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Kyungpook Mathematical Journal 2021; 61(1): 181-190

Published online March 31, 2021 https://doi.org/10.5666/KMJ.2021.61.1.181

Copyright © Kyungpook Mathematical Journal.

Certain Characterization of Real Hypersurfaces of type A in a Nonflat Complex Space Form

U-Hang Ki

The National Academy of Siences, Seoul 06579, Korea
e-mail : uhangki2005@naver.com

Received: April 8, 2020; Revised: July 3, 2020; Accepted: August 18, 2020

Let M be a real hypersurface with almost contact metric structure (ϕ,ξ,η,g) in a nonflat complex space form Mn(c). We denote S and Rξ by the Ricci tensor of M and by the structure Jacobi operator with respect to the vector field ξ respectively. In this paper, we prove that M is a Hopf hypersurface of type A in Mn(c) if it satisfies Rξϕ=ϕRξ and at the same time Rξ(SϕϕS)=0.

Keywords: real hypersurface, structure Jacobi operator, Ricci tensor, Hopf hypersurface.

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 Pn, or a complex hyperbolic space Hn 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 Pn, 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 Hn, the classfication of homogenous real hypersurfaces in Hn 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 Pn or Hn (0kn1). These and a horosphere in Hn 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 Pn, n2. If it satisfies

g((AϕϕA)X,Y)=0

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 Pn1, where 0<r<π/2,

  • (A2) a totally geodesic Pk (1kn2), where 0<r<π/2.

Theorem MR. Let M be a real hypersurface of Hn, n2. If it satisfies (1.1), then M is locally congruent to one of the following hypersurface :

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

  • (A1) a geodesic hypersphere, or a tube over a hyperplane Hn1,

  • (A2) a tube over a totally geodesic Hk (1kn2).

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 Rξ 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 c0, 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), c0. 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) (c0,n2) if M satisfies ϕRξ=Rξϕ and at the same time Rξ(SϕϕS)=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 Mn (c), c ≠ 0 with almost complex structure J, and N be a unit normal vector field on M. The Riemannian connection ˜ in Mn (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 tensor of M induced from that of Mn (c), and A denotes the shape operator of M in the 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 Kaehler condition ˜J=0, and taking account of above equations, we see that

Xξ=ϕAX,
(Xϕ)Y=η(Y)AXg(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 the Gauss and Codazzi are respectively given by

R(X,Y)Z=c{g(Y,Z)Xg(X,Z)Y+g(ϕY,Z)ϕXg(ϕX,Z)ϕY      2g(ϕX,Y)ϕZ}+g(AY,Z)AXg(AX,Z)AY,
(XA)Y(YA)X=c{η(X)ϕYη(Y)ϕX2g(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), 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

SX=c{(2n+1)X3η(X)ξ}+hAXA2X

for any vector field X on M, which implies

Sξ=2c(n1)ξ+hAξA2ξ.

Now, we put

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

ϕU=Aξ+αξ,

which shows that g(U,U)=βα2. Thus it is seen that

μ2=βα2.

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

μg(XW,ξ)=g(AU,X),
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

(XA)ξ=ϕXU+g(AU+α,X)ξAϕAX+αϕAX,

which together with (2.4) implies that

(ξA)ξ=2AU+α.

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

ϕ(XA)ξ=XU+μg(AW,X)ξϕAϕAXαAX+αg(Aξ,X)ξ,

which connected to (2.1) and (2.13) gives

ξU=3ϕAU+αAξβξ+ϕα.

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

Rξ(X)=R(X,ξ)ξ=c{Xη(X)ξ}+αAXη(AX)Aξ

for any vector field X on M, which implies that

Rξξ=0,
RξU=cU+αAU,RξAU=cAU+αA2U.

From (2.5) we obtain

SU=c(2n+1)U+hAUA2U,
SAξ=c{(2n+1)Aξ3αξ}+hA2ξA3ξ.

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

μSW=hA2ξA3ξα(hAξA2ξ)+c(2n+1)(Aξαξ).

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

α(ϕAXAϕX)=g(Aξ,X)U+g(U,X)Aξ.

We set Ω={pM: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

ϕAU=λAξA2ξ.

Applying by ϕ, we have

ϕA2ξ=AU+λU,

which together with (2.7) yields

μϕAW=AU+(λα)U.

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

g(AW,U)=0.

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

then we find

αϕA2Uα(λA2ξA3ξ)=g(AU,U)Aξ,

which enables us to obtain

g(AU,U)=γαλ2.

Further, we assume in the sequel that

Rξ(SϕϕS)X=0

for any vector field X on M.

Applying this by ξ, we have RξϕSξ=0, which together with (2.6) gives Rξϕ(hAξA2ξ)=0. Thus, it follows that

RξAU=(hλ)RξU

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

hAUA2U=(λ+cα)AUcα(hλ)U

since α0 on Ω. Applying this by ϕ and using (2.8) and (3.2), we find

αϕA2U={α(hλ)c}(λAξA2ξ)c(hλ)(Aξαξ).

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

αA3ξ=(αhc)A2ξ+(γαhλ+ch)Aξ+cα(λh)ξ.

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

α(hA2ξA3ξ)=cA2ξ+(αλhγch)Aξ+cα(hλ)ξ.

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

α(hA3ξA4ξ)={λαhγc2α}A2ξ      +c{γαλh+chα+α(hλ)}Aξ+c2(λh)ξ.

From (2.20) and (3.12) we get

αSAξ=cA2ξ+{c(2n+1)αγ+αλhch}Aξ+cα(hλ3α)ξ.

Combining (2.19) to (3.10), we find

SU=(λ+cα)AU+{c(2n+1)+cα(λh)}U.

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

μSW=(α+cα)A2ξ+{c(2n+1)+h(λα)+1α(γ+ch)}Aξ        +{c(hλ)+c(2n+1)α}ξ,

which connected to (3.6) implies that

μϕSW=(α+cα)AU+{αλ+cλα+c(2n+1)+h(λα)1α(γ+ch)}U.

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

Rξ{(α+cα)AU+(α+cα+h(λα)1α(γ+ch))U}=Rξ{(λ+cα)AUcα(hλ)U}.

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

(γαλ2)RξU=0,

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 α0 on Ω because of (3.1), we can write this as AU=cαU. Thus (3.2) turns out to be

A2ξ=ρAξ+cξ

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

In fact, from (2.16) we have

g(RξY,AX)g(RξX,AY)=g(A2ξ,Y)g(Aξ,X)g(A2ξ,X)g(Aξ,Y)          +c{g(Aξ,Y)η(X)g(Aξ,X)η(Y)},

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

γ=αλ2.

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

A3ξ=(hcα)A2ξ+(λ2hλ+cα)Aξ+c(λh)ξ.

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

hA3ξA4ξ={λhλ2(cα)2}A2ξ    +{cα(λ2λh+cαh)+c(hλ)}Aξ+c2α(λh)ξ.

Because of (2.5) and (3.2) we have

(SϕϕS)AU=λSAξSA2ξϕASU,

which connected to (3.8) gives

Rξ(λSAξSA2ξ)=RξϕSAU.

On the other hand, we have from (2.5)

λSAξSA2ξ=λ{(2n+1)cAξ3cαξ+hA2ξA3ξ}      (2n+1)cA2ξ+3cαλξ(hA3ξA4ξ),

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

SϕAU={cαλ(2n+1)cλh+λ2+( c α 2)}A2ξ    +3cαλξc2α(λh)ξ+{(2n+1)cλ    +λ(λhλ2cαh)cα(λ2λh+cαh)c(hλ)}Aξ.

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

ϕSAU=(λ+cα)(λA2ξA3ξ)+{(2n+1)c+cα(λh)(λAξA2ξ),

or, using (3.20)

ϕSAU={λ(λ+cα)λcα+cα(hλ)(2n+1)c}A2ξ+{(2n+1)cλ+cαλ(λh)(λ+cα)(λ2hλ+chα)}Aξ.

Combining (3.23) to this, it follows that

SϕAUϕSAU={cαλ(hλ)c(hλ)}Aξ.

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

In fact, if not, then we have RξAξ=0, which together with (2.16) implies that αA2ξ=(βc)Aξcαξ on this open subset. Applying by ϕ and using (3.2), we find α(AU+λU)=(βc)U on the set. If we apply this by U and taking account of the fact g(AU,U)=0, then we have αλ=βc, a contradiction. Thus, hλ=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

AU=0

on Ω and thus (3.2) becomes

A2ξ=λAξ.

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

and taking account of (2.1), we find

g((XA)Aξ,Y)+g(A(XA)ξ,Y)+g(A2ϕAX,Y)=(Xλ)g(Aξ,Y)+λg((XA)ξ,Y)+λg(AϕAX,Y),

which together with (2.13) and (3.24) yields

2g((XA)ξ,Aξ)=λ(Xα)+α(Xλ),

or, using (2.4),

(ξA)Aξ=12βcU.

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

12β=Aα+λα+(ξλ)Aξ+cU,

which together with (3.24) implies that

12Uβ=λ(Uα)+cμ2.

Thus, it follows that

α(Uλ)λ(Uα)=2cμ2

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

12(Aβλβ)+(α2+μ2)λ=g(Aξ,λ)Aξ+c(3α2λ)U.

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

λ(α(Uλ)λ(Uα))=c(3αλ)μ2,

which together with (3.28) gives c(λα)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(AϕϕA)=0. Thus our Theorem follows from Theorem O and Theorem MR. This completes the proof.

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