# Real Hypersurfaces of Nonflat Complex Projective Planes Whose Jacobi Structure Operator Satisfies a Generalized Commutative Condition

Authors: Theocharis Theofanidis
Journal: International Journal of Mathematics and Mathematical Sciences
 (2016)

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## Abstract

Real hypersurfaces satisfying the conditionϕl=lϕ(l=R(·,ξ)ξ) have been studied by many authors under at least one more condition, since the class of these hypersurfaces is quite tough to be classified. The aim of the present paper is the classification of real hypersurfaces in complex projective plane CP2 satisfying a generalization of ϕl=lϕ under an additional restriction on a specific function.


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## Body

## 1. Introduction

Ann-dimensional Kaehlerian manifold of constant holomorphic sectional curvature c is called complex space form, which is denoted by Mn(c). A complete and simply connected complex space form is a projective space CPn if c>0, a hyperbolic space CHn if c<0, or a Euclidean space Cn if c=0. The induced almost contact metric structure of a real hypersurface M of Mn(c) will be denoted by (ϕ,ξ,η,g).Real hypersurfaces inCPn which are homogeneous were classified by Takagi [1]. The same author classified real hypersurfaces in CPn, with constant principal curvatures in [2], but only when the number g of distinct principal curvatures satisfies g=3. Kimura showed in [3] that if a Hopf real hypersurface M in CPn has constant principal curvatures, then the number of distinct principal curvatures of M is 2, 3, or 5. Berndt gave the equivalent result for Hopf hypersurfaces in CHn [4], where he divided real hypersurfaces into four model spaces, named A0, A1, A2, and B. Real hypersurfaces of types A1 and A2 in CPn and of types A0, A1, and A2 in CHn are said to be hypersurfaces oftype A for simplicity. Another class of real hypersurfaces that appears quite often is theHopf hypersurfaces where the structure vector field is a principal vector field. For more details and examples on real hypersurfaces of type A and Hopf, we refer to [5].A Jacobi field along geodesics of a given Riemannian manifold (M,g) plays an important role in the study of differential geometry. It satisfies a well-known differential equation which inspires Jacobi operators. For any vector field X, the Jacobi operator is defined by RX:RX(Y)=R(Y,X)X, where R denotes the curvature tensor and Y is a vector field on M. RX is a self-adjoint endomorphism in the tangent space of M and is related to the Jacobi differential equation, which is given by ∇γ´(∇γ´Y)+R(Y,γ´)γ´=0 along a geodesic γ on M, where γ´ denotes the velocity vector along γ on M.In a real hypersurfaceM of a complex space form Mn(c), c≠0, the Jacobi operator on M with respect to the structure vector field ξ is called the structure Jacobi operator and is denoted by Rξ(X)=R(X,ξ)ξ=lX.Real hypersurfaces have been studied from many points of view. Certain authors have studied real hypersurfaces under conditions which include the operatorl [6–9]. Other authors have studied real hypersurfaces under the condition ϕl=lϕ, equipped with one or two additional conditions [10–15], proving that these hypersurfaces are Hopf and classifying them as type A.In the present paper we classify real hypersurfaces of complex projective planes, satisfying(1)ϕl-lϕX=ψXlX,restricted in the subspace D=ker⁡η of TpM for every point p∈M, where ker⁡(η) consists of all vector fields orthogonal to the Reeb flow vector field ξ and the form ψ is assumed to be nonlinear with respect to scalar product. If ψ is linear, then by replacing X with 2X, we obtain (ϕl-lϕ)X=2ψ(X)lX which implies ψ(X)lX=0. So (1) takes the simpler form ϕl=lϕ.Since this class is rather difficult to classify, a second condition is imposed. However it is not a condition acting in vector fields, but only in the functionα=g(Aξ,ξ):∇ξξ·α=0, where A is the shape operator. Geometrically speaking, we demand the function α to be constant in the direction of the integral curves of ξ. Namely, we prove the following.Main Theorem.A real hypersurfaceMof a complex projective planeCP2, satisfying(ϕl-lϕ)X=ψ(X)lX,∀X∈D(ψis nonlinear), and∇ξξ·α=0, is Hopf. Furthermore, ifα=g(Aξ,ξ)≠0thenMis locally congruent to a model space of typeAandψ(X)lX=0.For the case ofCPn in order to determine real hypersurface of type A, the technical assumption η(Aξ)≠0 is needed. Actually, there is a nonhomogeneous tube with Aξ=0 (of radius π/4) over a certain Kaehler submanifold in CPn, when its focal map has constant rank on M [16].


## 2. Preliminaries

LetMn be a Kaehlerian manifold of real dimension 2n, equipped with an almost complex structure J and a Hermitian metric tensor G. Then for any vector fields X and Y on Mn(c), the following relations hold:(2)J2X=-X,GJX,JY=GX,Y,∇~J=0,where ∇~ denotes the Riemannian connection of G of Mn.Now, letM2n-1 be a real (2n-1)-dimensional hypersurface of Mn(c), and denote by N a unit normal vector field on a neighborhood of a point in M2n-1 (from now on we will writeM instead of M2n-1). For any vector field X tangent to M we have JX=ϕX+η(X)N, where ϕX is the tangent component of JX, η(X)N is the normal component, and (3)ξ=-JN,ηX=gX,ξ,g=GM.By properties of the almost complex structureJ and the definitions of η and g, the following relations hold [17]:(4)ϕ2=-I+η⊗ξ,η∘ϕ=0,ϕξ=0,ηξ=1,(5)gϕX,ϕY=gX,Y-ηXηY,gX,ϕY=-gϕX,Y.The above relations define analmost contact metric structure on M which is denoted by (ϕ,ξ,g,η). By virtue of this structure, we can define a local orthonormal basis {e1,e2,…,en-1,ϕe1,ϕe2,…,ϕen-1,ξ}, called ϕ-basis. Furthermore, let A be the shape operator in the direction of N, and denote by ∇ the Riemannian connection of g on M. Then A is symmetric and the following equations are satisfied:(6)∇Xξ=ϕAX,∇XϕY=ηYAX-gAX,Yξ.As the ambient spaceMn(c) is of constant holomorphic sectional curvature c, the equations of Gauss and Codazzi are, respectively, given by(7)RX,YZ=c4gY,ZX-gX,ZY+gϕY,ZϕX-gϕX,ZϕY-2gϕX,YϕZ+gAY,ZAX-gAX,ZAY,(8)∇XAY-∇YAX=c4ηXϕY-ηYϕX-2gϕX,Yξ.The tangent spaceTpM, for every point p∈M, is decomposed as follows: (9)TpM=ker⁡η⊥⊕ker⁡η,where ker⁡(η)⊥=span⁡{ξ} and ker⁡(η) is defined as follows:(10)ker⁡η=X∈TpM:ηX=0.Based on the above decomposition, by virtue of (6), we decompose the vector field Aξ in the following way:(11)Aξ=αξ+βU,where β=|ϕ∇ξξ| and U=-1/βϕ∇ξξ∈ker⁡(η), provided that β≠0.As stated before, if the vector fieldξ is a principal vector field, the real hypersurface is called Hopf hypersurface. In this case the vector field Aξ is expressed as Aξ=αξ, α=g(Aξ,ξ).Finally, differentiation of a functionf along a vector field X will be denoted by (Xf). All manifolds of this paper are assumed to be connected and of class C∞.


## 3. Auxiliary Relations

In the study of real hypersurfaces of a complex space formMn(c), c≠0, it is a crucial condition that the structure vector field ξ is principal. The purpose of this paragraph is to establish relations that will help us prove this condition.LetN={p∈M:β≠0 in a neighborhood of p}. If we had at least one point of N, where α=0, then from (7) we would obtain lU=(c/4-β2)U and lϕU=c/4ϕU. Combining the last two equations with (1) we would take β=0 which is a contradiction. Therefore α≠0 in N.Lemma 1.
Let M be a real hypersurface of a complex projective planeCP2, satisfying (1). Then the following relations hold in N:(12)AU=γα-c4α+β2αU+βξ,AϕU=γα-c4αϕU,(13)∇ξξ=βϕU,∇Uξ=γα-c4α+β2αϕU,∇ϕUξ=c4α-γαU,(14)∇ξU=κ1ϕU,∇UU=κ2ϕU,∇ϕUU=κ3ϕU+γα-c4αξ,(15)∇ξϕU=-κ1U-βξ,∇UϕU=-κ2U-γα-c4α+β2αξ,∇ϕUϕU=-κ3U,where κ1, κ2, and κ3 are smooth functions in N.Proof.
From (7) we get (16)lX=c4X-ηXξ+αAX-gAX,ξAξwhich for X=U and X=ϕU yields(17)ilU=c4U+αAU-βAξ,iilϕU=c4ϕU+αAϕU.The scalar products of (17)(i) with U and ϕU yield, respectively,(18)gAU,U=γα-c4α+β2α,(19)gAU,ϕU=gAϕU,U=δα,where γ=g(lU,U) and δ=g(lU,ϕU). From (18), (19), and g(AU,ξ)=g(Aξ,U)=β we obtain AU=(γ/α+β2/α-c/4α)U+βξ. From (19), g(AϕU,ξ)=g(Aξ,ϕU)=0, and ϵ=g(lϕU,ϕU) we obtain AϕU=(ϵ/α-c/4α)ϕU+δ/αU. In order to prove (12) we need to show that γ=ϵ and δ=0. Combining the analysis of AU and AϕU with (11) and (17) we obtain lU=γU+δϕU and lϕU=δU+ϵϕU. The last two equations and ϕlU-lϕU=ψ(U)lU which holds due to (1) yield(20)iγ-ϵ=ψUδ,ii-2δ=ψUγ.Moreover, the decompositions of lU and lϕU combined with ϕlϕU+lU=ψ(ϕU)lϕU which holds due to (1) and (4) yield(21)iγ-ϵ=ψϕUδ,ii2δ=ψϕUϵ.
Let us assume thatδ≠0 in a neighborhood of a point in N. Then (20)(i) and (21)(i) give ψ(U)=ψ(ϕU). Apparently ψ(U)γ≠0; otherwise (20)(ii) would yield δ=0. As a result, (20) and (21) lead, respectively, to -γ(γ-ϵ)=2δ2 and ϵ(γ-ϵ)=2δ2. The last two relations are added and result in (γ-ϵ)2=-4δ2 which is a contradiction. This means that δ=0 holds, and (20) and (21) imply γ=ϵ.
Equation (13) is obtained from (12) and relation (6) for X=ξ, X=U, and X=ϕU. Next we recall the rule(22)XgY,Z=g∇XY,Z+gY,∇XZ.
By virtue of (22) for X=Z=ξ, Y=U and for X=ξ, Y=Z=U, it is shown, respectively, that ∇ξU⊥ξ and ∇ξU⊥U, which means ∇ξU=κ1ϕU. In a similar way, (22) for X=Y=Z=U and X=Z=U, Y=ξ yields, respectively, ∇UU⊥U and ∇UU⊥ξ. So ∇UU=κ2ϕU holds. Finally, (22) for X=ϕU, Y=Z=U and X=ϕU, Y=U, Z=ξ (with the aid of (13)) yields, respectively, ∇ϕUU⊥U and g(∇ϕUU,ξ)=γ/α-c/4α. Therefore we have ∇ϕUU=κ3ϕU+(γ/α-c/4α)ξ and (14) has been proved. In order to prove (15) we use the second of (6) with the combinations, (i) X=ξ, Y=U, (ii) X=Y=U, and (iii) X=ϕU, Y=U, and make use of (11), (12), and (14).Lemma 2.
Let M be a real hypersurface of a complex projective planeCP2, satisfying (1). Then in N one has (ϕU(γ/α-c/4α))=3β/α[(γ/α-c/4α)2-c/4].Proof.
PuttingX=U, Y=ξ in (8), we obtain (∇UA)ξ-(∇ξA)U=-c/4ϕU. Combining the last equation with (11) and Lemma 1, it follows that(23)Uα-ξβξ+Uβ-ξγα-c4α+β2αU+γ-c4+κ2β-γα-c4αγα-c4α+β2α-κ1β2αϕU=-c4ϕU.The last equation because of the linear independency of U, ϕU, and ξ yields(24)Uα=ξβ,(25)Uβ=ξγα-c4α+β2α,(26)γ+κ2β-γα-c4αγα-c4α+β2α-κ1β2α=0.In the same way, putting X=ϕU, Y=ξ in (8), we obtain (∇ϕUA)ξ-(∇ξA)ϕU=c/4U. Combining the last equation with (11) and Lemma 1, we have(27)ϕUβ+γα-c4αγα-c4α+β2α-κ1β2α-β2-γ=0,(28)κ3β=ξγα-c4α,(29)ϕUα+3βγα-c4α-κ1β-αβ=0.Similarly, putting X=U, Y=ϕU in (8), we get (∇UA)ϕU-(∇ϕUA)U=-c/2ξ, which, by use of (11) and Lemma 1, implies that(30)-κ2β2α-3βγα-c4α-β3α+ϕUγα-c4α+β2α=0,(31)Uγα-c4α=κ3β2α.We expand (30) and then replace the terms κ2, (ϕUβ), and (ϕUα) from (26), (27), and (29), respectively. The final equation is(32)ϕUγα-c4α=3βαγα-c4α2-c4.Lemma 3.
LetM be a real hypersurface of a complex projective plane CP2, satisfying (1). Then, κ3=0 holds in N.Proof.
Because of (14), (15), (28), (31), and Lemma 2, the well-known relation [U,ϕU]=∇UϕU-∇ϕUU takes the form (33)U,ϕUγα-c4α=-κ2κ3β2α-κ3βγα-c4α+β2α-3βκ3αγα-c4α2-c4-κ3βγα-c4α.On the other hand (27), (29), (31), and Lemma 2 yield(34)U,ϕUγα-c4α=UϕUγα-c4α-ϕUUγα-c4α=3Uβαγα-c4α2-c4-3βUαα2γα-c4α2-c4+6κ3β3α2γα-c4α-β2αϕUκ3+2κ3βαγα-c4αγα-c4α+β2α-2κ3βγα-κ1κ3β3α2-κ3β3α-3κ3β3γα3+3κ3cβ34α3.The last equations using (24), (25), and (28) yield(35)3αγα-c4α2-c4ξβ-3βα2γα-c4α2-c4ξα-βϕUκ3=2c-βκ2+β2ακ1-8γα-c4α2-5β2αγα-c4ακ3.
Following a similar way, we calculate[ξ,ϕU](γ/α-c/4α)=(∇ξϕU-∇ϕUξ)(γ/α-c/4α) and then [ξ,ϕU](γ/α-c/4α)=ξ(ϕU(γ/α-c/4α))-ϕU(ξ(γ/α-c/4α)). By equalizing the results we obtain(36)3αγα-c4α2-c4ξβ-3βα2γα-c4α2-c4ξα-βϕUκ3=γ-γα-c4α2-6β2αγα-c4ακ3.
Comparing (35) with (36) and by making use of (26) we obtain(37)κ3γα-c4α2-c4=0.
Due to (32), we have γ≠c/4. So if we had κ3≠0 in neighborhood of point in N then (37) would imply (γ/α-c/4α)2=c/4. By differentiating the last relation with respect to ξ, in combination with γ≠c/4 and (28), we would obtain κ3=0 which is a contradiction.
Thereforeκ3 holds in N.


## 4. Proof of Main Theorem

We first prove the following proposition.Proposition 4.
LetM be a real hypersurface of a complex projective plane CP2(c≠0), satisfying (1) and (∇ξξα)=0. Then M is Hopf.Proof.
We keep working inN. By virtue of Lemma 3 and (11), (28), and (31), we obtain [Aξ,ξ](γ/α-c/4α)=Aξ(ξ(γ/α-c/4α))-ξ(Aξ(γ/α-c/4α))=0. However, from Lemmas 1, 2, and 3 and (11) we calculate [Aξ,ξ](γ/α-c/4α)=3β/α(∇Aξξ-∇ξAξ)(γ/α-c/4α)=(γ/α-c/4α+β2/α-κ1)((γ/α-c/4α)2-c/4). The two expressions of [Aξ,ξ](γ/α-c/4α) yield(38)γα-c4α+β2α-κ1γα-c4α2-c4=0.
Let us assume that there exists a point inN, such that (γ/α-c/4α)2-c/4≠0 in a neighborhood of this point. Then (38) implies(39)κ1=γα-c4α+β2α.Based on the definition of the vector field U in Section 2, condition (∇ξξα)=0 is equivalent to ϕUα=0. The last relation, (39), and (29) yield(40)γ-c4=α2+β22.By virtue of (39) and (40) we simplify (27) and obtain(41)ϕUβ=3β44α2+β2+α24+c4.We modify (32) by replacing the term γ/α-c/4α from (40). Then we make calculations with the aid of (41), (ϕUα)=0, to obtain β2+α2=2c. The last equation is differentiated with respect to ϕU, giving (ϕUβ)=0. As a result, (41) yields c<0 which is a contradiction and (γ/α-c/4α)2=c/4 holds in N.
Combining(ϕUα)=0 and (29) we acquire(42)κ1=3γα-c4α-α.The combination of (26) and (42) leads to(43)κ2β=4β2αγα-c4α-γ-c4-β2.In addition, (27), (γ/α-c/4α)2=c/4, and (42) yield(44)ϕUβ=γα-c4α2β2+α2α.
We estimate the vector fieldR(ϕU,ξ)U from (7) and Lemma 1 as R(ϕU,ξ)U=β(γ/α-c/4α)ϕU. The same vector field is calculated from R(ϕU,ξ)U=∇ϕU∇ξU-∇ξ∇ϕUU-∇[ϕU,ξ]U, (42), (γ/α-c/4α)2=c/4, and (ϕUα)=0:R(ϕU,ξ)U=-4β(γ/α-c/4α)ϕU-2(γ/α-c/4α)κ2ϕU+ακ2ϕU+αβ. The two expressions of R(ϕU,ξ)U are equalized and the outcome is modified as follows: first we multiply with β, then we replace the term κ2β from (43), and finally we divide with γ-c/4. After these steps, we are led to (45)β2α-8β2α2γα-c4α+2γα-c4α-α=0.We differentiate the above relation and utilize (ϕUβ)≠0 (otherwise (44) yields c<0 which is a contradiction) in order to get γ/α-c/4α=α/8. The last two equations give α=0 which is a contradiction. Therefore we have a contradiction in N; hence N=∅ and M is Hopf.From Proposition4 we have on M(46)Aξ=αξ,α=gAξ,ξand α is a constant [5]. We consider ϕ-basis{e,ϕe,ξ} which satisfies(47)Ae=λ1e,Aϕe=λ2ϕe,Aξ=αξ.From (7) and (47) we obtain(48)le=c4e+αλ1e,lϕe=c4ϕe+αλ2ϕe.By making use of (1) with X=e, in combination with (48), we obtain(49)αλ1-λ2=0.If α≠0 then λ1=λ2=λ and λ is the root of the quadratic t2-αt-c/4 [5] and consequently a constant. The classification follows from [3]. Since M is of type A then (due to [5]) M satisfies ϕA=Aϕ. The last condition and (1) and (15) lead to ψ(X)lX=0, ∀X∈D, which is combined with lξ=0, resulting in ψ(X)lX=0, ∀X∈TpM=0.



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