.D.)

5 Ruled real hypersurfaces

A ruled real hypersurface M of P_{n}(C) is defined by a foliated one by complex hyperplanes P_{n−1}C and the shape operator A of M is given by (cf.[6])

(5.1)  Aξ = αξ + νV (ν ≠ 0), AV = νξ, AX = 0 for any X ⊥ ξ, V,

where V is a unit vector orthogonal to ξ, and where α, ν are differentiable functions on M. From (5.1) we immediately see that ϕ̄Ā + Āϕ̄ = 0, which is equivalent to the condition that the distribution D is integrable (cf. Proposition 5 in [6]).

From (3.3) and (5.1), we have

(5.2)  Sξ = aξ + bV, SV = cV + bξ,
SX = (2n + 1)X for any X ⊥ ξ, V,

where a = 2(n − 1) + αh − α^{2} − ν^{2}, b = ν(h − α) and c = 2n + 1 − ν^{2}. We have Proposition 7. There does not exist a ruled real hypersurface with ˇ∇^{(k)}-parallel Ricci tensor in P_{n}(C). Proof. Suppose that a ruled real hypersurface M satisfy ˇ∇^{(k)}S = 0. Then (5.2), ˇ∇^{(k)}S = 0 and straightforward calculations yield the following:

(5.3)  g(( ˇ∇_{X}^{(k)}S)Y, V) = ν^{2}g( ˇ∇_{X}^{(k)}Y, V) = 0,

(5.4)  g(( ˇ∇_{ξ}^{(k)}S)Y, V) = ν^{2}g( ˇ∇_{ξ}^{(k)}Y, V) = 0,

(5.5)  g(( ˇ∇_{V}^{(k)}S)Y, V) = ν^{2}g( ˇ∇_{V}^{(k)}Y, V) = 0,

(5.6)  g(( ˇ∇_{X}^{(k)}S)V, V) = Xν^{2} = 0,

(5.7)  g(( ˇ∇_{ξ}^{(k)}S)V, V) = ξν^{2} = 0,

(5.8)  g(( ˇ∇_{V}^{(k)}S)V, V) = V ν^{2} = 0.

From (5.6)-(5.8) we see that ν is a non-vanishing constant on M. From (2.5) and g(V, V) = 1, we see that ˇ∇_{X}^{(k)}V, ˇ∇_{V}^{(k)}V and ˇ∇_{ξ}^{(k)}V are all orthogonal to V and ξ, and from (5.3)-(5.5) we see that ˇ∇^{(k)}V = 0.

Thus from (2.4) we have

(5.9)  ∇_{ξ}V = kϕV, ∇_{V} V = 0, ∇_{X}V = 0

for any X ⊥ ξ, V. Then from (2.2), (5.1) and (5.9), we get R(V, ϕV)ϕV = ∇_{V} (∇_{ϕV} ϕV) − ∇_{ϕV} (∇_{V} ϕV) − ∇_{[V,ϕV]}ϕV = 0. But, from (3.1) and (5.1) we see that R(V, ϕV)ϕV = 4V. Thus we obtain V = 0, which is impossible. Therefore we have our conclusion. (Q.E.D.)

From the above Proposition 7, we see also that there does not exist a ruled real hypersurface with ˇ∇^{(k)}-parallel curvature tensor. Remark 2. We will discuss on real hypersurfaces of a complex hyperbolic space H_{n}(C) of constant holomorphic sectional curvature -4 by using the extended generalized Tanaka connection in a forthcoming paper. Acknowledgements. Research of the first author was supported by TGRC-KOSEF and BSRI-97-1425. Research of the second author was supported by TGRC-KOSEF. A version of this paper was presented at the First Conference of Balkan Society of Geometers, Politehnica University of Bucharest, September 23-27, 1996.

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Chonnam University Department of Mathematics Kwangju 500-757, Korea e-mail:jtcho@chonnam.chonnam.ac.kr

Topology and Geometry Research Center Kyungpook University Taegu 702-701, Korea