Real hypersurfaces equipped with pseudo-parallel structure Jacobi operator in CP^2 and CH^2

0.09 MB·English

by  K. Panagiotidou

REAL HYPERSURFACESEQUIPPEDWITHPSEUDO-PARALLEL
STRUCTURE JACOBIOPERATOR INCP2 AND CH2
KonstantinaPanagiotidouandPhilipposJ.Xenos
MathematicsDivision-SchoolofTechnology,AristotleUniversityofThessaloniki,Greece
E-mail:[email protected],[email protected]
2
1
0
2 ABSTRACT. Motivatedbytheworkdonein[4],[5],[12]and[15],weclassifyrealhypersurfacesin
n CP2andCH2equippedwithpseudo-parallelstructureJacobioperator.
a
J
Keywords:Realhypersurface,Pseudo-parallelstructureJacobioperator,Complexprojectivespace,
1
1 Complexhyperbolicspace.

]
G MathematicsSubjectClassification(2000):Primary53B25;Secondary53C15,53D15.
D
h. 1 Introduction
t
a
m A complex n-dimensional Kaehler manifold of constant holomorphic sectional curvature
[ c is called a complex space form, which is denoted by M (c). A complete and simply
n

1 connected complex space form is complex analytically isometric to a complex projective
v
0 space CPn, a complex Euclidean space Cn or a complex hyperbolic space CHn if c >
7
0,c = 0orc< 0respectively.
2
2 Let M be a real hypersurface in a complex space form M (c), c 6= 0. Then an almost
n
.
1 contact metric structure (ϕ,ξ,η,g) can be defined on M induced from the Kaehler metric
0
2 andcomplex structure J on M (c). Thestructure vector fieldξ iscalled principal ifAξ =
n
1
αξ, where A is the shape operator of M and α = η(Aξ) is a smooth function. A real
:
v
hypersurface issaidtobeaHopfhypersurface ifξ isprincipal.
i
X
The classification problem of real hypersurfaces in complex space forms is of great
r
a importance in Differential Geometry. The study of this was initiated by Takagi [18], [17],
who classified all homogenous real hypersurfaces in CPn into six types, which are said
to be of type A , A , B, C, D and E. In [3] Hopf hypersurfaces were considered as
1 2
tubes over certain submanifolds in CPn. In [9] the local classification theorem for Hopf
hypersurfaces withconstantprincipalcurvatures inCPn wasgiven. Inthecaseofcomplex
hyperbolic space CHn, the classification theorem for Hopf hypersurfaces with constant
principal curvatures wasgivenbyBerndt[1].
Okumura[13],inCPn,and MontielandRomero[10],inCHn,gavetheclassification
ofrealhypersurfaces satisfying relationAϕ = ϕA.
Theorem1.1 Let M be a real hypersurface of M (c) , n ≥ 2 (c 6= 0). If it satisfies
n
Aϕ−ϕA= 0,thenMislocally congruent tooneofthefollowinghypersurfaces:
1
• IncaseCPn
(A )ageodesic hypersphere ofradiusr,where0 < r < π,
1 2
(A )atubeofradiusroveratotallygeodesicCPk,(1 ≤ k ≤ n−2),where0< r <
2
π.
2
• IncaseCHn
(A )ahorosphere inCHn,i.eaMontieltube,
0
(A )ageodesic hypersphere oratubeoverahyperplane CHn−1,
1
(A )atubeoveratotallygeodesic CHk (1 ≤ k ≤ n−2).
2
The Jacobi operator with respect to X on M is defined by R(·,X)X, where R is the
RiemmaniancurvatureofM.ForX = ξ theJacobioperatoriscalledstructureJacobioper-
atorand isdenoted byl = R(·,ξ)ξ. Ithasafundamental role inalmost contact manifolds.
ManydifferentialgeometershavestudiedrealhypersurfacesintermsofthestructureJacobi
operator.
The study of real hypersurfaces whose structure Jacobi operator satisfies conditions
concernedtotheparallelnessofitisaproblemofgreatimportance. In[14]thenonexistence
ofrealhypersurfaces innonflat complexspace form withparallel structure Jacobi operator
(∇l = 0) was proved. In [16] a weaker condition (D-parallelness), that is ∇ l = 0 for
X
anyvectorfieldX orthogonal toξ,wasstudied anditwasprovedthenonexistence ofsuch
realhypersurfaces incaseofCPn (n ≥ 3). Theξ-parallelness ofstructure Jacobi operator
incombination with other conditions wasanother problem that wasstudied by manyother
authorssuchasKi,Perez,Santos,Suh([8]).
Atensor fieldP oftype(1, s)issaidtobesemi-parallel ifR·P = 0,whereRactson
P asaderivation.
Moregenerally, itissaidtobepseudo-parallel ifthereexistsafunctionLsuchthat
R·P = L{(X ∧Y)·P},
where (X ∧Y)Z = g(Y,Z)X −g(Z,X)Y. If L 6= 0, then the pseudo-parallel tensor is
calledproper.
A Riemannian manifold M is said to be semi-symmetric if R·R = 0, where the Rie-
mannian curvature tensor R acts on R asa derivation. Deszcz in [6] introduced the notion
of pseudo-symmetry. A Riemannian manifold is said to be pseudo-symmetric if there ex-
ists a function L such that R(X,Y) · R = L{(X ∧ Y) · R}. If L is a constant then
the pseudo-symmetric space is called a pseudo-symmetric space of constant type. Both of
these notions werestudied inthecase ofreal hypersurfaces incomplexspace forms. More
precisely, in [12] Niebergall and Ryan proved the non-existence of semi-symmetric Hopf
realhypersurfaces andrecently in[5]Cho,HamadaandInoguchi gavetheclassification of
pseudo-symmetric Hopfrealhypersurfaces inCP2 andCH2.
Recently, in [15] Perez and Santos proved that there exist no real hypersurfaces in
complex projective space CPn, n ≥ 3, with semi-parallel structure Jacobi operator, (i.e.
2
R · l = 0). Cho and Kimura in [4] generalized the previous work and proved the non-
existence ofrealhypersurfaces incomplex space forms, whose structure Jacobi operator is
semi-parallel.
Fromtheaboveraisesnaturally thequestion:
”Dothereexistrealhypersurfaces withpseudo-parallel structure Jacobioperator?”
In this paper, we study real hypersurfaces in CP2 and CH2 equipped with pseudo-
parallel structure Jacobi operator, i.e. the structure Jacobi operator satisfies the following
condition:
R(X,Y)·l = L{(X ∧Y)·l},
moreprecisely:
R(X,Y)lZ −l(R(X,Y)Z) = L{(X ∧Y)lZ −l((X ∧Y)Z)}, (1.1)
withL 6= 0.
EventhoughChoandKuriharaprovedin[4]thenon-existence ofrealhypersurfaces in
complex space form, whose structure Jacobi operator is semi-parallel, in the present paper
we prove the existence of real hypersurfaces, whose structure Jacobi operator is pseudo-
parallelandweclassifythem. Moreprecisely:
Main Theorem: Every real hypersurface M in CP2 or CH2, equipped with pseudo-
parallelstructure JacobioperatorisaHopfhypersurface.
IncaseofCP2,Mislocallycongruent to:
• ageodesic hypersphere ofradiusr,where0 < r < π,
2
• or toanon-homogeneous real hypersurface, whichis considered as atube of radius
π overaholomorphic curveinCP2.
4
IncaseofCH2,Mislocallycongruent to:
• ahorosphere,
• ortoageodesic hypersphere,
• ortoatubeoverCH1,
• ortoaHopfhypersurface withη(Aξ) = 0inCH2.
2 Preliminaries
∞
Throughout this paper all manifolds, vector fields e.t.c. are assumed to be of class C
and all manifolds are assumed to be connected. Furthermore, the real hypersurfaces are
supposed to be oriented and without boundary. Let M be a real hypersurface immersed
in a nonflat complex space form (M (c),G) with almost complex structure J of constant
n
3
holomorphicsectionalcurvaturec. LetN beaunitnormalvectorfieldonM andξ = −JN.
For a vector field X tangent to M we can write JX = ϕ(X) +η(X)N, where ϕX and
η(X)N arethetangential andthenormalcomponent ofJX respectively. TheRiemannian
connection ∇inM (c)and∇inM arerelatedforanyvectorfieldsX,Y onM:
n
∇ X = ∇ X +g(AY,X)N,
Y Y
∇ N = −AX,
X
where g is the Riemannian metric on M induced from G of M (c) and A is the shape
n
operator of M in M (c). M has an almost contact metric structure (ϕ,ξ,η) induced from
n
J onM (c)whereϕisa(1,1)tensorfieldandη a1-formonM suchthat([2])
n
g(ϕX,Y)= G(JX,Y), η(X) = g(X,ξ) = G(JX,N).
Thenwehave
ϕ2X = −X +η(X)ξ, η◦ϕ= 0, ϕξ = 0, η(ξ) = 1, (2.1)
g(ϕX,ϕY)= g(X,Y)−η(X)η(Y), g(X,ϕY)= −g(ϕX,Y), (2.2)
∇ ξ = ϕAX, (∇ ϕ)Y = η(Y)AX −g(AX,Y)ξ. (2.3)
X X
Since the ambient space is of constant holomorphic sectional curvature c, the equations of
GaussandCodazziforanyvectorfieldsX,Y ,Z onM arerespectively givenby
c
R(X,Y)Z = [g(Y,Z)X −g(X,Z)Y +g(ϕY,Z)ϕX (2.4)
4
−g(ϕX,Z)ϕY −2g(ϕX,Y)ϕZ]+g(AY,Z)AX −g(AX,Z)AY,
c
(∇ A)Y −(∇ A)X = [η(X)ϕY −η(Y)ϕX −2g(ϕX,Y)ξ], (2.5)
X Y
4
whereRdenotestheRiemanniancurvature tensoronM.
Relation(2.4)impliesthatthestructure Jacobioperatorlisgivenby:
c
lX = [X −η(X)ξ]+αAX −η(AX)Aξ. (2.6)
4
ForeverypointP ǫ M,thetangent spaceT M canbedecomposed asfollowing:
P
T M = span{ξ}⊕D
P
where D = {X ǫ T M : η(X) = 0}. Due to the above decomposition,the vector field
P
4
Aξ canbewritten:
Aξ = αξ+βU, (2.7)
whereβ = |ϕ∇ ξ|andU = −1ϕ∇ ξ ǫ ker(η),provided thatβ 6= 0.
ξ β ξ
3 Some previousresults
Intherestofthispaper, weusethenotionM (c),c6= 0,todenote CP2 orCH2.
2
Let M be a non-Hopf hypersurface in M (c). Then the following relations holds on
2
everythree-dimensional realhypersurface inM (c).
2
Lemma3.1 Let M be a real hypersurface in M (c). Then the following relations hold on
2
M:
AU = γU +δϕU +βξ, AϕU = δU +µϕU, (3.1)
∇ ξ = −δU +γϕU, ∇ ξ = −µU +δϕU, ∇ ξ = βϕU, (3.2)
U ϕU ξ
∇ U = κ ϕU +δξ, ∇ U = κ ϕU +µξ, ∇ U = κ ϕU, (3.3)
U 1 ϕU 2 ξ 3
∇ ϕU = −κ U −γξ, ∇ ϕU = −κ U −δξ, ∇ ϕU = −κ U −βξ, (3.4)
U 1 ϕU 2 ξ 3
whereγ,δ,µ,κ ,κ ,κ aresmoothfunctions onM.
1 2 3
Proof: Let{U,ϕU,ξ}beanorthonormal basisofM. Thenwehave:
AU = γU +δϕU +βξ AϕU = δU +µϕU,
where γ,δ,µ are smooth functions, since g(AU,ξ) = g(U,Aξ) = β and g(AϕU,ξ) =
g(ϕU,Aξ) = 0.
Thefirstrelation of(2.3), because of(2.6)and(3.1), forX = U,X = ϕU andX = ξ
implies(3.2).
From the well known relation: Xg(Y,Z) = g(∇ Y,Z)+g(Y,∇ Z) for X,Y,Z ǫ
X X
{ξ,U,ϕU}weobtain(3.3)and(3.4),whereκ ,κ andκ aresmoothfunctions. (cid:3)
1 2 3
In[7], T.A.IveyandP.J.Ryanproved thenon-existence ofrealhypersurfaces inM (c),
2
whose structure Jacobi operator vanishes. Inourcontext, wegiveadifferent proof oftheir
Proposition 8(non-Hopfcase)andLemma9.
Proposition3.2 Theredoesnotexist realnon-flat hypersurface inM (c), whosestructure
2
Jacobioperatorvanishes.
Proof: Let M be a non-Hopf real hypersurface in M (c), so the vector field Aξ can be
2
writtenAξ = αξ +βU (i.e. αβ 6= 0).
5
Let {U,ϕU,ξ} denote an orthonormal basis of M. Since the structure Jacobi operator
of M vanishes, from relation (2.6) for X = U and X = ϕU, we obtain: AU = (β2 −
α
c )U +βξ andAϕU = − c ϕU. Conversely, ifwehavearealhypersurface, whoseshape
4α 4α
operatorsatisfiesthelastrelationsthenl = 0. Relations(3.2),(3.3)and(3.4)becauseofthe
latterbecomerespectively:
β2 c c
∇ ξ = ( − )ϕU, ∇ ξ = U, ∇ ξ = βϕU, (3.5)
U ϕU ξ
α 4α 4α
c
∇ U = κ ϕU, ∇ U = κ ϕU − ξ, ∇ U = κ ϕU, (3.6)
U 1 ϕU 2 ξ 3
4α
β2 c
∇ ϕU = −κ U −( − )ξ, ∇ ϕU = −κ U, ∇ ϕU = −κ U −βξ, (3.7)
U 1 ϕU 2 ξ 3
α 4α
whereκ ,κ ,κ aresmoothfunctions onM.
1 2 3
On M the Codazzi equation for X, Y ǫ {U,ϕU ξ}, because of (3.5), (3.6) and (3.7)
yields:
4β2
Uβ = βκ ( +1), (3.8)
2
c
β2κ c β2 c
3
= βκ + ( − ), (3.9)
1
α 4α α 4α
4αβ2κ
2
Uα = ξβ = , (3.10)
c
4α2βκ
2
ξα = , (3.11)
c
3c
(ϕU)α = β(α+κ + ), (3.12)
3
4α
c β2 c
(ϕU)β = β2+βκ + ( − ), (3.13)
1
2α α 4α
β2 c β2 βκ 3c
1
(ϕU)( − ) = β( + − ). (3.14)
α 4α α α 4α
The Riemannian curvature on M satisfies (2.4) and on the other hand is given by the
relation R(X,Y)Z = ∇ ∇ Z − ∇ ∇ Z − ∇ Z. The combination of these two
X Y Y X [X,Y]
relations implies:
β2 c
Uκ −ξκ = κ ( − −κ ), (3.15)
3 1 2 3
α 4α
c c
(ϕU)κ −ξκ = κ (κ + )+β(κ − ). (3.16)
3 2 1 3 3
4α 2α
Relation(3.14),becauseof(3.9),(3.12)and(3.13),yields:
κ = −4α, (3.17)
3
6
andsorelation (3.9)becomes:
c c β2
βκ = ( − )−4β2. (3.18)
1
4α 4α α
Differentiating the relations (3.17) and (3.18) with respect to U and ξ respectively and
substituting in(3.15)anddueto(3.10),(3.11)and(3.17)weobtain:
κ (c−2β2−4α2) = 0. (3.19)
2
Owingto(3.19), weconsider M the open subset of points P ǫ M, whereκ 6= 0ina
1 2
neighborhood ofeveryP. Dueto(3.19)weobtain: 2β2 +4α2 = conM . Differentiation
1
ofthelastrelationalongξ andtakingintoaccount(3.10),(3.11)and2β2+4α2 = cyields:
c = 0,whichisacontradiction. Therefore, M isempty. Thus,κ = 0onM andrelations
1 2
(3.8),(3.10)and(3.11)become:
Uα = Uβ = ξα = ξβ = 0.
Usingtheaboverelationsweobtain:
[U,ξ]α = Uξα−ξUα= 0,
1
[U,ξ]α = (∇ ξ−∇ U)α = (4β2 +16α2 −c)(ϕU)α.
U ξ
4α
Combiningthelasttworelationswehave:
(4β2 +16α2 −c)(ϕU)α = 0. (3.20)
Let M be the set of points P ǫ M, for which there exists a neighborhood of every P
2
such that (ϕU)α 6= 0. So in M from (3.20) we have: 16α2 +4β2 = c. Differentiating
2
thelastrelationwithrespecttoϕU andtakingintoaccount(3.12),(3.13),(3.17),(3.18)and
16α2 +4β2 = c,weobtain: 4α2 +β2 = 0, whichisimpossible. SoM isempty. Hence,
2
on M we have (ϕU)α = 0. Then, relations (3.12), (3.17) and (3.18) imply: c = 4α2 and
βκ = α2 − 5β2. On the other hand from relation (3.16), because of (3.17) we obtain:
1
κ = −2β. Substitution ofκ inβκ = α2 −5β2 yields: 3β2 = α2. Takingthecovariant
1 1 1
derivative along ϕU of 3β2 = α2, because of (3.13), we conclude: β = 0, which is a
contradiction.
Suppose that Aξ = βξ (i.e. α = 0 and β 6= 0). Since the structure Jacobi operator of
M vanishes, fromrelation (2.6)forX = ϕU,weobtain: c = 0,whichisimpossilbe.
Hence, there do not exist non-Hopf hypersurfaces withl = 0. Usingthis and the Hopf
case([7]),wecompletetheproofofthepresent Proposition. (cid:3)
7
4 AuxiliaryRelations
IfM isarealhypersurface inM (c),weconsidertheopensubsetNofM suchthat:
2
N = {P ǫ M : β 6= 0, inneighborhood ofP}.
Furthermore, weconsider V,ΩopensubsetsofNsuchthat:
V= {P ǫ N :α = 0, in a neighborhood of P},
Ω = {P ǫ N : α6= 0, in a neighborhood of P},
whereV∪ΩisopenanddenseintheclosureofN.
Lemma4.1 Let M be a real hypersurface in M (c), equipped with pseudo-parallel struc-
2
tureJacobioperator. ThenVisempty.
Proof: Let{U,ϕU,ξ}bealocalorthonormal basisonV. Therelation (2.7)takes theform
Aξ = βU andweconsider:
′ ′ ′ ′
AU = γ U +δ ϕU +βξ, AϕU = δ U +µϕU, (4.1)
′ ′ ′
since g(AU,ξ) = g(U,Aξ) = β, g(AϕU,ξ) = g(ϕU,Aξ) = 0 and γ ,δ ,µ are smooth
functions.
From(2.6)forX = U andX = ϕU,takingintoaccount (4.1),weobtain:
c c
lϕU = ϕU lU = ( −β2)U. (4.2)
4 4
Relation(1.1)forX = U,Y = ξ andZ = ϕU,becauseof(2.4),(4.1)and(4.2)yields:
′
δ = 0,sinceβ 6= 0.
Furthermore, relation (1.1) for X = U and Y = Z = ϕU, owing to (2.4), (4.1), (4.2)
′
andδ = 0implies:
′
µ = 0 c= L, (4.3)
andforX = ξ andY = Z = ϕU,because of(4.3), gives: c = 0,whichisacontradiction.
Therefore, Visempty. (cid:3)
InwhatfollowsweworkonΩ,whereα 6= 0andβ 6= 0.
Byusing(2.6)andrelations (3.1)weobtain:
c c
lU = ( +αγ −β2)U +αδϕU lϕU = αδU +(αµ+ )ϕU (4.4)
4 4
Therelation(1.1)because of(2.4),(3.1)and(4.4),implies:
δ = 0, forX = U, Y = ξ and Z=ϕU, (4.5)
8
andadditional dueto(4.5)yields:
c
µ(αµ+ ) = 0, forX = U, Y = ϕU and Z = ξ. (4.6)
4
Owingto(4.6),weconsider Ω theopensubsetofΩ,suchthat:
1
c
Ω = {P ǫ Ω : µ 6= − , in a neighborhood of P}.
1
4α
Therefore, inΩ from(4.6)wehave: µ = 0.
1
Lemma4.2 Let M be a real hypersurface in M (c), equipped with pseudo-parallel struc-
2
tureJacobioperator. ThenΩ isempty.
1
Proof: InΩ ,relation (1.1)forX = U,Y = ϕU andZ = U,becauseof(2.4),(3.1),(4.4)
1
and(4.5)yields:
(β2−αγ)(c−L)= 0. (4.7)
Dueto(4.7),weconsidertheopensubsetΩ ofΩ ,suchthat:
11 1
Ω = {P ǫ Ω :c 6= L, in a neighborhood of P}.
11 1
SoinΩ ,weobtain: γ = β2.
11 α
InΩ ,therelation(2.5),because ofLemma3.1and(4.5),yields:
11
β2κ c
3
= βκ + , forX = U and Y = ξ (4.8)
1
α 4
(ϕU)α = β(α+κ ), forX = ϕU and Y = ξ (4.9)
3
c
(ϕU)β = β2+βκ + , forX = ϕU and Y = ξ (4.10)
1
2
β2 β2
(ϕU) = (κ +β), forX = U and Y = ϕU. (4.11)
1
α α
Substituting in (4.11) the relations (4.9), (4.10) and taking into account (4.8) we obtain:
3cβ = 0,whichisacontradiction. Therefore,Ω isemptyandL = cinΩ .
4α 11 1
In Ω , relation (1.1) for X = ξ and Y = Z = ϕU, because of (2.4), (3.1) and (4.4)
1
implies: c =0,whichisimpossible. Therefore,Ω isempty. (cid:3)
1
FromLemma4.1,weconclude thatµ = − c inΩ.
4α
Lemma4.3 Let M be a real hypersurface in M (c), equipped with pseudo-parallel struc-
2
tureJacobioperator. ThenΩisempty.
Proof: In Ω, relation (1.1) for X = ϕU, Y = ξ and Z = U, due to (2.4), (3.1), (4.4) and
(4.5) yields: γ = β2 − c . Owing to µ = − c and γ = β2 − c and (4.5), relation (4.4)
α 4α 4α α 4α
implies: lU = lϕU = 0 and since lξ = 0, we obtain that the structure Jacobi operator
9
vanishesinΩ. DuetoProposition 3.2,weconclude thatΩisempty. (cid:3)
From Lemmas 4.1 and 4.3, we conclude that N is empty and we lead to the following
result:
Proposition4.4 Every real hypersurface in M (c), equipped with pseudo-parallel struc-
2
tureJacobioperator, isaHopfhypersurface.
5 ProofofMainTheorem
Since M is a Hopf hypersurface, due to Theorem 2.1 ([11]) we have that α is a constant.
We consider a unit vector field e ǫ D, such that Ae = λe, then Aϕe = νϕe at some point
P ǫM,where{e,ϕe,ξ} isalocalorthonormal basis. Thenthefollowingrelation holdson
M,(Corollary 2.3[11]):
α c
λν = (λ+ν)+ . (5.1)
2 4
Therelation(2.6)implies:
c c
le = ( +αλ)e and lϕe = ( +αν)ϕe. (5.2)
4 4
Relation(1.1)forX = eandY = Z = ϕe,because of(2.4)and(5.2)yields:
α(c+λν −L)(ν −λ) =0. (5.3)
Relation (1.1) for X = Z = e, Y = ξ and for X = Z = ϕe, Y = ξ,because of(2.4) and
(5.2)impliesrespectively:
c c
( +αλ)(L−αλ− )= 0, (5.4)
4 4
c c
( +αν)(L−αν − )= 0. (5.5)
4 4
Becauseof(5.3),weconsider M theopensubsetofM,suchthat:
1
M = {P ǫ M : α(ν −λ) 6= 0 in a neighborhood of P}.
1
SoinM ,wehave: L = c+λν.
1
Proposition5.1 Let M be a real Hopf hypersurface in M (c), equipped with pseudo-
2
parallelstructure Jacobioperator. ThenM isempty.
1
Proof: Becauseof(5.4),weconsider M theopensubsetofM ,suchthat:
11 1
c
M = {P ǫ M : L 6= αλ+ , in a neighborhood of P}.
11 1
4
10