Parallel *-Ricci tensor of real hypersurfaces in CP^2 and CH^2

0.07 MB·

by  Georgios Kaimakamis

CP2 CH2
Parallel *-Ricci tensor of real hypersurfaces in and
GEORGE KAIMAKAMIS AND KONSTANTINA PANAGIOTIDOU
ABSTRACT:Inthispapertheideaofstudyingrealhypersurfacesinnon-flatcomplexspaceforms,whose*-Ricci
4
tensor satisfies geometric conditions is presented. More precisely, the non-existence of three dimensional real
1
0 hypersurfacesinnon-flatcomplexspaceformswithparallel*-Riccitensorisproved.Attheendofthepaperideas
2 forfurtherresearchon∗-Riccitensoraregiven.

n
a
Keywords: Realhypersurface,Parallel,*-Riccitensor,Complexprojectiveplane,Complexhyperbolicplane.
J

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2 MathematicsSubjectClassification(2010):Primary53B20;Secondary53C15,53C25.

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1 Introduction
D
.
h Acomplexspaceformisann-dimensionalKaehlermanifoldofconstantholomorphicsectionalcurvature
t
a c. Acompleteandsimplyconnected complexspaceformiscomplexanalytically isometricto
m
[ • acomplexprojective spaceCPn ifc > 0,

1
v • acomplexEuclidean spaceCn ifc= 0,
4
9 • oracomplexhyperbolic spaceCHn ifc < 0.
7
6
. The symbol Mn(c) is used to denote the complex projective space CPn and complex hyperbolic space
1
0 Cn, when it is not necessary to distinguish them. Furthermore, since c 6= 0 in previous two cases the
4 notion ofnon-flatcomplexspaceformreferstoboththem.
1
: Let M be a real hypersurface in a non-flat complex space form. An almost contact metric structure
v
i (ϕ,ξ,η,g) is defined on M induced from the Kaehler metric G and the complex structure J on Mn(c).
X
The structure vector field ξ is called principal if Aξ = αξ, where A is the shape operator of M and
r
a α = η(Aξ)isasmoothfunction. ArealhypersurfaceiscalledHopfhypersurface, ifξ isprincipalandα
iscalledHopfprincipal curvature.
TheRiccitensor S ofaRiemannianmanifoldisatensorfieldoftype(1,1)andisgivenby
g(SX,Y)= trace{Z 7→ R(Z,X)Y}.
IftheRiccitensorofaRiemannianmanifoldsatisfiestherelation
S = λg,
whereλisaconstant, thenitiscalledEinstein.
Realhypersurfaces innon-flatcomplex space formshavebeenstudied intermsoftheirRiccitensor
S,whenitsatisfiescertaingeometricconditions extensively. Differenttypesofparallelism orinvariance
oftheRiccitensorareissuesofgreatimportance inthestudyofrealhypersurfaces.
1
2 G.KaimakamisandK.Panagiotidou
In[4]itwasprovedthenon-existence ofrealhypersurfaces innon-flatcomplexspaceformsM (c),
n
n ≥ 3 with parallel Ricci tensor, i.e. (∇ S)Y = 0, for any X, Y ∈ TM. In [5] Kim extended
X
the result of non-existence of real hypersurfaces with parallel Ricci tensor in case of three dimensional
real hypersurfaces. Another type of parallelism which was studied is that of ξ-parallel Ricci tensor, i.e.
(∇ S)Y = 0 for any Y ∈ TM. More precisely in [6] Hopf hypersurfaces in non-flat complex space
ξ
formswithconstantmeancurvatureandξ-parallelRiccitensorwereclassified. Moredetailsonthestudy
ofRiccitensorofrealhypersurfaces areincluded inSection6of[7].
Motivated by Tachibana, who in [9] introduced the notion of *-Ricci tensor on almost Hermitian
manifolds, in [2] Hamada defined the *-Ricci tensor of real hypersurfaces in non-flat complex space
formsby
1
∗
g(S X,Y) = (trace{ϕ◦R(X,ϕY)}), forX,Y ∈TM.
2
∗ ∗
The -Ricci tensor S is a tensor field of type (1,1) defined on real hypersurfaces. Taking into account
the work that so far has been done in the area of studying real hypersurfaces in non-flat complex space
formsintermsoftheirtensorfields,thefollowingissueraisesnaturally:
∗ ∗
Thestudyofrealhypersurfacesintermsoftheir -RiccitensorS ,whenitsatisfiescertaingeometric
conditions.
In this paper three dimensional real hypersurfaces in CP2 and CH2 equipped with parallel ∗-Ricci
tensorarestudied. Therefore, thefollowingcondition issatisfied
∗
(∇ S )Y = 0, X,Y ∈TM. (1.1)
X
Moreprecisely thefollowingTheoremisproved.
MainTheorem: Theredonotexistrealhypersurfaces inCP2 andinCH2,whose*-Riccitensor is
parallel.
Thepaperisorganizedasfollows: InSection2preliminaries relationsforrealhypersurfaces innon-
flatcomplexspaceformsarepresented. InSection3theproofofMainTheorem isprovided. Finally,in
Section4ideasforfurtherresearchontheaboveissueareincluded.
2 Preliminaries
∞
Throughoutthispaperallmanifolds,vectorfieldsetcareassumedtobeofclassC andallmanifoldsare
assumedtobeconnected. Furthermore, therealhypersurfaces M aresupposed tobewithoutboundary.
LetM bearealhypersurface immersedinanon-flatcomplexspaceform(M (c),G)withcomplex
n
structureJ ofconstantholomorphic sectionalcurvature c. LetN bealocallydefinedunitnormalvector
fieldonM andξ = −JN thestructure vectorfieldofM.
ForavectorfieldX tangenttoM thefollowingrelation holds
JX = ϕX +η(X)N,
where ϕX and η(X)N are the tangential and the normal component of JX respectively. The Rieman-
nianconnections ∇inM (c)and∇inM arerelatedforanyvectorfieldsX,Y onM by
n
∇ Y = ∇ Y +g(AX,Y)N,
X X
whereg istheRiemannianmetricinduced fromthemetricG.
Parallel*-Riccitensor 3
Theshapeoperator Aoftherealhypersurface M inM (c)withrespecttoN isgivenby
n
∇ N = −AX.
X
The real hypersurface M has an almost contact metric structure (ϕ,ξ,η,g) induced from the complex
structureJ onM (c),whereϕisthestructuretensoranditisatensorfieldoftype(1,1). Moreover,ηis
n
an1-formonM suchthat
g(ϕX,Y)= G(JX,Y), η(X) = g(X,ξ) = G(JX,N).
Furthermore, thefollowingrelations hold
2
ϕ X = −X +η(X)ξ, η◦ϕ= 0, ϕξ = 0, η(ξ) = 1,
g(ϕX,ϕY) = g(X,Y)−η(X)η(Y), g(X,ϕY) = −g(ϕX,Y).
SinceJ iscomplexstructure implies∇J = 0. Thelastrelationleadsto
∇ ξ = ϕAX, (∇ ϕ)Y = η(Y)AX −g(AX,Y)ξ. (2.1)
X X
Theambient space M (c)isofconstant holomorphic sectional curvature cand thisresults intheGauss
n
andCodazziequations tobegivenrespectively by
c
R(X,Y)Z = [g(Y,Z)X −g(X,Z)Y +g(ϕY,Z)ϕX (2.2)
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−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]
X Y
4
whereRdenotestheRiemanniancurvature tensoronM andX,Y,Z areanyvectorfieldsonM.
ThetangentspaceT M ateverypointP ∈M canbedecomposed as
P
T M = span{ξ}⊕D,
P
whereD = kerη = {X ∈ T M : η(X) = 0}andiscalledholomorphicdistribution. Duetotheabove
P
decomposition thevectorfieldAξ canbewritten
Aξ = αξ +βU, (2.3)
1
whereβ = |ϕ∇ ξ|andU = − ϕ∇ ξ ∈ ker(η)providedthatβ 6= 0.
ξ β ξ
Since the ambient space M (c) is of constant holomorphic sectional curvature c following similar
n
calculations to those in Theorem 2 in [3] and taking into account relation (2.2), it is proved that the
∗
*-RiccitensorS ofM isgivenby
cn
∗ 2 2
S = −[ ϕ +(ϕA) ]. (2.4)
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4 G.KaimakamisandK.Panagiotidou
3 ProofofMainTheorem
Let M be a non-Hopf hypersurface in CP2 or CH2, i.e. M2(c). Then the following relations hold on
everynon-Hopfthree-dimensional realhypersurface inM2(c).
Lemma3.1 LetMbearealhypersurface inM2(c). Thenthefollowingrelations holdonM
AU = γU +δϕU +βξ, AϕU = δU +µϕU, (3.1)
∇ ξ = −δU +γϕU, ∇ ξ = −µU +δϕU, ∇ ξ = βϕU, (3.2)
U ϕU ξ
∇UU = κ1ϕU +δξ, ∇ϕUU = κ2ϕU +µξ, ∇ξU = κ3ϕU, (3.3)
∇UϕU = −κ1U −γξ, ∇ϕUϕU = −κ2U −δξ, ∇ξϕU = −κ3U −βξ, (3.4)
whereγ,δ,µ,κ1,κ2,κ3 aresmoothfunctions onMand{U,ϕU,ξ}isanorthonormal basisofM.
FortheproofoftheaboveLemmasee[8]
Let M be a real hypersurface in M2(c), i.e. CP2 or CH2, whose ∗-Ricci tensor satisfies relation
(1.1),whichismoreanalytically written
∗ ∗
∇ (S Y) = S (∇ Y), X,Y ∈TM. (3.5)
X X
Weconsider theopensubsetNofM suchthat
N = {P ∈ M : β 6= 0, inaneighborhood ofP}.
InwhatfollowsweworkontheopensubsetN.
OnNrelation(2.3)andrelations(3.1)-(3.4)ofLemma3.1hold. Sorelation(2.4)forX ∈{U,ϕU,ξ}
takingintoaccountn = 2andrelations (2.3)and(3.1)yields
∗ ∗ 2 ∗ 2
S ξ = βµU −βδϕU, S U = (c+γµ−δ )U and S ϕU = (c+γµ−δ )ϕU. (3.6)
Theinner product ofrelation (3.5)forX = Y = ξ withξ duetothe firstandthe third of(3.6),the first
of(2.1)forX = ξ andthethirdofrelations (3.3)and(3.4)implies
δ = 0. (3.7)
Moreover,theinnerproductofrelation(3.5)forX = ϕU andY = ξ withξ becauseof(3.7),thefirstof
(2.1)forX = ϕU,thefirstandthesecondof(3.6)andthesecondof(3.3)resultsin
µ = 0.
Finally,theinnerproductofrelation(3.5)forX = ξandY = ϕU withξ takingintoaccountµ = δ = 0,
thefirstandthethirdof(3.6)andthelastrelation of(3.4)leadsto
c= 0,
whichisacontradiction. SotheopensubsetNisemptyandweleadtothefollowingProposition.
∗
Proposition3.2 Everyrealhypersurface inM2(c)whose -Riccitensorisparallel, isaHopfhypersur-
face.
Parallel*-Riccitensor 5
SinceM isaHopfhypersurface, the structure vector fieldξ is aneigenvector oftheshape operator,
i.e. Aξ = αξ. Due to Theorem 2.1 in [7] α is constant. We consider a point P ∈ M and choose a unit
principalvectorfieldW ∈DatP,suchthatAW = λW andAϕW = νϕW. Then{W,ϕW,ξ}isalocal
orthonormal basisandthefollowingrelationholds(Corollary 2.3[7])
α c
λν = (λ+ν)+ . (3.8)
2 4
Thefirstof relation (2.1)and relation (2.4)for X ∈{W,ϕW,ξ} because of Aξ = αξ, AW = λW
andAϕW = νϕW impliesrespectively
∇ ξ = λϕW and ∇ ξ = −νW (3.9)
W ϕW
∗ ∗ ∗
S ξ = 0, S W = (c+λν)W and S ϕW = (c+λν)ϕW. (3.10)
Relation(3.5)forX = W andY = ξ becauseofthefirstof(3.9)andthefirstandthirdrelationof(3.10)
yields
λ(c+λν)= 0.
Suppose that (c+ λν) 6= 0 then the above relation results in λ = 0. Moreover, relation (3.5) for
X = ϕW andY = ξ becauseofthesecond of(3.9)andthefirstandsecondrelation of(3.10)yields
ν = 0.
Substitution ofλ = ν = 0in(3.8)resultsinc= 0,whichisacontradiction. Sorelationc = −λν holds.
Thelastoneimpliesλν 6= 0sincec6= 0.
Supposethatλ = −c. Substitution ofthelastonein(3.8)leadsto
ν
2
2αν +5cν −2αc = 0. (3.11)
In case of CP2 we have that c = 4 and from equation (3.11) there is always a solution for ν. So
ν is constant and λ will be also constant. Therefore, the real hypersurface has three distinct constant
eigenvalues. The latter results in M being a real hypersurface of type (B), i.e. a tube of radius r over
complexquadric. Substitution oftheeigenvalues oftype(B)inλν = −cleadstoacontradiction. Sono
realhypersurface inCP2 hasparallel ∗-Riccitensor(eigenvalues canbefoundin[7]).
In case of CH2 we have that c = −4 and from equation (3.11) there is a solution for ν if 0 ≤
2 25
α ≤ . If α = 0 equation (3.11) implies cν = 0, which is impossible. So there is a solution for
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2 25
ν if 0 < α ≤ and ν will be constant. The latter results in that λ is also constant and so the real
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hypersurface is of type (B), i.e. a tube of radius r around totally geodesic RHn. Substitution of the
eigenvalues of type (B) in λν = −c leads to a contradiction and this completes the proof of our Main
Theorem(eigenvalues canbefoundin[1]).
4 Discussion-OpenProblems
∗
Inthispaperthreedimensional realhypersurfaces innon-flat complexspaceformswithparallel -Ricci
tensorarestudiedandthenon-existenceofthemisproved. Therefore,aquestionwhichraisesinanatural
wayis
Are there real hypersurfaces in non-flat complex space forms of dimension greater than three with
6 G.KaimakamisandK.Panagiotidou
∗
parallel -Riccitensor?
Generally, the next step in the study of real hypersurfaces in non-flat complex space forms is to
study them when a tensor field P type (1,1) of them satisfies other types of parallelism such as the D-
parallelismorξ-parallelism. ThefirstoneimpliesthatP isparallelinthedirectionofanyvectorfieldX
orthogonal toξ, i.e. (∇ P)Y = 0, forany X ∈D,and thesecond one implies that P isparallel inthe
X
directionofthestructurevectorξ,i.e. (∇ P)Y = 0. Sothequestionswhichshouldbeansweredarethe
ξ
following
∗
Arethererealhypersurfaces innon-flat complex space formswhose -Riccitensor satisfies thecon-
ditionofD-parallelism orξ-parallelism?
Finally, other types of parallelism play important role in the study of real hypersurfaces is that of
semi-parallelism and pseudo-parallelism. A tensor field P of type (1, s) is said to be semi-parallel if it
satsfiesR·P = 0,whereRistheRiemanniancurvaturetensorandactsasaderivationonP. Moreover,
P is said to be pseudo-parallel if there exists a function L such that R(X,Y)·P = L{(X ∧Y)·P},
where(X ∧Y)Z = g(Y,Z)X −g(Z,X)Y. Sothequestions are:
Are there real hypersurfaces in non-flat complex space forms with semi-parallel or pseudo-parallel
∗
-Riccitensor?
Theimportance of answering the above question lays in the fact that the class ofreal hypersurfaces
∗
withparallel∗-Riccitensorisincludedintheclassofrealhypersurfaceswithsemi-parallel -Riccitensor.
∗
Furthermore,thelastoneisincludedintheclassofrealhypersurfaceswithpseudo-parallel -Riccitensor.
References
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[3] T.IveyandP.J.Ryan,”The*-RiccitensorforhypersurfacesinCPnandCHn”,TokyoJ.Math.34(2011),445-471.
[4] U-H.Ki,”RealhypersurfaceswithparallelRiccitensorofacomplexspaceform”,TsukubaJ.Math,13(1989),73-81.
[5] U.K.Kim,”NonexistenceofRicci-parallelrealhypersurfacesinCP2andCH2”,Bull.KoreanMath.Soc.41(2004),699-708.
[6] M.KimuraandS.Maeda,”OnrealhypersurfacesofacomplexprojectivespaceII”,TsukubaJ.Math,15(1991),547-561.
[7] R.NiebergallandP.J.Ryan,”Realhypersurfacesincomplexspaceforms”,Math.Sci.Res.Inst.Publ.32(1997),233-305.
[8] K.PanagiotidouandPh.J.Xenos,RealhypersurfacesinCP2andCH2whosestructureJacobioperatorisLieD-parallel,NoteMat.
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G.KAIMAKAMIS,FACULTYOFMATHEMATICSANDENGINEERINGSCIENCES,HELLENICMILITARYACADEMY,VARI,ATTIKI,
GREECE
E-MAIL:[e-mail]
K.PANAGIOTIDOU,FACULTYOFENGINEERING,ARISTOTLEUNIVERSITYOFTHESSALONIKI,THESSALONIKI54124,GREECE
E-MAIL:[e-mail]