An n-dimensional complex space form Mn(c) is a Kaehlerian manifold of constant holomorphic sectional curvature 4c. As is well known, complete and simply connected complex space forms are isometric to a complex projective space Pnℂ, or a complex hyperbolic space Hnℂ according as c>0 or c<0.

In this paper we consider a real hypersurface M in a complex space form Mn(c) for c≠0. Such an M has an almost contact metric structure (ϕ,ξ,η,g) induced from the Kaehlerian metric and complex sturcture J of Mn(c). The structure vector ξ is said to be principal if Aξ=αξ, where A is the shape operator in the direction of the unit normal N and α=η(Aξ). In this case, it is known that α is locally constant [9] and that M is called a Hopf hypersurface [12].

In the study of real hypersurfaces in Pnℂ, Takagi [14, 15] classified all homogeneous Hopf hypersurfaces, and Cecil-Ryan [2] and Kimura [10] showed that they can be regarded as the tubes of constant radius over Kaehlian submanifolds. Such tubes can be divided into six types: A₁, A₂, B, C, D and E.

In the case of real hypersurfaces in Hnℂ, the classfication of homogenous real hypersurfaces in Hnℂ is obtained by Berndt [1]. He showed that they are realized as the tubes of constant radius over certain submanifolds. Such tubes are said to be real hypersurfaces of type A₀, A₁, A₂ or B. Among the several types of real hypersurfaces appearing in Takagi's list or Berndt's list, several are tubes over totally geodesic Pnℂ or Hnℂ (0≤k≤n−1). These and a horosphere in Hnℂ are together said to be of type A. By a theorem due to Okumura[13] and to Montiel and Romero[11]we have

Theorem O. Let M be a real hypersurface of Pnℂ, n≥2. If it satisfies

for any vector fields X and Y, then M is locally congruent to a tube of radius r over one of the following Kaehlerian submanifolds :

• (A₁) a hyperplane Pn−1ℂ, where 0<r<π/2,
  

• (A₂) a totally geodesic Pkℂ (1≤k≤n−2), where 0<r<π/2.
  

Theorem MR. Let M be a real hypersurface of Hnℂ, n≥2. If it satisfies (1.1), then M is locally congruent to one of the following hypersurface :

• (A₀) a horosphere in Hnℂ, i.e., a Montiel tube,
  

• (A₁) a geodesic hypersphere, or a tube over a hyperplane Hn−1ℂ,
  

• (A₂) a tube over a totally geodesic Hkℂ (1≤k≤n−2).
  

Characterization problems for a real hypersurface of type A in a complex space form were studied by many authors (cf. [3, 4, 5], etc.).

We denote by S and R_ξ be the Ricci tensor and the structure Jacobi operator with respect to the vector field ξ of M respectively. To investigate of real hypersurfaces with respect to the structure Jacobi operator it is a very important problem to study real hypersurfaces satisfying Rξϕ=ϕRξ in Mn(c). Real hypersurfaces in a complex space form Mn(c) for c≠0, which satisfies both Rξϕ=ϕRξ and RξS=SRξ, have been studied in [6, 7, 8].

Under the condition RξA=ARξ we know that the following theorem ([3,4]):

Theorem CK.([4]) Let M be a real hypersurface of Mn(c), c≠0. If M satisfies Rξϕ=ϕRξ and at the same time satisfies RξA=ARξ, then M is a Hopf hypersurface. Further, M is of type A or a Hopf hypersurface with g(Aξ,ξ)=0.

The purpose of this paper is, using the hypothesis concerned with the Ricci tensor S and the structure Jacobi operator Rξ, to establish the following theorem another characterizing homogenous real hypersurfaces of type A and some special classes of Hopf hypersurfaces:

Theorem 1.1. Let M be a real hypersurface in a nonflat complex space form Mn(c) (c≠0,n≥2) if M satisfies ϕRξ=Rξϕ and at the same time Rξ(Sϕ−ϕS)=0, then M is a Hopf hypersurface. Further, M is locally congruent to one of homogenous real hypersurfaces of type A or a Hopf hypersurface with g(Aξ,ξ)=0.}

All manifolds in the present paper are assume to be connected and of class C∞ and the real hypersurfaces supposed to be orientable.

Let M be a real hyperusurface 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 :

where g denotes the Riemannian metric tensor of M induced from that of M_n (c), and A denotes the shape operator of M in the direction N.

For any vector field X tangent to M, we put

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

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

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 the Gauss and Codazzi are respectively given by

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), γ=η(A3ξ) 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

for any vector field X on M, which implies

Now, we put

where W is a unit vector field orthogonal to ξ. In the sequel, we put U=∇ξξ, then by (2.1) we see that U=μϕW and hence U is orthogonal to W. So we have g(U,U)=μ2. Using (2.7), it is clear that

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

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

because W is orthogonal to ξ.

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

which together with (2.4) implies that

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

which connected to (2.1) and (2.13) gives

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

for any vector field X on M, which implies that

From (2.5) we obtain

Because of (2.5) and (2.7), we also have

Let M be a real hypersurface in complex space form M_n (c), c ≠ 0 satisfying Rξϕ=ϕRξ, which means that the eigenspace of R_ξ is invariant by the structure operator ϕ. Then by (2.16) we have

We set Ω={p∈M:μ(p)≠0}, and suppose that Ω is nonvoid, that is, ξ is not principal curvature vector on M. In the sequel, we discuss our arguments on the open subset Ω of M unless otherwise stated. Then, it is, using (3.1), clear that α≠0 on Ω. So a function λ given by β=αλ is defined. Thus, replacing X by U in (3.1) and using (2.8), we find

Applying by ϕ, we have

which together with (2.7) yields

Since W is orthogonal to U, we see from the last equation

If we replace X by AU in (3.1) and take account of (3.2),

then we find

which enables us to obtain

Further, we assume in the sequel that

for any vector field X on M.

Applying this by ξ, we have RξϕSξ=0, which together with (2.6) gives Rξϕ(hAξ−A2ξ)=0. Thus, it follows that

by virtue of (3.6). Because of (2.18) we can write (3.9) as

since α≠0 on Ω. Applying this by ϕ and using (2.8) and (3.2), we find

If we combine this to (3.6), then we get

where we have used (3.7), which tells us that

Tranforming this to A and make use of (3.11), we have

From (2.20) and (3.12) we get

Combining (2.19) to (3.10), we find

Now, we see from (2.21) and (3.12) that

which connected to (3.6) implies that

In the next step, if we apply by μW to (3.8), then we find Rξ(μϕSW−SU)=0, which together with (3.15) and (3.16) gives

If we use (3.8) to this, then we obtain (hλ−γα)RξU=λ(h−λ)RξU. Hence we have

which together with (3.7) yields g(AU,U)RξU=0.

Now, suppose that g(AU,U)≠0 on Ω. Then we have RξU=0 on this open subset. We discuss our arguments on such a place. So we have αAU+cU=0. Since α≠0 on Ω because of (3.1), we can write this as AU=−cαU. Thus (3.2) turns out to be

because of (2.8), where we have put ρ=λ−cα. Therefore we verify that RξA=ARξ.

In fact, from (2.16) we have

which together with (3.18) gives the requied relationship. According to Theorem CK, we conclude that μ=0, a contradiction. Therefore g(AU,U)=0 is proved. Hence, from (3.7) we have

We are now going to prove that AU=0 on Ω. If we use (3.19), then (3.11) can be written as

Using (3.19), we also have from (3.13)

Because of (2.5) and (3.2) we have

which connected to (3.8) gives

On the other hand, we have from (2.5)

which together with (3.2), (3.12), (3.19) and (3.21) yields

We also, using (3.2) and (3.6) with g(AU,U)=0, verify that

or, using (3.20)

Combining (3.23) to this, it follows that

Using (3.23) and this, (3.22) reformed as (h−λ)(λ−α)RξAξ=0 and hence (h−λ)RξAξ=0 by virtue of (2.9) and (2.17). Accordingly we obtain h−λ=0 on Ω.

In fact, if not, then we have RξAξ=0, which together with (2.16) implies that αA2ξ=(β−c)Aξ−cαξ on this open subset. Applying by ϕ and using (3.2), we find α(AU+λU)=(β−c)U on the set. If we apply this by U and taking account of the fact g(AU,U)=0, then we have αλ=β−c, a contradiction. Thus, h−λ=0 on Ω is proved.

Therefore (3.8) tells us that RξAU=0, that is, αA2U+cAU=0, which shows g(A2U,U)=0 because of g(AU,U)=0. Consequently we have

on Ω and thus (3.2) becomes

We are now going to prove Ω=∅. Differentiating (3.25) covariantly along Ω

and taking account of (2.1), we find

which together with (2.13) and (3.24) yields

or, using (2.4),

Putting X=ξ in (3.27) and taking account of (2.13) and (3.24) and the last relationship, we obtain

which together with (3.24) implies that

Thus, it follows that

by virtue of β=αλ.

On the other hand, if we put X=Aξ in (3.26) and make use of (2.4), (2.7), (2.13) and (3.24), then we get

Taking the inner product with U to this and using (3.27), we find

which together with (3.28) gives c(λ−α)μ2=0, a contradiction. Thus, Ω=∅ is proved.

Proof of Theorem 1.1. Since we know that Ω=∅, M is a Hopf hypersurface. Hence (3.1) turns out to be α(Aϕ−ϕA)=0. Thus our Theorem follows from Theorem O and Theorem MR. This completes the proof.