Jacobi Operators with Respect to the Reeb Vector Fields on Real Hypersurfaces in a Nonat Complex Space Form

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doi:10.5666/KMJ.2016.56.2.541

Article

Kyungpook Mathematical Journal 2016; 56(2): 541-575

Published online June 1, 2016 https://doi.org/10.5666/KMJ.2016.56.2.541

Jacobi Operators with Respect to the Reeb Vector Fields on Real Hypersurfaces in a Nonat Complex Space Form

U-Hang Ki¹, Soo Jin Kim², Hiroyuki Kurihara³

The National Academy of Siences, Seoul 06579, Korea¹
Department of Mathematics Chosun University, Gwangju 61452, Korea²
The College of Education, Ibaraki University, Mito 310-8512, Japan³

Received: August 18, 2015; Accepted: January 19, 2016

Abstract

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Abstract
1. Introduction
2. Preliminaries
3. Real Hypersurfaces Satisfying R_ξφ = φR_ξ
4. Real Hypersurfaces SatisfyingR_ξφ = φR_ξand ∇_{φ∇_ξξ}R_ξ = 0
5. The Exterior Derivative of 1-form u
6. Lemmas
7. Proof of the Main Theorem
References

Let M be a real hypersurface of a complex space form with almost contact metric structure (φ, ξ, η, g). In this paper, we prove that if the structure Jacobi operator R_ξ = R(·, ξ)ξ is φ∇_ξξ-parallel and R_ξ commute with the structure tensor φ, then M is a homogeneous real hypersurface of Type A provided that TrR_ξ is constant.

Keywords: complex space form, real hypersurface, structure Jacobi operator, structure tensor, Ricci tensor.

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
3. Real Hypersurfaces Satisfying R_ξφ = φR_ξ
4. Real Hypersurfaces SatisfyingR_ξφ = φR_ξand ∇_{φ∇_ξξ}R_ξ = 0
5. The Exterior Derivative of 1-form u
6. Lemmas
7. Proof of the Main Theorem
References

A complex n-dimensional Kähler manifold of constant holomorphic sectional curvature 4c ≠= 0 is called a complex space form, which is denoted by M_n(c). So naturally there exists a Kähler structure J and Kähler metric g̃ on M_n(c). As is well known, complete and simply connected complex space forms are isometric to a complex projective space P_n(ℂ), or complex hyperbolic space H_n(ℂ) as c > 0 or c < 0. Now let us consider a real hypersurface M in M_n(c). Then we also denote by g the induced Riemannian metric of Mand by N a local unit normal vector field of M in M_n(c). Further, A denotes by the shape operator of M in M_n(c). Then, an almost contact metric structure (φ, ξ, η, g) of M is naturally induced from the Kähler structure of M_n(c) as follows:

φX=(JX)T, ξ=-JN, η(X)=g(X,ξ),X∈TM,

where TM denotes the tangent bundle of M and ( )^T the tangential component of a vector. The Reeb vector ξ is said to be principal if Aξ = αξ, where α = η(Aξ). A real hypersurface is said to a Hopf hypersurface if the Reeb vector ξ of M is principal. Hopf hypersurfaces is realized as tubes over certain submanifolds in P_nℂ, by using its focal map (see Cecil and Ryan [2]). By making use of those results and the mentioned work of Takagi ([17], [18]), Kimura [11] proved the local classification theorem for Hopf hypersurfaces of P_nℂ whose all principal curvatures are constant. For the case H_nℂ, Berndt [1] proved the classification theorem for Hopf hypersurfaces whose all principal curvatures are constant. Among the several types of real hypersurfaces appeared in Takagi’s list or Berndt’s list, a particular type of tubes over totally geodesic P_kℂ or H_kℂ (0 ≤ k ≤ n–1) adding a horosphere in H_nℂ, which is called type A, has a lot of nice geometric properties. For example, Okumura [13](resp. Montiel and Romero [12]) showed that a real hypersurface in P_nℂ (resp. H_nℂ) is locally congruent to one of real hypersurfaces of type A if and only if the Reeb flow ξ is isometric or equivalently the structure operator φ commutes with the shape operator A.

The Reeb vector field ξ plays an important role in the theory of real hypersurfaces in a complex space form M_n(c). Related to the Reeb vector field ξ the Jacobi operator R_ξ defined by R_ξ = R(·, ξ)ξ for the curvature tensor R on a real hypersurface M in M_n(c) is said to be a structure Jacobi operator on M. The structure Jacobi operator has a fundamental role in contact geometry. In [3], Cho and first author started the study on real hypersurfaces in complex space form by using the operator R_ξ. In particular the structure Jacobi operator has been studied under the various commutative conditions ([4], [5], [7], [16]). For example, Pérez et al. [16] called that real hypersurfaces M has commuting structure Jacobi operator if R_ξR_X = R_XR_ξ for any vector field X on M, and proved that there exist no real hypersurfaces in M_n(c) with commuting structure Jacobi operator. On the other hand Ortega et al. [14] have proved that there are no real hypersurfaces in M_n(c) with parallel structure Jacobi operator R_ξ, that is, ∇_XR_ξ = 0 for any vector field X on M. More generally, such a result has been extended by [15]. In this situation, if naturally leads us to be consider another condition weaker than parallelness. In the preceding work, we investigate the weaker condition ξ-parallelness, that is, ∇_ξR_ξ = 0 (cf. [4], [7], [8]). Moreover some works have studied several conditions on the structure Jacobi operator R_ξ ([3], [5], [7] and [8]). The following facts are used in this paper without proof.

Theorem 1.1. (Ki, Kim and Lim [5])

Let M be a real hypersurface in a nonat complex space form M_n(c), c ≠= 0 which satisfies R_ξ(Aφ − φA) = 0. Then M is a Hopf hypersurface in M_n(c). Further, M is locally congruent to one of the following hypersurfaces:

(I) In cases that M_n(c) = P_nℂ with η(Aξ) ≠= 0,
- (A₁) a geodesic hypersphere of radius r, where 0 < r < π/2 and r ≠= π/4;
- (A₂) a tube of radius r over a totally geodesic P_kℂ for some k ∈ {1, . . . , n–2}, where 0 < r < π/2 and r ≠= π/4.
(II) In cases M_n(c) = H_nℂ,
- (A₀) a horosphere;
- (A₁) a geodesic hypersphere or a tube over a complex hyperbolic hyperplane H_n_–1ℂ;
- (A₂) a tube over a totally geodesic H_kℂ for some k ∈ {1, . . . , n – 2}.

Theorem 1.2. (Ki, Nagai and Takagi [9])

Let M be a real hypersurface in a nonat complex space form M_n(c), c ≠= 0 If M satisfies R_ξφ = φR_ξand at the same time R_ξS = SR_ξ. Then M is the same types as those in Theorem 1.1, where S denotes the Ricci tensor of M.

In [6], the authors started the study on real hypersurfaces in a complex space form with φ∇_ξξ-parallel structure Jacobi operator R_ξ, that is, ∇_{φ∇_ξξ}R_ξ = 0 for the vector φ∇_ξξ orthogonal to ξ. In this paper we invetigate the structure Jacobi operator is φ∇_ξξ-parallel under the condition that the structure Jacobi operator commute with the structure tensor φ. We prove that if the structure Jacobi operator R_ξ is φ∇_ξξ-parallel and R_ξ commute with the structure tensor φ, then M is homogeneous real hypersurfaces of Type A provided that TrR_ξ is constant.

All manifolds in this paper are assumed to be connected and of class C^∞ and the real hypersurfaces are supposed to be oriented.

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
3. Real Hypersurfaces Satisfying R_ξφ = φR_ξ
4. Real Hypersurfaces SatisfyingR_ξφ = φR_ξand ∇_{φ∇_ξξ}R_ξ = 0
5. The Exterior Derivative of 1-form u
6. Lemmas
7. Proof of the Main Theorem
References

Let M be a real hypersurface immersed in a complex space form M_n(c), c ≠= 0 with almost complex structure J, and N be a unit normal vector field on M. The Riemannian connection ∇̃ in M_n(c) and ∇ in M are related by the following formulas for any vector fields X and Y on M:

∇˜XY=∇XY+g(AX,Y)N, ∇˜XN=-AX

where g denotes the Riemannian metric of M induced from that of M_n(c) and A denotes the shape operator of M in direction N. For any vector field X tangent to M, we put

JX=φX+η(X)N, JN=-ξ.

We call ξ the structure vector field (or the Reeb vector field) and its flow also denoted by the same latter ξ. The Reeb vector field ξ is said to be principal if Aξ = αξ, where α = η(Aξ).

A real hypersurface M is said to be a Hopf hypersurface if the Reeb vector field ξ is principal. It is known that the aggregate (φ, ξ, η, g) is an almost contact metric structure on M, that is, we have

φ2X=-X+η(X)ξ, g(φX,φY)=g(X,Y)-η(X)η(Y),η(ξ)=1, φξ=0, η(X)=g(x,ξ)

for any vector fields X and Y on M. From Kähler condition ∇̃J = 0, and taking account of above equations, we see that

(2.1)∇Xξ=φAX,

(2.2)(∇Xφ)Y=η(Y)AX-g(AX,Y)ξ

for any vector fields X and Y tangent to M.

Since we consider that the ambient space is of constant holomorphic sectional curvature 4c, equations of Gauss and Codazzi are respectively given by

(2.3)R(X,Y)Z=c{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,

(2.4)(∇XA)Y-(∇YA)X=c{η(X)φY-η(Y)φX-2g(φX,Y)ξ}

for any vector fields X, Y and Z on M, where R denotes the Riemannian curvature tensor of M.

In what follows, to write our formulas in convention forms, we denote by α = η(Aξ), β = η(A²ξ) and h = TrA, and for a function f we denote by ∇f the gradient vector field of f.

From the Gauss equation (2.3), the Ricci tensor S of M is given by

(2.5)SX=c{(2n+1)X-3η(X)ξ}+hAX-A2X

for any vector field X on M.

Now, we put

(2.6)Aξ=αξ+μW,

where W is a unit vector field orthogonal to ξ. In the sequel, we put U = ∇_ξξ, then by (2.1) we see that

(2.7)U=μφW

and hence U is orthogonal to W. So we have g(U,U) = μ². Using (2.7), it is clear that

(2.8)φU=-Aξ+αξ,

which shows that g(U,U) = β − α². Thus it is seen that

(2.9)μ2=β-α2.

Making use of (2.1), (2.7) and (2.8), it is verified that

(2.10)μg(∇XW,ξ)=g(AU,X),

(2.11)g(∇Xξ,U)=μg(AW,X)

because W is orthogonal to ξ.

Now, differentiating (2.8) covariantly and taking account of (2.1) and (2.2), we find

(2.12)(∇XA)ξ=-φ∇XU+g(AU+∇α,X)ξ-AφAX+αφAX,

which together with (2.4) implies that

(2.13)(∇ξA)ξ=2AU+∇α.

Applying (2.12) by φ and making use of (2.11), we obtain

(2.14)φ(∇XA)ξ=∇XU+μg(AW,X)ξ-φAφAX-αAX+αg(Aξ,X)ξ,

which connected to (2.1), (2.9) and (2.13) gives

(2.15)∇ξU=3φAU+αAξ-βξ+φ∇α.

Using (2.3), the structure Jacobi operator R_ξ is given by

(2.16)Rξ(X)=R(X,ξ)ξ=c{X-η(X)ξ}+αAX-η(AX)Aξ

for any vector field X on M. Differentiating this covariantly along M, we find

(2.17)g((∇XRξ)Y,Z)=g(∇X(RξY)-Rξ(∇XY),Z)=-c(η(Z)g(∇Xξ,Y)+η(Z)g(∇Xξ,Z))+(Xα)g(AY,Z)+αg((∇XA)Y,Z)-η(AZ){g((∇XA)ξ,Y)+g(AφAX,Y)}-η(AY){g((∇XA)ξ,Z)+g(AφAX,Z)}.

From (2.5) and (2.16), we have

(2.18)(RξS-SRξ)(X)=-η(AX)A3ξ+η(A3X)Aξ-η(A2X)(hAξ-cξ)+(hη(AX)-cη(X))A2ξ-ch(η(AX)ξ-η(X)Aξ).

Let Ω be the open subset of M defined by

Ω={p∈M;Aξ-αξ≠0}.

At each point of Ω, the Reeb vector field ξ is not principal. That is, ξ is not an eigenvector of the shape operator A of M if Ω ≠= ∅︀.

In what follows we assume that Ω is not an empty set in order to prove our main theorem by reductio ad absurdum, unless otherwise stated, all discussion concerns the set Ω.

3. Real Hypersurfaces Satisfying R_ξφ = φR_ξ

Go to

Abstract
1. Introduction
2. Preliminaries
3. Real Hypersurfaces Satisfying R_ξφ = φR_ξ
4. Real Hypersurfaces SatisfyingR_ξφ = φR_ξand ∇_{φ∇_ξξ}R_ξ = 0
5. The Exterior Derivative of 1-form u
6. Lemmas
7. Proof of the Main Theorem
References

Let M be a real hypersurface in M_n(c), c ≠= 0. We suppose that R_ξφ = φR_ξ. Then by using (2.16) we have

(3.1)α(φAX-AφX)=g(Aξ,X)U+g(U,X)Aξ.

Then, using (3.1), it is clear that α ≠= 0 on Ω. So a function λ given by β = αλ is defined. Because of (2.9), we have

(3.2)μ2=αλ-α2.

Replacing X by U in (3.1) and taking account of (2.8), we find

(3.3)φAU=λAξ-A2ξ,

which implies

(3.4)φA2ξ=AU+λU

because U is orthogonal to Aξ. From this and (2.6) we have

(3.5)μφAW=AU+(λ-α)U,

which together with (2.7) yields

(3.6)g(AW,U)=0.

Using (2.6) and (3.3), we can write (2.15) as

(3.7)∇ξU=(3λ-2α)Aξ-3μAW-αλξ+φ∇α.

Since α ≠= 0 on Ω, (3.1) reformed as

(3.8)(φA-Aφ)X=η(X)U+u(X)ξ+τ(u(X)W+w(X)U),

where a 1-form u is defined by u(X) = g(U,X) and w by w(X) = g(W,X), where we put

(3.9)ατ=μ, λ-α=μτ.

Differentiating (3.8) covariantly and taking the inner product with any vector field Z, we find

(3.10)g(φ(∇YA)X,Z)+g(φ(∇YA)Z,X)=-η(AX)g(AY,Z)-g(AX,Y)η(AZ)+g(A2X,Y)η(Z)+η(X)g(A2Y,Z)+(η(X)+τω(X))g(∇YU,Z)+g(∇YU,X)(η(Z)+τω(Z))+u(X)g(∇Yξ,Z)+g(∇Yξ,X)u(Z)+(Yτ)(u(X)w(Z)+u(Z)w(X))+τ(u(X)g(∇YW,Z)+g(∇YW,X)u(Z))

because of (2.1) and (2.2). From this, taking the skew-symmetric part with respect to X and Y, and making use of the Codazzi equation (2.4), we find

(3.11)c(η(X)g(Y,Z)-η(Y)g(X,Z))+g((∇XA)φY,Z)-g((∇YA)φX,Z)=-η(AX)g(AY,Z)+η(AY)g(AX,Z)+η(X)g(A2Y,Z)-η(Y)g(A2X,Z)+(η(X)+τw(X))g(∇YU,Z)-(η(Y)+τw(Y))g(∇XU,Z)+(g(∇YU,X)-g(∇XU,Y))(η(Z)+τw(Z))+u(X)g(∇Yξ,Z)-u(Y)g(∇Xξ,Z)+(g(∇Yξ,X)-g(∇Xξ,Y))u(Z)+(Yτ)(u(X)w(Z)+u(Z)w(X))-(Xτ)(u(Y)w(Z)+u(Z)w(Y))+τ{u(X)g(∇YW,Z)-u(Y)g(∇XW,Z)}+τ{(g(∇YW,X)-g(∇XW,Y))u(Z)}.

Interchanging Y and Z in (3.10), we obtain

g(φ(∇ZA)X,Y)+g(φ(∇ZA)Y,X)=-η(AX)g(AY,Z)-g(AX,Z)η(AY)+g(A2X,Z)η(Y)+η(X)g(A2Y,Z)+(η(X)+τw(X))g(∇ZU,Y)+g(∇ZU,X)(η(Y)+τw(Y))+u(X)g(∇Zξ,Y)+g(∇Zξ,X)u(Y)+(Zτ)(u(X)w(Y)+u(Y)w(X))+τ(u(X)g(∇ZW,Y)+g(∇ZW,X)u(Y)),

or, using (2.4)

g(φ(∇XA)Z,Y)+g(φ(∇YA)Z,X)+c(η(X)g(Z,Y)+η(Y)g(Z,X)-2η(Z)g(X,Y))=-η(AX)g(AY,Z)-g(AX,Z)η(AY)+g(A2X,Z)η(Y)+η(X)g(A2Y,Z)+(η(X)+τw(X))g(∇ZU,Y)+g(∇ZU,X)(η(Y)+τw(Y))+u(X)g(∇Zξ,Y)+g(∇Zξ,X)u(Y)+(Zτ)(u(X)w(Y)+u(Y)w(X))+τ(u(X)g(∇ZW,Y)+g(∇ZW,X)u(Y)).

Combining this to (3.11), we have

(3.12)2g(∇YA)φX,Z)+2c(η(Z)g(X,Y)-η(X)g(Y,Z))+2η(X)g(A2Z,Y)-2η(AX)g(AZ,Y)+(g(∇ZU,X)-g(∇XU,Z))(η(Y)+τw(Y))+(g(∇YU,X)-g(∇XU,Y))(η(Z)+τw(Z))+(g(∇ZU,Y)+g(∇YU,Z))(η(X)+τw(X))+(g(∇Zξ,X)-g(∇Xξ,Z))u(Y)+(g(∇Yξ,X)-g(∇Xξ,Y))u(Z)+(g(∇Zξ,Y)+g(∇Yξ,Z))u(X)+(Yτ)(u(X)w(Z)+u(Z)w(X))+(Zτ)(u(X)w(Y)+u(Y)w(X))-(Xτ)(u(Y)w(Z)+u(Z)w(Y))+τ{u(X)(g(∇ZW,Y)+g(∇YW,Z))+u(Z)(g(∇XW,Y)-g(∇YW,X))+u(Y)(g(∇ZW,X))-g(∇XW,Z))}=0.

If we put X = ξ in (3.12), then we have

(3.13)g(∇YU,Z)+g(∇ZU,Y)+2c(η(Z)η(Y)-g(Z,Y))+2g(A2Y,Z)-2αg(AY,Z)-du(ξ,Z)(η(Y)+τw(Y))-du(ξ,Y)(η(Z)+τw(Z))-2u(Y)u(Z)-(ξτ)(u(Y)w(Z)+u(Z)w(Y))-τ{u(Z)dw(ξ,Y)+u(Y)dw(ξ,Z)}=0,

where d denotes the operator of the exterior derivative.

4. Real Hypersurfaces SatisfyingR_ξφ = φR_ξand ∇_{φ∇_ξξ}R_ξ = 0

Go to

Abstract
1. Introduction
2. Preliminaries
3. Real Hypersurfaces Satisfying R_ξφ = φR_ξ
4. Real Hypersurfaces SatisfyingR_ξφ = φR_ξand ∇_{φ∇_ξξ}R_ξ = 0
5. The Exterior Derivative of 1-form u
6. Lemmas
7. Proof of the Main Theorem
References

We will continue our discussions under the same hypothesis R_ξφ = φR_ξ as in Section 3. Furthermore, suppose that ∇_{φ∇_ξξ}R_ξ = 0 and then ∇_WR_ξ = 0 since we assume that μ ≠= 0. Replacing X by W in (2.17), we find

(4.1)(Wα)g(AY,Z)-c(η(Z)g(φAW,Y)+η(Y)g(φAW,Z))+αg((∇WA)Y,Z)-η(AZ){g((∇WA)ξ,Y)+g(AφAW,Y)}-η(AY){g((∇WA)ξ,Z)+g(AφAW,Z)}=0

by virtue of ∇_WR_ξ = 0. Putting Y = ξ in this and making use of (2.13) and (3.6), we obtain

(4.2)αAφAW+cφAW=0

because U and W are mutually orthogonal. From this and (2.16), it is seen that R_ξφAW = 0 by virtue of (3.6), and hence R_ξAW = 0 which together with (2.16) implies that

(4.3)αA2W=-cAW+cμξ+μ(α+g(AW,W))Aξ,

which tells us that

(4.4)αg(A2W,W)=(μ2-c)g(AW,W)αμ2.

Since α ≠= 0, β = αλ and (3.2), we see that

(4.5)g(A2W,W)=(λ-α-cα) g(AW,W)+μ2.

Combining (3.5) to (4.2), we get

(4.6)αA2U=-(μ2+c)AU-c(λ-α)U.

If we apply μW to (3.3) and make use of (2.6), then we find

(4.7)g(AU,U)=μ2(g(AW,W)+α-λ).

Using (4.2), we see from (4.1)

α(∇WA)X=-(Wα)AX+η(AX)(∇WA)ξ+g((∇WA)ξ,X)Aξ-cσμ(w(X)φAW+g(φAW,X)W)

for any vector field X, which together with (3.5) yields

(4.8)α(∇WA)X=-(Wα)AX+η(AX)(∇WA)ξ+g((∇WA)ξ,X)Aξ-cα{w(X)AU+u(AX)W+(λ-α)(w(X)U+u(X)W)}.

Now, if we put X = W in (2.12), and make use of (3.5) and (4.2), then we find

(4.9)(∇WA)ξ=-φ∇WU+(Wα)ξ+1μ(α+cα) {AU+(λ-α)U)}.

Also, if we take the inner product (2.12) with Aξ and take account of (2.6), (3.2) and (3.4), then we obtain

α(Xα)+μ(Xμ)=g(αξ+μW,(∇XA)ξ)-g(A2U+λAU,X),

which together with (2.4), (2.13) and (4.6) yields

(4.10)μ(∇WA)ξ=-(α+cα) AU-cα(λ+α)U+μ∇μ.

If we take the inner product (4.10) with ξ and make use of (2.13) and (3.6), then we find

(4.11)Wα=ξμ.

Using (4.10), we can write (4.8) as

(4.12)α(∇WA)X+(Wα)AX+1μη(AX) {(α+cα) AU+cα(λ+α)U-μ∇μ}+1μ {(α+cα) u(AX)+cα(λ+α)u(X)-μ(Xμ)} Aξ+cα{w(X)AU+u(AX)W+(λ-α)(w(X)U+u(X)W)}=0.

Putting X = W in this, we get

(4.13)α(∇WA)W+(Wα)AW-(Wμ)Aξ+ (α+2cα) AU+2cλαU-μ∇μ=0.

Combining (4.9) to (4.10), we obtain

μφ∇WU-μ(Wα)ξ+μ∇μ=2 (α+cα) AU+(μ2+2cαλ) U.

If we apply φ to this and make use of (2.8), (2.11) and (3.3), then we find

-μ∇WU-μ2g(AW,W)ξ+μφ∇μ=2 (α+cα) (λAξ-A2ξ)-μ (μ2+2cαλ) W,

which together with (2.6) yields

(4.14)μ∇WU=μφ∇μ+(2c-μ2)Aξ+2μ (α+cα) AW-(αμ2+2cλ+μ2g(AW,W))ξ.

Now, we can take a orthonormal frame field {e₀ = ξ, e₁ = W, e₂, . . . , e_n, e_n₊₁ = φe₁ = (1/μ)U, e_n₊₂ = φe₂, . . . , e₂_n = φe_n} of M. Differentiating (2.6) covariantly and making use of (2.1), we find

(4.15)(∇XA)ξ+AφAX=(Xα)ξ+αφAX+(Xμ)W+μ∇XW,

which implies

(4.16)μdivW=μ∑i=02ng(∇eiW,ei)=ξh-ξα-Wμ.

Taking the inner product with Y to (4.15) and taking the skew-symmetric part, we have

(4.17)-2cg(φX,Y)+2g(AφAX,Y)=(Xα)η(Y)-(Yα)η(X)+αg((φA+Aφ)X,Y)+(Xμ)w(Y)-(Yμ)w(X)+μ(g(∇XW,Y)-g(∇YW,X)).

Putting X = ξ in this and using (2.10) and (4.11), we have

(4.18)μ∇ξW=3AU-αU+∇α-(ξα)ξ-(Wα)W.

Putting X = μW in (4.15) and taking account of (4.10), we get

-(α+cα) AU-cα(λ+α)U+μ∇μ+μAφAW=μ(Wα)ξ+μ(Wμ)W+μαφAW+μ2∇WW,

or, using (3.5) and (4.2),

(4.19)μ2∇WW=-2 (α+cα) AU-(μ2+2cαλ) U+μ∇μ-μ(Wα)ξ-μ(Wμ)W.

Now, putting X = U in (4.17) and making use of (2.6) and (3.3), we have

μ(g)(∇UW,Y)-g(∇YW,U))=(2cμ-Uμ)w(Y)-(Uα)η(Y)+μ2η(AY)+2λμw(AY)-2μw(A2Y),

which together with (4.3) gives

(4.20)μdw(U,Y)=(2cμ-Uμ)w(Y)-{Uα+2c(λ-α)}η(Y)-{μ2+2(λ-α)g(AW,W)}η(AY)+2μ (λ+cα) w(AY).

Because of (2.10) and (4.18), it is verified that

(4.21)μdw(ξ,X)=2u(AX)-αu(X)-(ξα)η(X)-(Wα)w(X)+Xα.

Using (2.11) and (3.7), we obtain

(4.22)du(ξ,X)=(3λ-2α)η(AX)-2μw(AX)-αλη(X)+g(φ∇α,X).

Using above two equations, (3.13) is reduced to

(4.23)g(∇XU,Y)+g(∇YU,X))=2c(g(X,Y)-η(X)η(Y))-2g(A2X,Y)+2αg(AX,Y)+(ξτ)(u(X)w(Y)+u(Y)w(X))+1α(2u(AX)+Xα-(ξα)η(X)-(Wα)w(X))u(Y)+1α(2u(AY)+Yα-(ξα)η(Y)-(Wα)w(Y))u(X)+{(3λ-2α)η(AX)-2μw(AX)-αλη(X)+g(φ∇α,X)}(η(Y)+τw(Y))+{(3λ-2α)η(AY)-2μw(AY)-αλη(Y)+g(φ∇α,Y)}(η(X)+τw(X)),

where we have used (4.21) and (4.22). Taking the trace of this and using (4.7), we find

(4.24)divU=2c(n-1)+αh-TrA2+λ(λ-α).

Replacing X by U in (4.23) and using (4.6) and (4.7), we find

g(∇UU,Y)+g(∇YU,U)=(λ-α)(Yα)+2 (2λ-α+cα) u(AY)+{Uαα+2cλα+2(λ-α)(g(AW,W)+α-λ)} u(Y)+{μ(Wα)-(λ-α)ξα}η(Y)+μ2(ξτ)w(Y).

Since g(∇_XU,U) = μ(Xμ), it follows that

(4.25)du(U,X)=-2μ(Xμ)+(λ-α)(Xα)+2 (2λ-α+cα) u(AX)+{Uαα+2cλα+2(λ-α)(g(AW,W)+α-λ)} u(X)+{μ(Wα)-(λ-α)ξα}η(X)+μ2(ξτ)w(X),

which implies that

(4.26)du(U,W)=-2μ(Wμ)+(λ-α)Wα+μ2(ξτ).

5. The Exterior Derivative of 1-form u

Go to

Abstract
1. Introduction
2. Preliminaries
3. Real Hypersurfaces Satisfying R_ξφ = φR_ξ
4. Real Hypersurfaces SatisfyingR_ξφ = φR_ξand ∇_{φ∇_ξξ}R_ξ = 0
5. The Exterior Derivative of 1-form u
6. Lemmas
7. Proof of the Main Theorem
References

We will continue our discussions under the hypotheses as those stated in Section 4.

Putting Z = U in (3.12), we find

-2μg((∇YA)X,W)+2cη(X)u(Y)-du(U,X)(η(Y)+τw(Y))-du(U,Y)(η(X)+τw(X))-dη(U,X)u(Y)-dη(U,Y)u(X)+μ2(g(∇Xξ,Y)+g(∇Yξ,X))+μ2((Xτ)w(Y)+(Yτ)w(X))+τ{μ2(g(∇XW,Y)+g(∇YW,X))-dw(U,Y)u(X)-dw(U,X)u(Y)}-(Uτ)(u(X)w(Y)+u(Y)w(X))=0.

Because of (2.1), (2.11) and (3.3), we see

dη(U,X)=(λ-α)η(AX)-2μw(AX).

Using this and (2.4), above equation reformed as

-2μg((∇WA)Y,X)-2c(η(Y)u(X)+η(X)u(Y))-du(U,X)(η(Y)+τw(Y))-du(U,Y)(η(X)+τw(X))+μ2((Xτ)w(Y)+(Yτ)w(X))-(Uτ)(u(X)w(Y)+u(Y)w(X))-{(λ-α)η(AX)-2μwAX)}u(Y)-{(λ-α)η(AY)-2μw(AY)}u(X)+μ2(g(∇Xξ,Y)+g(∇Yξ,X))+τ{μ2(g(∇XW,Y)+g(∇YW,X))-dw(U,Y)u(X)-dw(U,X)u(Y)}=0.

Substituting (4.20) into this, we obtain

2μg((∇WA)Y,X)=-2c(η(Y)u(X)+η(X)u(Y))-du(U,X)(η(Y)+τw(Y))-du(U,Y)(η(X)+τw(X))+μ2((Xτ)w(Y)+(Yτ)w(X))-(Uτ)(u(X)w(Y)+u(Y)w(X))-{(λ-α)η(AX)-2μw(AX)}u(Y)-{(λ-α)η(AY)-2μw(AY)}u(X)+μ2(g(∇Xξ,Y)+g(∇Yξ,X))+τμ2(g(∇XW,Y)+g(∇YW,X))-1αu(X){(2cμ-Uμ)w(Y)-(Uα+2c(λ-α))η(Y)-{μ2+2(λ-α)g(AW,W)}η(AY)+2μ (λ+cα) w(AY)}-1αu(Y){(2cμ-Uμ)w(X)-{Uα+2c(λ-α)}η(X)-{μ2+2(λ-α)g(AW,W)}η(AX)+2μ (λ+cα) w(AX)}.

Combining this to (4.12), we have

(5.1)-2μ(Wα)g(AY,X)+2η(AY) {-(α+cα) u(AX)-cα(α+λ)u(X)+μXμ}+2 {-(α+cα) u(AY)-cα(α+λ)u(Y)+μ(Yμ)} η(AX)-2cμα{u(AX)w(Y)+u(AY)w(X)+(λ-α)(w(X)u(Y)+w(Y)u(X))}=-2αc(η(Y)u(X)+η(X)u(Y))-αdu(U,X)(η(Y)+τw(Y))-αdu(U,Y)(η(X)+τw(X))+αμ2((Xτ)w(Y)+(Yτ)w(X))-α(Uτ)(u(X)w(Y)+u(Y)w(X))-μ2(η(AX)u(Y)+η(AY)u(X))+2αμ(w(AY)u(X)+w(AX)u(Y))+αμ2(g(∇Xξ,Y)+g(∇Yξ,X))+μ3(g(∇XW,Y)+g(∇YW,X))-u(X){(2cμ-Uμ)w(Y)-(Uα+2c(λ-α))η(Y)-(μ2+2(λ-α)g(AW,W))η(AY)+2μ (λ+cα) w(AY)}-u(Y){(2cμ-Uμ)w(X)-(Uα+2c(λ-α))η(X)-(μ2+2(λ-α)g(AW,W))η(AX)+2μ (λ+cα) w(AX)}.

If we put Y = W in (5.1) and take account of (2.1), (3.5) and (4.19), then we find

-2μ(Wα)w(AX)+μ2(Xμ)+2μ(Wμ)η(AX)-2cμα{u(AX)+(λ-α)u(X)}=-μdu(U,X)-αdu(U,W)(η(X)+τw(X))+αμ2((Xτ)+(Wτ)w(X))-μ2{(Wα)η(X)+(Wμ)w(X)}+(Uμ-α(Uτ)-2cαμg(AW,W)) u(X),

or, using (4.25) and (4.26)

2μ(Wα)AW-2cμU+{μ(λ-α)ξα-3μ2Wα-αμ2(ξτ)}ξ-{μ2(Wμ)+τμ2(Wα)+2μ3(ξτ)}W+μ2∇μ-μ(λ-α)∇α-2μ(2λ-α)AU-μ {Uαα+2(λ-α)g(AW,W)-2(λ-α)2} U+αμ2((Wτ)W+∇τ)+{Uμ-α(Uτ)-2cμαg(AW,W)} U=0.

By the way, since ατ = μ, we find

(5.2)αμ∇τ=μ∇μ-(λ-α)∇α.

Using this, above equation is reduced to

(5.3)μ∇μ-(λ-α)∇α=(2λ-α)AU+{(λ-α+cα) g(AW,W)-(λ-α)2+c} U-(Wα)AW+{2μ(Wα)-(λ-α)ξα}ξ+(λ-α)(2Wα-τ(ξα))W.

If we take the inner product (5.3) with W, then we get

(5.4)μ(Wμ)={3(λ-α)-g(AW,W)}Wα-τ(λ-α)ξα.

Also, taking the inner product (5.3) with U and making use of (4.7), we obtain

(5.5)Uμμ-Uαα=(3λ-2α+cα) g(AW,W)+(λ-α)(2α-3λ)+c.

On the other hand, replacing Y by W in (4.23) and using (4.3), we find

g(∇XU,W)+g(∇WU,X)-μαg(φ∇α,X)-(ξτ)u(X)-{μ(3λ-2α)-2μg(AW,W)+g(φ∇α,W)}(η(X)+τw(X))+2 (λ-2α-cα) g(AW,X)-2cw(X)+μα(4α-3λ+2g(AW,W))η(AX)+μ (λ+2cα) η(X)=0,

or using (4.14),

(5.6)g(∇XU,W)+g(φ∇μ,X)-λ-αμg(φ∇α,X)-(ξτ)u(X)+2(λ-α)w(AX)+{Uαα+(λ-α)(5α-6λ+4g(AW,W))} w(X)+{Uαμ+μ(4α-5λ+3g(AW,W))} η(X)=0.

By the way, applying (5.3) by φ and making use of (2.6), (3.3) and (3.5), we have

μφ∇μ-(λ-α)φ∇α=-1μ(Wα)AU+μ(ξτ)U+μ2(2λ-α)ξ-μ(2λ-α)AW-μ {(λ-α+cα) g(AW,W)-(λ-α)(3λ-2α)+c} W.

Substituting this into (5.6), we find

(5.7)g(∇XU,W)=Wαμ2u(AX)+αw(AX)+{3(λ-α)2+cαg(AW,W)+c-Uαα-3(λ-α)g(AW,W)} w(X)+{3μ(λ-α-g(AW,W))-Uαμ} η(X).

On the other hand, (4.12) turns out, using (2.4), to be

α(∇XA)W=cαμ(η(X)U+2u(X)ξ)-(Wα)AX+1μη(AX) {μ∇μ-(α+cα) AU-cα(λ+α)U}+1μ {μ(Xμ)-(α+cα) u(AX)-cα(λ+α)u(X)} Aξ-cα {w(X)AU+u(AX)W+(λ-α)(u(X)W+w(X)U)}.

If we apply by φ to this and make use of (3.3), then we find

(5.8)-αφ(∇XA)W=(Wα)φAX+cαη(X)W-(Xμ)U+1μη(AX) {(α+cα) {(λ-α)Aξ-μAW}-cαμ(λ+α)W-μφ∇μ}+1μ{(α+cα) u(AX)+2cλαu(X)} U+cαw(X)(μ2ξ-μAW).

Now, if we put Z = W in (3.12), then we find

2g(φ(∇YA)W,X)=2{(w(A2Y)-cw(Y))η(X)-w(AY)η(AX)}+du(W,X)(η(Y)+τw(Y))+τdu(Y,X)+(Wτ)(w(Y)u(X)+w(X)u(Y))+(g(∇WU,Y)+g(∇YU,W))(η(X)+τw(X))+2μ{u(AX)+(λ-α)u(X)}u(Y)+(Yτ)u(X)-(Xτ)u(Y)+τ(u(Y)g(∇WW,X)+u(X)g(∇WW,Y)).

Using (2.1), (2.10), (3.5) and (3.8), we can write the above equation as

2αg(φ(∇YA)W,X)=μdu(Y,X)-2cη(X)w(AY)+2μ(c+α2+αg(AW,W))η(X)η(Y)+2(αμ2+μ2g(AW,W)-cα)η(X)w(Y)-2αη(AX)w(AY)+α(Wτ)(w(X)u(Y)+w(Y)u(X))+αg(∇WU,X)(η(Y)+τw(Y))+αg(∇WU,Y)(η(X)+τw(X))-g(∇XU,W)η(AY)+g(∇YU,W)η(AX)+2αμ{u(AX)+(λ-α)u(X)}u(Y)+α((Yτ)u(X)-(Xτ)u(Y))+μ(u(X)g(∇WW,Y)+u(Y)g(∇WW,X)),

or using (5.8),

μdu(X,Y)=(Wα)g((φA+Aφ)X,Y)+2cαμ(w(X)w(AY)-w(Y)w(AX))+η(AX)g(φ∇μ,Y)-η(AY)g(φ∇μ,X)+2cμα(u(X)u(AY)-u(Y)u(AX))-(Xμ)u(Y)+(Yμ)u(X)+α((Xτ)u(Y)-(Yτ)u(X))+g(∇YU,W)η(AX)-g(∇XU,W)η(AY)+{2cα-2cλ-μ2(α+g(AW,W))}(η(X)w(Y)-η(Y)w(X)),

which together with (5.2) and (5.7) yields

(5.9)μdu(X,Y)=(Wα)g((φA+Aφ)X,Y)+2cμα(w(X)w(AY)-w(Y)w(AX))+Wαμ2(η(AX)u(AY)-η(AY)u(AX))+η(AX)g(φ∇μ,Y)-η(AY)g(φ∇μ,X)+α(η(AX)w(AY)-η(AY)w(AX))+2cμα(u(X)u(AY)-u(Y)u(AX))+μα((Xα)u(Y)-(Yα)u(X))+{(μ2+c)g(AW,W)+αμ2-cα+2cλ}(η(X)w(Y)-η(Y)w(X)).

Putting X = φe_i and Y = e_i in this and summing up for i = 1, 2, · · ·, n, we obtain

μ∑i=02ndu(φei,ei)=(h-α-g(AW,W))Wα-μ(Wμ),

where we have used (2.6)–(2.8), (3.5) and (4.7). Taking the trace of (2.12), we obtain

∑i=02ng(φ∇eiU,ei)=ξα-ξh.

Thus, it follows that

(5.10)μ(ξh-ξα)=μ(Wμ)+(g(AW,W)+α-h)Wα,

which together with (4.16) gives

(5.11)μ2(divW)=(g(AW,W)+α-h)Wα,

We notice here that

Remark 5.1

If AU = σU for some function σ on Ω, then AW ∈ span{ξ,W} on Ω, where span{ξ,W} is a linear subspace spanned by ξ and W.

In fact, because of the hypothesis AU = σU, (3.5) reformed as

μφAW=(σ+λ-α)U,

which implies that AW = μξ + (σ + λ − α)W ∈ span{ξ,W}.

Now, we prepare the following lemma for later use.

Lemma 5.2

Let M be a real hypersurface of M_n(c), c ≠= 0 which satisfies R_ξφ = φR_ξand ∇_{φ∇_ξξ}R_ξ = 0. If AW ∈ span{ξ,W}, then Ω = ∅︀.

Proof

Since (3.5) and AW = μξ + g(AW,W)W, we have

(5.12)AU=(g(AW,W)+α-λ)U.

From (4.2) we also have

g(AW,W)(αAU+cU)=0.

Now, suppose that g(AW,W) ≠= 0 on Ω. Then we have αAU + cU = 0 on this subset, which together with (5.12) gives

(5.13)μ2=αg(AW,W)+c.

From this and (2.16) we have R_ξW = 0 and consequently R_ξAξ = 0 on the subset because of (2.6) and (2.16). If we take (3.1) by R_ξ and using R_ξU = 0 and R_ξAξ = 0, we obtain R_ξ(Aφ − φA) = 0, that is, R_ξ(ℒ_ξg) = 0 on the subset, where ℒ_ξ denotes the operator of the Lie derivative with respect to ξ. Owing to Theorem 5.1 of [5], it is verified that Aξ = αξ, a contradiction. Therefore we have the following

(5.14)g(AW,W)=0

on Ω. So we have

(5.15)AW=μξ.

From (5.12) and (5.14), we get

(5.16)AU=(α-λ)U.

Differentiating (5.15) covariantly, we find

(∇XA)W+A∇XW=(Xμ)ξ+μ∇Xξ.

Taking the inner product with W and making use of (2.11) and (5.16), we have

g((∇XA)W,W)=2(λ-α)u(X)

Using (2.4) it reformed as

(5.17)(∇WA)W=2(λ-α)U.

On the other hand, (4.13) is reduced, using (5.16) and (5.17), to

(5.18)(μ2+2c)U=-μ(Wα)ξ+(Wμ)Aξ+μ∇μ.

Taking the inner product with W, we have

(5.19)Wμ=0.

Hence, it follows from (5.18) that

(5.20)μ∇μ=μ(Wα)ξ+(μ2+2c)U,

which shows that for any vector fields X

μ(Xμ)=μ(Wα)η(X)+(μ2+2c)u(X).

Differentiating this covariantly and using (2.1), we have

(Yμ)(Xμ)+μ(Y(Xμ))=Y(μ(Wα))η(X)+μ(Wα)g(φAY,X)+(2μ(Wα)η(Y)+2(μ2+2c)u(Y))u(X)+(μ2+2c)g(∇YU,X)+{μ(Wα)η(∇YX)+(μ2+2c)u(∇YU)}.

Taking the skew-symmtric part of this, we find

(5.21)Y(μ(Wα))η(X)-X(μ(Wα))η(Y)+(μ(Wα))g((φA+Aφ)Y,X)+2μ(Wα)(η(Y)u(X)-η(X)u(Y))+(μ2+2c)(g(∇YU,X)-g(∇XU,Y))=0.

Replacing Y by ξ in this, and using (2.10) and (5.17), we have

X(μ(Wα))-2μ(Wα)u(X)=ξ(μ(Wα))η(X)+(μ(Wα))u(X)+(μ2+c)(g(∇ξU,X)-μ2η(X)).

Substituting this into (5.21), we obtain

μ(Wα)(u(Y)η(X)-u(X)η(Y))+(μ2+2c)(g(∇ξU,Y)η(X)-g(∇ξU,X)η(Y))+μ(Wα)g((φA+Aφ)Y,X)+(μ2+2c)(g(∇YU,X)-g(∇XU,Y))=0.

Putting Y = U in this, and using (2.8), (3.3), (4.11), (5.15) and (5.16), we obtain

(5.22)(μ2+2c)(g(∇UU,X)-μ(Xμ))+μ(μ2+2c)(Wα)η(X)+μ2(λ-α)(Wα)w(X)=0.

On the other hand, putting Y = U in (5.9) and making use of (5.14), (5.15) and (5.19), we have

g(∇UU,X)-μ(Xμ)=2(λ-α)(Wα)w(X)+Uααu(X)-(λ-α)Xα.

Combining this to (5.22), we have

(μ2+2c) {2(λ-α)(Wα)w(X)+Uααu(X)-(λ-α)Xα}+μ(μ2+2c)(Wα)η(X)-μ2(λ-α)(Wα)w(X)=0.

If we put X = W in this, then we have

(μ2+2c)(λ-α)Wα=(μ2+4c)(λ-α)Wα,

which, together with λ ≠= α, shows that

(5.23)Wα=0.

Thus, (5.20) becomes

(5.24)μ∇μ=(μ2+2c)U,

which implies

(5.25)φ∇μ=-(μ2+2c)W.

Using (5.23), we can write (5.21) as

(μ2+2c)(g(∇YU,X)-g(∇XU,Y))=0.

Now, suppose that μ² + 2c ≠= 0. Then we have g(∇_YU,X) − g(∇_XU, Y ) = 0. Using (5.14)–(5.16), (5.20), (5.23) and (5.25), we can write (5.9) as

(μ2+c)(w(X)η(Y)-w(Y)η(X))=0,

which implies μ² + c = 0. So μ is constant. Thus, (5.23) becomes μ² + 2c = 0, a contradiction. Therefore, we see that μ² + 2c = 0.

Accordingly we see that μ is constant, which together with (5.4) yields ξα = 0. Hence (5.3) is reduced to

(5.26)μ2∇α={μ2(3λ-2α)-cα}U.

Taking the inner product this to X and differentiating covariantly, we find

μ2(Y(Xα))={μ2(3Yλ-2Yα)-cα}u(X)+{μ2(3λ-2α)-cα}(g(∇YU,X)+g(U,∇YX)).

The skew-symmetric part of this is given by

3μ2((Yλ)u(X)-(Xλ)u(Y))+(2μ2+c)((Xα)u(Y)-(Yα)u(X))+{μ2(3λ-2α)-cα}(g(∇YU,X)-g(∇XU,Y))=0,

which implies that ∇λ = χU for some function χ, where we have used (4.24) and (5.26). Thus it follows that

{μ2(3λ-α)-cα}(g(∇YU,X)-g(∇XU,Y))=0.

If g(∇_YU,X) − g(∇_XU, Y ) = 0, then similarly as above we have a contradiction. Thus we have μ²(3λ −2α) −cα = 0, which together with μ²+2c = 0 gives 2λ −α = 0. i.e. 2μ² + α² = 0, a contradiction. Therefore Lemma 5.2 is proved.

Lemma 5.3

α2φ(∇λ-∇h)=-4μ(μ2+c)(AW-μξ)+αμ(h-λ)(Wα)U+fW,

for some function f on Ω.

Proof

Putting X = Y = e_i in (3.12), summing up for i = 0, 1, · · ·, 2n and using (2.1) and (2.4), we find

Tr(∇φZA)-2c(n-1)η(Z)+(TrA2)η(Z)-hη(AZ)+g(∇ξU,Z)-g(∇ZU,ξ)+τ(g(∇WU,Z)+g(∇UW,Z))+(divU)(η(Z)+τw(Z))+g((φA+Aφ)U,Z)+(Wτ)u(Z)+(Uτ)w(Z)+τ(divW)u(Z)=0,

or using (2.10), (3.3), (3.7) and (4.24)

(5.27)φ∇α-φ∇h+μα(∇WU+∇UW)-4μAW+(Wτ+τ(divW))U+(Uτ+τ(divU)+μ(4λ-3α-h))W+(λ-α)(λ+3α)ξ=0.

On the other hand, combining (4.20) to (5.8) and making use of (5.7), we find

∇UW=1μ{μφ∇μ-(λ-α)φ∇α}-(ξτ)U+2 (2λ-α+cα) AW+{(λ-α)(2α-3λ)+c-(λ+cα) g(AW,W)} W+μ {g(AW,W)+3α-5λ-2cα} ξ.

Substituting this and (4.14) into (5.27), we find

(5.28)αφ(∇α-∇h)+2μφ∇μ-(λ-α)φ∇α=4μ (α-λ-cα) AW+(μ(ξτ)-α(Wτ)-μ(divW)) U-4(λ-α)(μ2+c)ξ-(α(Uτ)+μ(divW)) W-μ{3c+(λ-α)(α-3λ)-(λ+cα) g(AW,W)+4αλ-3α2-hα}W.

From (4.11), (4.16) and (5.2) we have

αμ (μ(ξτ)-α(Wτ)-μ(divW))=2μ2(Wα)-μ(λ-2α)ξα-αμ(ξh).

By the way, using (5.4) and (5.10) we have

μ(λ-2α)ξα+αμ(ξh)=α(3λ-2α-h)Wα.

Thus, we have

(5.29)αμ (μ(ξτ)-α(Wτ)-μ(divW))=α(h-λ)Wα.

Differentiating (3.2) covariantly, we find

(5.30)2μ∇μ=(λ-2α)∇α+α∇λ.

Using this and (5.29), the equation (5.28) reformed as

α2φ(∇λ-∇h)=-4μ(μ2+c)(AW-μξ)+αμ(h-λ)(Wα)U+fW,

where we have put

f=αμ{hα+4α2-8αλ+3λ2-3c+(λ+cα) g(AW,W)-divU-αμ(Uτ)}.

This completes the proof of Lemma 5.3.

6. Lemmas

Go to

Abstract
1. Introduction
2. Preliminaries
3. Real Hypersurfaces Satisfying R_ξφ = φR_ξ
4. Real Hypersurfaces SatisfyingR_ξφ = φR_ξand ∇_{φ∇_ξξ}R_ξ = 0
5. The Exterior Derivative of 1-form u
6. Lemmas
7. Proof of the Main Theorem
References

We will continue our discussions under the same hypotheses as those in Section 4. Further we assume that TrR_ξ is constant, that is, g(Sξ, ξ) is constant. Then, from (2.5) we see that β − hα is constant, i.e.

(6.1)α(h-λ)=C,

where C is some constant. Differentiating this covariantly, we have

(6.2)(λ-h)∇α+α(∇λ-∇h)=0.

So we have αφ(∇λ − ∇h) = (h − λ)φ∇α. Thus, from Lemma 5.3 we find

α(h-λ)μφ(∇α-(Wα)W)=-4(μ2+c)(AW-μξ)+αμfW,

which tells us that

α(h-λ)μ2(Uα)=4(μ2+c)g(AW,W)-αμf.

Combining the last two equations, it follows that

(6.3)α(h-λ)μφ (∇α-(Wα)W-Uαμ2U)=-4(μ2+c)(AW-μξ-g(AW,W)W).

Applying this by φ and using (3.5), we find

(6.4)α(h-λ) (∇α-(ξα)ξ-(Wα)W-Uαμ2U)=4(μ2+c){AU+(λ-α)U-g(AW,W)U}.

Taking the inner product with AW to this, and using (4.6), (5.4) and α ≠= 0, we see

(6.5)(h-λ)(g(AW,∇α)-μ(ξα)-g(AW,W)(Wα))=0.

First of all, we prove the following:

Lemma 6.1

h − λ ≠= 0 on Ω.

Proof

If not, then we have from (6.4)

(μ2+c){AU-(g(AW,W)+α-λ)U}=0

on this subset. Because of Remark 5.1 and Lemma 5.2, it is verified that μ² +c = 0 on the set and hence μ is constant. Accordingly we see that Wα = 0 because of (4.11) and hence ξα = 0 and ξτ = 0 by virtue of (5.2) and (5.4). Thus, (5.3) reformed as

(λ-α)∇α+(2λ-α)AU+{c-(λ-α)2}U=0,

which together with μ² + c = 0 implies that

(6.6)Xα=λu(X)+ɛg(AU,X)

for any vector field X, where we have put cɛ = α² − 2c. Differentiating (6.6) covariantly with respect to a vector field Y and taking skew-symmetric part, we get

(Yλ)u(X)-(Xλ)u(Y)+λ(g(∇YU,X)-g(∇XU,Y))+(Yɛ)u(AX)-(Xɛ)u(AY)+ɛ{cμ(η(Y)w(X)-η(X)w(Y))+g(A∇YU,X)-g(A∇XU,Y)}=0.

where we have used the Codazzi equation (2.4). Since ξα = 0 and (6.1), by replacing X by ξ in this, we get

-λ(g(∇ξU,Y)+g(∇Yξ,U))+ɛ(g(∇YU,αξ+μW)-cμw(Y)-g(∇ξU,AY))=0,

where we have used (2.6), which together with (2.10) and (5.9) implies that

(6.7)ɛA∇ξU+λ∇ξU+μλAW∈span{ξ,W}.

On the other hand, we can write (3.7) as

∇ξU=-μ(ɛ+3)AW+(λ-α)(ɛ+2)Aξ,

where we have used (3.3) and (6.6), which together with (2.6), (4.3) and the fact that μ² + c = 0 yields

A∇ξU=-μ(λ-α)AW+{c-(λ-α)(ɛ+3)g(AW,W)}Aξ-c(λ-α)(ɛ+3)ξ.

Combining the last three equations, it is seen that

{(2λ-α)ɛ+2λ}AW∈span{ξ,W},

which shows that (2λ − α)ɛ + 2λ = 0 by Lemma 5.2. So we have (2λ − α)(α² − 2c) + 2cλ = 0, a contradiction because of μ² + c = 0. This completes the proof.

If we combine (6.2) to (5.30), then we have

(6.8)2μ∇μ=(h-2α)∇α+α∇h.

If we apply this by ξ, then we find

(6.9)2μ(ξμ)=(h-2α)ξα+α(ξh).

From (4.11), (5.6) and (5.12) we get (h − λ)(μ(ξα) − α(Wα)) = 0 and hence

(6.10)μ(ξα)=α(Wα)

by virtue of Lemma 6.1, which together with (6.9) yields

(6.11)μ(ξh)=(2λ-h)Wα.

From (5.2) and (6.10) we have ξτ = 0. Thus, using (6.8) and (6.10) we verify from (5.3)

(6.12)12(h∇α+α∇h)-λ∇α+(Wα)AW=(2λ-α)AU+{(λ-α+cα) g(AW,W)-(λ-α)2+c} U+(Wα){μξ+(λ-α)W}.

Because of Lemma 6.1, (6.5) implies that

(6.13)g(AW,∇α)=(α+g(AW,W))Wα,

with the aid of (6.10). Applying (5.3) by AW and making use of (4.3), (6.10), (6.13) and ξτ = 0, we find

μg(AW,∇μ)-(λ-α)(α+g(AW,W))Wα+g(A2W,W)Wα={μ2+(λ-α)g(AW,W)}Wα,

which together with (4.5) gives

(6.14)μαg(AW,∇μ)={(μ2+c)g(AW,W)+αμ2}Wα.

In the next place, we will prove that

Lemma 6.2

ξα = Wα = Wμ = ξh = ξλ = Wλ = 0 and ξ(g(AW,W)) = 0 on Ω.

Proof

Differentiating (4.4) covariantly, we get

(6.15)g(A2W,W)(Xα)+α(X(g(A2W,W)))=2μg(AW,W)(Xμ)+(μ2-c)(X(g(AW,W)))+μ2(Xα)+2μα(Xμ).

Replacing X by ξ in this, and using (4.11) and (6.10), we find

(6.16)α(ξ(g(A2W,W)))=αμ(μ2-g(A2W,W))Wα+2μ(α+g(AW,W))Wα+(μ2-c)(ξ(g(AW,W))).

By the way, using (4.10), (4.18), (6.10) and (6.13), we verify that ξ(g(AW,W)) = Wμ, which together with (5.4) and (6.10) yields

ξ(g(AW,W))=1μ{2(λ-α)-g(AW,W)}Wα.

Substituting this and (4.5) into (6.14), we find

(6.17)α2ξ(g(A2W,W))={cμg(AW,W)+μα+μα(μ2-c)} Wα.

On the other hand, we have

12(X(g(A2W,W)))=g((∇XA)W,AW)+g(A2W,∇XW),

which implies

(6.18)12α(X(g(A2W,W)))=αg((∇WA)X,AW)+2cαu(X)-cg(AW,∇XW)+cu(AX)+α(α+g(AW,W))u(AX),

where we have used (2.6), (2.11) and (4.3).

By the way, putting X = AW in (4.12) and making use of (2.6) and (6.14), we obtain

α(∇WA)AW=-(Wα)A2W+(α+g(AW,W)) {-(α+cα) AU-cα(λ+α)U+μ∇μ}+1μα{(μ2+c)g(AW,W)+αμ2} (Wα)Aξ-cαg(AW,W){AU+(λ-α)U},

which implies

αg((∇WA)AW,ξ)=1μ{αμ2+(μ2+c)g(AW,W)}

because of (2.6) and (4.11). If we replace X by ξ in (6.16) and make use of (4.11), (4.18) and (6.17), then we obtain

(μ2-c-αg(AW,W))(Wα)=0

because of λ − α ≠= 0.

Now, suppose that Wα ≠= 0 on Ω. Then since λ ≠= α, we have αg(AW,W) = μ² − c, which together with (3.2) and (4.7) gives αg(AU,U) = −cμ². From this and (4.6) we verify that α²g(A²U,U) = c²μ². Using the last two equations it is seen that ||αAU + cU||² = 0 and hence αAU + cU = 0. Thus, (3.5) is reduced to μφAW = (λ − α − c/α)U, which shows that AW = μξ + g(AW,W)W on this subset. According to Lemma 5.2, we have Ω = ∅︀, and hence Wα = 0 on Ω. Thus, it is clear that Wμ = 0, ξα = 0, ξh = 0 and ξλ = 0, where we have used (4.11), (5.6), (6.2), (6.9), (6.10) and (6.11). Since (3.2), Wα = 0 and Wμ = 0, we have Wλ = 0. Hence Lemma 6.2 is proved.

Because of Lemma 6.2, we can write (6.4) as

α(h-λ) (∇α-Uαμ2U)=4(μ2+c){AU-(λ-α-g(AW,W))U},

which tells us that

(6.19)14α(h-λ)∇α=(μ2+c)AU+θU,

where the function θ is defined by

μ2θ=α(h-λ)4(Uα)-(μ2+c)g(AU,U).

We also have from (5.4)

(6.20)μ∇μ-(λ-α)∇α=(2λ-α)AU+ρU,

where we have put

(6.21)ρ=(λ-α+cα) g(AW,W)-(λ-α)2+c.

Remark 6.3

μ² + c ≠= 0 on Ω.

If not, then we have μ²+c = 0 and hence μ is constant on this subset. So (6.19) and (6.20) are reduced respectively to

μ2∇α=(Uα)U,(λ-α)∇α+(2λ-α)AU+{c-(λ-α)2}U=0

because of Lemma 5.2. Combining these two equations, we obtain

(2λ-α)AU={(λ-α)2-c-Uαα} U.

Suppose that 2λ − α = 0 on this subset. Then, the equation μ² + c = 0 becomes α² − 2c = 0, a contradiction. Thus we have 2λ − α ≠= 0. Owing to Remark 5.1 and Lemma 5.2, above equation produces a contradiction. Hence μ² + c ≠= 0 on Ω is proved.

Lemma 6.4

(2λ − α)θ = (μ² + c)ρ on Ω.

Proof

From (6.17) and (6.18) we have

14α(h-λ)(2λ-α)∇α-(μ2+c) {12∇μ2-(λ-α)∇α}={(2λ-α)θ-(μ2+c)ρ}U.

Using the same method as that used to derive (6.7) from (6.6), we can deduce from this that

(2λ-α)(ξθ)U+{(2λ-α)θ-(μ2+c)ρ}(∇ξU+μAW)=0,

where, we have used (2.10), (6.1) and Lemma 6.2. If we take the inner product with U to this and make use of ξμ = 0, then we get (2λ − α)ξθ = 0 and hence

{(2λ-α)θ-(μ2+c)ρ}(∇ξU+μAW)=0.

If (2λ − α)θ − (μ² + c)ρ ≠= 0 on Ω, then we have

∇ξU+μAW=0.

We discuss our arguments on such a place. Using (3.7), the last equation can be written as

φ∇α=2μAW+(2α-3λ)Aξ+αλξ.

Applying this by φ and taking account of (3.5) and Lemma 6.2, we obtain

(6.22)∇α=-2AU+λU.

Combining this to (6.19), we obtain

{μ2+c+12α(h-λ)} AU={14αλ(h-λ)-θ} U.

Because of Remark 5.1 and Lemma 5.2, we conclude that μ²+c+(1/2)α(h−λ) = 0. Hence it follows from (6.1) that μ is constant. Thus, (6.20) reformed as

(λ-α)∇α=(α-2λ)AU-ρU,

which together with (6.22) implies that αAU = {λ(α − λ) − ρ}U. Therefore we verify that (2λ−α)θ−ρ(μ² +c) = 0 by virtue of Remark 5.1 and Lemma 5.2. This completes the proof.

Lemma 6.5

Let span{ξ,W} be the linear subspace spanned by ξ and W. Then there exists P ∈ span{ξ,W} such that

g(AW,∇XU)=cαw(A2X)-{μ2+(λ-α+cα) g(AW,W)} w(AX)+g(P,X).

Proof

Putting Y = AW in (5.9) and using (3.6), (4.3), (6.13) and Lemma 6.2, we find

μdu(X,AW)=2cαμ{g(A2W,W)w(X)-g(AW,W)w(AX)}+η(AX)g(φ∇μ,AW)-μ(α+g(AW,W))g(φ∇μ,X)+α{g(A2W,W)η(AX)-μ(α+g(AW,W))w(AX)}+{(μ2+c)g(AW,W)+αμ2-cα+2cλ}(g(AW,W)η(X)-μw(X)),

which enables us to obtain

g(AW,∇XU)-g(∇AWU,X)=-α (α+g(AW,W)+2cα2g(AW,W)) w(AX)-(α+g(AW,W))g(φ∇μ,X)+g(P1,X),

for some P_{1 ∈} span{ξ,W}. If we replace X by AW in (4.23) and make use of (3.5), (4.3), (6.14) and Lemma 6.2, then we get

g(∇XU,AW)+g(∇AWU,X)=2cw(AX)+2αw(A2X)-2w(A3X)+(μ+μαg(AW,W)){(3λ-2α)η(AX)-2μw(AX)-αλη(X)+g(φ∇α,X)}+{μ(3λ-2α)(α+g(AW,W))-2μg(AW,W)-αλμ-1μg(AU+(λ-α)U,∇α)}(η(X)+τw(X))-2cμη(X),

which shows that

g(∇XU,AW)+g(∇AWU,X)=-2w(A3X)+2αw(A2X)+2cw(AX)-2(λ-α)(α+g(AW,W))w(AX)+μα(α+g(AW,W))g(φ∇α,X)+g(P2,X),

for some P_{2 ∈} span{ξ,W}. Adding to the last two equations, we obtain

2g(AW,∇XU)=-2w(A3X)+2αw(A2X)+2cw(AX)-2(λ-α)(α+g(AW,W))w(AX)-α (α+g(AW,W)+2cα2g(AW,W)) w(AX)-(α+g(AW,W))(φ∇μ-μαφ∇α)+g(P3,X)

for some P_{3 ∈} span{ξ,W}.

By the way, applying (6.20) by φ, and using (2.8) and (3.4), we find

(6.23)φ∇μ-μαφ∇α=(2λ-α){-AW+μξ+(λ-α)W}-ρW.

Because of (4.3), we have

A3W=-cαA2W+(λ-α)(α+g(AW,W))AW+μ (α+cα+g(AW,W)) Aξ.

Combining the last three equations, we obtain

g(AW,∇XU)=cαw(A2X)-{μ2+(λ-α+cα) g(AW,W)}w(AX)+g(P4,X)

for some P_{4 ∈} span{ξ,W}. The completes the proof.

Remark 6.6

Wρ = 0 on Ω.

In fact, we have

W(g(AW,W))=g((∇WA)W,W)+2g(AW,∇WW),

which together with (4.13) and Lemma 6.2 yields

W(g(AW,W))=2g((AW,∇WW).

However, if we take the inner product with AW to (4.19) and make use of Lemma 6.2 and (6.14), then we obtain g(AW,∇_WW) = 0. So we have W(g(AW,W)) = 0, which connected to (6.21) and Lemma 6.2 gives Wρ = 0.

7. Proof of the Main Theorem

Go to

Abstract
1. Introduction
2. Preliminaries
3. Real Hypersurfaces Satisfying R_ξφ = φR_ξ
4. Real Hypersurfaces SatisfyingR_ξφ = φR_ξand ∇_{φ∇_ξξ}R_ξ = 0
5. The Exterior Derivative of 1-form u
6. Lemmas
7. Proof of the Main Theorem
References

We will continue our discussions under the same assumptions as those in Section 6. Taking the inner product X to (6.20) and differentiating covariantly, we have

(Yμ)(Xμ)+μ(Y(Xμ))-(Yλ-Yα)(Xα)-(λ-α)(Y(Xα))=(2(Yλ)-Yα)u(AX)+(2λ-α)(g((∇YA)U,X)+g(A∇YU,X))+(Yρ)u(X)+ρg(∇YU,X)+g((2λ-α)AU+ρU,∇YX).

Taking the skew-symmetric part of this and using (2.4), we find

(7.1)(Xλ)(Yα)-(Yλ)(Xα)+(2(Xλ)-Xα)u(AY)-(2(Yλ)-Yα)u(AX)=cμ(2λ-α)(η(Y)w(X)-η(X)w(Y))+(2λ-α)(g(A∇YU,X)-g(A∇XU,Y))+(Yρ)u(X)-(Xρ)u(Y)+ρ(g(∇YU,X)-g(∇XU,Y)),

where we have used (2.4) and (2.8). Differentiating (6.21) covariantly and taking the inner product ξ to this, it follows from Lemma 6.2 that ξρ = 0. Putting Y = ξ in (7.1) and using (2.6) and ξρ = 0, we find

cμ(2λ-α)w(X)-(2λ-α){g(αξ+μW,∇XU)+g(∇ξU,AX)}-ρ(g(∇XU,ξ)-g(∇ξU,X))=0,

or using (2.10), (5.7) and Lemma 6.2,

(7.2)(2λ-α)A∇ξU+ρ∇ξU+μρAW∈span{ξ,W}.

If we put Y = W in (7.1) and take account of Lemma 6.2 and Remark 6.6, then we have

(7.3)(2λ-α){g(∇XU,AW)-g(A∇WU,X)+cμη(X)}+ρ(g(∇XU,W)-g(∇WU,X))=0.

By the way, putting Y = W in (5.9), we have

g(∇XU,W)-g(∇WU,X)=-(α+2cα) w(AX)-g(φ∇μ,X)+g(P5,X)

for some P_{5 ∈} span{ξ,W}, which together with Lemma 6.5 and (7.3) implies that

(2λ-α){cαA2W-(μ2+(λ-α+cα) g(AW,W)) AW-A∇WU}-ρ{(α+2cα) AW+φ∇μ}∈span{ξ,W}.

It follows from this and (4.14) that

(2λ-α)Aφ∇μ+ρφ∇μ+(2λ-α){cαA2W+(λ-α+cα) g(AW,W)AW}+ρ (α+2cα) AW∈span{ξ,W}.

If we take account of (4.3), (6.21) and (6.23), then the last equation can be written as

(7.4)μα(2λ-α)Aφ∇μ+ρφ∇μ+(2λ-α)2(λ-α+cα) AW+(2λ-α){cαA2W+((λ-α)2-c)AW}+ρ (α+2cα) AW∈span{ξ,W}.

On the other hand, from (3.7) we have

A∇ξU=μ(3λ-2α)AW-3μA2W+2μ2Aξ+Aφ∇α,

where we have used (2.6). Substituting this into (7.2), we find

(2λ-α)Aφ∇α+ρφ∇α-2μρAW-(2λ-α)μ{3A2W+(2α-3λ)AW}∈span{ξ,W}.

Combining this to (7.4), we obtain

(λ-α){-ρμφ∇α+2ρAW+(2λ-α)(3A2W+(2α-3λ)AW)}+ρφ∇μ+(2λ-α)2(λ-α+cα) AW+cα(2λ-α)A2W+(2λ-α){(λ-α)2-c}AW+ρ (α+2cα) AW∈span{ξ,W},

which together with (4.3) and (6.23) implies that

{2ρα-(2λ-α)(μ2+c)}AW∈span{ξ,W},

that is,

{2ρα-(2λ-α)(μ2+c)}(AW-μξ-g(AW,W))=0.

According to Lemma 5.2, we see that

(7.5)2ρα=(2λ-α)(μ2+c).

From this fact and Lemma 6.4, we see that 2αθ = (μ²+c)² by virtue of 2λ−α ≠= 0. Thus, (6.19) is reduced to

(7.6)κ∇α=2αAU+(μ2+c)U

with the aid of Remark 6.3, where we have put

κ=α2(h-λ)2(μ2+c).

Differentiating this covariantly and taking the inner product with ξ, it follows from (6.1) and Lemma 6.2 that ξκ = 0.

As in the same method as that used from (6.6) to drive (6.7), we can deduce from (7.6) that

2αg(A∇ξU,X)+(μ2+c)g(∇ξU,X)=μ{-2cαw(X)-2α2w(AX)-(μ2+c)w(AX)+2αg(∇XU,W)},

which together with (5.7) implies that

(7.7)2αA∇ξU+(μ2+c)∇ξU+μ(μ2+c)AW∈span{ξ,W}.

On the other hand, applying (7.6) by φ and using (2.6) and (3.3), we find

κμφ∇α=-2αAW+(μ2+c)W+2αμξ,

which together with (4.3) yields

κμAφ∇α=(μ2+c)AW-2μg(AW,W)Aξ-2cμξ.

From Lemma 6.1 we have κ ≠= 0 and hence combining the last two equations, it is verified that

(7.8)2αAφ∇α+(μ2+c)φ∇α∈span{ξ,W}.

By the way, applying (3.7) by A and using (4.3), we find

2αA∇ξU+(μ2+c)∇ξU-2αμ (3λ-2α+3cα) AW+3μ(μ2+c)AW-2αAφ∇α-(μ2+c)φ∇α∈span{ξ,W},

which together with (7.7) and (7.8) gives

(2μ2+α2+2c)(AW-μξ-g(AW,W))=0.

Owing to Lemma 5.2, we see that 2μ² + α² + 2c = 0, which implies that 2μ∇μ + α∇α = 0. Hence (6.20) reformed as

(7.9)∇α+2AU+μ2+cαU=0

by virtue of 2λ − α ≠= 0 on Ω, where we have used (7.5). Combining this to (6.19), we have

{μ2+c+12α(h-λ)}AU=14{4θ+(h-λ)(μ2+c)}U.

According to Remark 5.1, it follows that μ²+c+(1/2)α(h−λ) = 0, which together with (6.1) gives μ is constant and hence α is constant. Thus (7.9) becomes AU = −{(μ² + c)/(2α)}U, a contradiction by virtue of Remark 5.1.

Therefore we verify that Ω = ∅︀, that is, Aξ = αξ on M. Thus, from (2.18) we see that R_ξS = SR_ξ. Hence from Theorem 1.2 ([9]) M is homogeneous real hypersurfaces of Type A.

Let M be of Type A. Then M always satisfies ∇_{φ∇_ξξ}R_ξ = 0. Since TrA is constant and (2.16), it is easy to see that φR_ξ = R_ξφ and TrR_ξ is constant.

Consequently we conclude that

Theorem 7.1

Let M be a real hypersurface of a complex space form M_n(c), c ≠= 0, n ≥ 3 which satisfies ∇_{φ∇_ξξ}R_ξ = 0 and TrR_ξis constant. Then M holds φR_ξ = R_ξφ if and only if Aξ = 0 or M is locally congruent to one of following:

(I) In cases that M_n(c) = P_nℂ with η(Aξ) ≠= 0,
- (A₁) a geodesic hypersphere of radius r, where 0 < r < π/2 and r ≠= π/4;
- (A₂) a tube of radius r over a totally geodesic P_kℂ for some k ∈ {1, . . . , n–2}, where 0 < r < π/2 and r ≠= π/4.
(II) In cases M_n(c) = H_nℂ,
- (A₀) a horosphere;
- (A₁) a geodesic hypersphere or a tube over a complex hyperbolic hyperplane H_n_–1ℂ;
- (A₂) a tube over a totally geodesic H_kℂ for some k ∈ {1, . . . , n – 2}.

References

Go to

Abstract
1. Introduction
2. Preliminaries
3. Real Hypersurfaces Satisfying R_ξφ = φR_ξ
4. Real Hypersurfaces SatisfyingR_ξφ = φR_ξand ∇_{φ∇_ξξ}R_ξ = 0
5. The Exterior Derivative of 1-form u
6. Lemmas
7. Proof of the Main Theorem
References

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Cho, JT, and Ki, U-H (1998). Real hypersurfaces in complex projective spaces in terms of Jacobi operators. Acta Math Hungar. 80, 155-167.
Cho, JT, and Ki, U-H (2008). Real hypersurfaces in complex space form with Reeb ow symmetric Jacobi operator. Canadian Math Bull. 51, 359-371.
Ki, U-H, Kim, I-B, and Lim, DH (2010). Characterizations of real hypersurfaces of type A in a complex space form. Bull Korean Math Soc. 47, 1-15.
Ki, U-H, and Kurihara, H (2009). Real hypersurfaces and ξ-parallel structure Jacobi operators in complex space forms. J Korean Academy Sciences, Sciences Series. 48, 53-78.
Ki, U-H, Kurihara, H, Nagai, S, and Takagi, R (2009). Characterizations of real hypersurfaces of type A in a complex space form in terms of the structure Jacobi operator. Toyama Math J. 32, 5-23.
Ki, U-H, Kurihara, H, and Takagi, R (2009). Jacobi operators along the structure ow on real hypersurfaces in a nonat complex space form. Tsukuba J Math. 33, 39-56.
Ki, U-H, Nagai, S, and Takagi, R (2010). The structure vector field and structure Jacobi operator of real hypersurfaces in nonat complex space forms. Geom Dedicata. 149, 161-176.
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Kimura, M (1986). Real hypersurfaces and complex submanifolds in complex projective space. Trans Amer Math Soc. 296, 137-149.
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Pérez, JD, Santos, FG, and Suh, YJ (2007). Real hypersurfaces in nonat complex space forms with commuting structure Jacobi operator. Houston J Math. 33, 1005-1009.
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Takagi, R (1975). Real hypersurfaces in a complex projective space with constant principal curvatures I, II. J Math Soc. 15, Array-Array.

Article

Jacobi Operators with Respect to the Reeb Vector Fields on Real Hypersurfaces in a Nonat Complex Space Form

Abstract

1. Introduction

Theorem 1.1. (Ki, Kim and Lim [5])

Theorem 1.2. (Ki, Nagai and Takagi [9])

2. Preliminaries

3. Real Hypersurfaces Satisfying Rξφ = φRξ

4. Real Hypersurfaces SatisfyingRξφ = φRξand ∇φ∇ξξRξ = 0

5. The Exterior Derivative of 1-form u

Remark 5.1

Lemma 5.2

Lemma 5.3

6. Lemmas

Lemma 6.1

Lemma 6.2

Remark 6.3

Lemma 6.4

Lemma 6.5

Remark 6.6

7. Proof of the Main Theorem

Theorem 7.1

References

3. Real Hypersurfaces Satisfying R_ξφ = φR_ξ

4. Real Hypersurfaces SatisfyingR_ξφ = φR_ξand ∇_{φ∇_ξξ}R_ξ = 0