Ricci curvature of double manifolds via isoparametric foliations

0.1 MB·

by  ChiaKuei Peng

RICCI CURVATURE OF DOUBLE MANIFOLDS VIA ISOPARAMETRIC
FOLIATIONS
CHIAKUEIPENGANDCHAOQIAN
6
1
0
2 Abstract. Givenaclosedmanifold Mandavectorbundleξofranknover M,bygluingtwo

n copiesofthediscbundleofξ,wecanobtainaclosedmanifoldD(ξ,M),theso-calleddouble
a
manifold.
J
Inthispaper,wefirstlyprovethateachspherebundleS (ξ)ofradiusr > 0isanisopara-
3 r
1 metrichypersurface inthetotalspaceofξ equippedwithaconnectionmetric, andforr > 0
small enough, the induced metric of Sr(ξ) has positive Ricci curvature under the additional
]
G assumptionsthatMhasametricwithpositiveRiccicurvatureandn 3.
≥
D Asanapplication, if M admitsametricwithpositiveRiccicurvatureandn 2, thenwe
≥
constructametricwithpositiveRiccicurvatureonD(ξ,M). Moreover,underthesamemetric,
.
h
D(ξ,M)admitsanaturalisoparametricfoliation.
t
a ForacompactminimalisoparametrichypersurfaceYn inSn+1(1),whichseparatesSn+1(1)
m
into Sn+1 and Sn+1, one can get double manifolds D(Sn+1) and D(Sn+1). Inspired by Tang,
+ +
[ − −
Xie and Yan’s work on scalar curvature of such manifolds with isoparametric foliations(cf.

1 [TXY12]),westudyRiccicurvatureofthemwithisoparametricfoliationsinthelastpart.
v
5
2
1. Introduction
1
3
0 Let N be a connected complete Riemannian manifold. A non-constant smooth function
.
1 f : N R is called transnormal, if there exists a smooth function b : R R such that the
0 → →
6 gradient of f satisfies f 2 = b(f). Moreover, if there exists another function a : R R
|∇ | →
1 such that the Laplacian of f satisfies f = a(f), then f is said to be isoparametric. The two
: △
v equations meanthatregular levelhypersurfaces areparallel andallhaveconstant meancurva-
i
X ture. Each regular level hypersurface is said to be an isoparametric hypersurface. In [Wa87],
r
a Wang proved that singular level sets are also smooth submanifolds, the so-called focal sub-
manifolds. Recall that an isoparametric foliation on a Riemannian manifold N is defined by
thewholefamilyofregularlevelhypersurfacestogetherwithfocalsubmanifoldsofanisopara-
metricfunction(cf. [Wa87],[GT13]and[QT15]). Equivalently, anisoparametric foliation isa
singularRiemannianfoliationallofwhoseregularleavesarehypersurfaceswithconstantmean
curvature(cf. [GQ15]). For instance, the orbits of an isometric cohomogeneity one action on
N form anisoparametric foliation, the so-called homogeneous isoparametric foliation. Hence,
2010MathematicsSubjectClassification. 53C12,53C40,53C07.
Keywordsandphrases. doublemanifold,isoparametricfoliation,Riccicurvature,vectorbundle,spherebundle.
Thesecondauthoristhecorrespondingauthor.
1
2 C.K.PENGANDC.QIAN
isoparametric foliations can be regarded as a geometric generalization of cohomogeneity one
actions.
E. Cartan was the pioneer who made a comprehensive study of isoparametric hypersur-
faces in real space forms, especially in the unit spheres. Particularly, he proved that isopara-
metrichypersurfacesinrealspaceformsareequivalentlyhypersurfaceswithconstantprincipal
curvatures and obtained the classification results for the Euclidean and hyperbolic cases. For
the spherical case, if the number of distinct principal curvatures is no more than 3, then he
showed that the isoparametric hypersurfaces must be homogeneous. Given an isoparametric
hypersurfaceYninSn+1(1),letηbeaunitnormalvectorfield,gthenumberofdistinctprincipal
curvatures, cotθ (α = 1,2, ,g;0 < θ < θ < < θ < π) the principal curvatures with
α 1 2 g
··· ···
respecttoηandm themultiplicityofcotθ . Underthissetting,H.F.Mu¨nzner[Mu¨80]proved
α α
that θα = θ1 + α−g1π, mα = mα+2(indicesmod g), and there exists a homogeneous polynomial
F :Rn+2 Rofdegreeg,theso-called Cartan-Mu¨nzner polynomial, satisfying
→
EF2 = g2r2g 2,r = x,
 −
 |∇EF| = m2 m1g2rg 2,| |
 △ −2 −
wherem andm arethetwomultiplicities, and E, E areEuclidean gradient andLaplacian,
1 2
∇ △
respectively. It follows that f = FSn+1(1) is an isoparametric function on Sn+1(1) with Imf =
|
[ 1,1], and M = f 1( ) are two focal submanifolds with codimension m +1 and m + 1
− 1 2
in−Sn+1(1) resp±ectively.±Moreover, based on this global structure, he obtained the splendid
result that g = 1,2,3,4 or 6. Owing to E. Cartan and H. F. Mu¨nzner, the classification of
isoparametric hypersurfaces with 4 or 6 principal curvatures in unit spheres is an intriguing
problem in submanifold geometry. Up to now, isoparametric hypersurfaces with 4 principal
curvatures in unit spheres have been classified except for one case(cf. [CCJ07], [Im08] and
[Ch13]). For recent progress and application in this subject, we refer to [GX10], [TY13],
[QT14]and[TY15].
One of the fundamental problems in Riemannian geometry is to investigate Riemannian
manifolds withspecialcurvatureproperties, especially manifolds withpositivescalar,Riccior
sectional curvatures. For the scalar curvature case, due to the surgery theory of Schoen-Yau
[SY79]andGromov-Lawson [GL80], simply connected manifolds with positive scalar curva-
turearewellunderstood. MotivatedbySchoen-YauandGromov-Lawsonsurgerytheory,Tang,
XieandYanin[TXY12]obtained thefollowingtheorem based onthetheory ofisoparametric
hypersurfaces inunitspheres.
Theorem1.1. ([TXY12])LetYnbeacompactminimalisoparametrichypersurfaceinSn+1(1),
n 3, which separates Sn+1(1) into Sn+1 and Sn+1. Then each of the doubles D(Sn+1) and
+ +
D(≥Sn+1) admits a metric ofpositive scalar curvatu−re. Moreover, there is still an isoparametric
folia−tion in D(Sn+1)(orD(Sn+1)).
+
−
In this paper, we will study Ricci curvature properties of double manifolds D(Sn+1) and
+
D(Sn+1)togetherwithisoparametricfoliations. Westartwithageneralsetting. Letξbeavector
−
RICCICURVATUREOFDOUBLEMANIFOLDSVIAISOPARAMETRICFOLIATIONS 3
bundle of rank nover a closed manifold M. Given twocopies of the disc bundle B(ξ)over M
withboundarythespherebundleS(ξ),wedefinethedoublemanifoldD(ξ,M):= B(ξ) B(ξ),
Fid
where id : S(ξ) S(ξ) is the identity map. For instance, D(S+n+1) (cid:27) D(ν(M+),M+) and
→
D(Sn+1) (cid:27) D(ν(M ),M ), where ν(M+) and ν(M ) are the normal bundles over the focal
subm−anifolds M+ an−d M−ofisoparametric hypersur−face Yn inSn+1(1),respectively.
−
To consider the Ricci curvature properties of double manifolds, we firstly obtain the fol-
lowingresult.
Theorem1.2. Letξ beaRiemannianvectorbundleofranknoveraclosedRiemannianmani-
fold (M,ds2 ). Givenaconnection metricon E,then eachsphere bundle S (ξ)ofradius r > 0
M r
isanisoparametrichypersurfaceinE. Moreover,assume(M,ds2 )haspositiveRiccicurvature
M
and n 3, then, for sufficient small r > 0, the sphere bundle S (ξ) of radius r > 0 with the
r
≥
induced metricinE haspositiveRiccicurvature.
Remark 1.1. Actually, for sufficient small r > 0, the sphere bundle S (ξ) of radius r with
r
theinducedmetricinE haspositivescalarcurvaturewithouttheassumptionthat(M,ds2 )has
M
positiveRiccicurvature.
Remark1.2. Theorem1.2andRemark1.1generalizeTheorem1andTheorem3in[KS00](also
see[Na79]).
Asanapplication ofTheorem1.2,wecanprove
Theorem 1.3. Assume n 2, and the closed manifold M has a metric with positive Ricci
≥
curvature. Then the double manifold D(ξ,M) admits a metric with positive Ricci curvature,
andmeanwhileanaturalisoparametric foliation.
Remark 1.3. By the assumption that M has a metric with positive Ricci curvature, it follows
fromBonnet-Myerstheoremthatπ Misfinite. Then,byVanKampen’stheorem,π D(ξ,M)is
1 1
isomorphic toπ M,andisalsofinite.
1
Atlast,combining withTheoremAin[GZ02]andTheorem1.3,wecaninferthefollow-
ingresult, whichimprovesTheorem1.1in[TXY12].
Theorem 1.4. Let Yn be a compact isoparametric hypersurface in a unit sphere Sn+1(1) with
4 distinct principal curvatures and multiplicities (m ,m )(assume m m ), and M the focal
1 2 1 2
submanifolds ofYn withcodimension m +1andm +1inSn+1(1)re≤spectively. ±
1 2
(1). (m1,m2) = (1,k): The double manifold D(S+n+1), associated with M+ which is diffeo-
morphic to the Stiefel manifold Vk+2,2, admits a metric with positive Ricci curvature and a
homogeneous isoparametric foliation. However,thedoublemanifold D(Sn+1),associatedwith
M which is diffeomorphic to (S1 Sk+1)/Z , only admits a metric wit−h nonnegative Ricci
2
− ×
curvature andahomogeneous isoparametric foliation.
(2). 2 m m : Bothoftheresulting double manifolds D(Sn+1)and D(Sn+1)admitmetrics
1 2 +
≤ ≤ −
withpositiveRiccicurvature, andalsonaturalisoparametric foliations.
4 C.K.PENGANDC.QIAN
Remark 1.4. For case (m ,m ) = (1,k), by Van Kampen’s theorem, π D(Sn+1) = π (S1
1 2 1 1
Sk+1)/Z = Z(cf. [CR85], [QT14a]). Hence, according to Bonnet-Myers the−orem, D(Sn+1×)
2
−
cannotadmitametricwithpositiveRiccicurvature inthiscase.
Remark1.5. Thenaturalisoparametric foliations constructed inTheorem1.4arefoliateddif-
feomorphic tothe ones constructed in Theorem 1.1of[TXY12]. Forthe definition offoliated
diffeomorphism, wereferto[Ge14]and[GQ15].
2. Geometryofvectorbundlesandassociatedspherebundles
Inthispaper, weshallmakeuseofthefollowingconvention ontherangesofindices:
1 A,B,C,... m+n,1 i, j,k,... m,
≤ ≤ ≤ ≤
m+1 α,β,γ,... m+n,m+1 a,b,c,... m+n 1.
≤ ≤ ≤ ≤ −
Generally, let (N,ds2) be a Riemannian manifold of dimension m + n. Choose a lo-
N
cal orthonormal frame e1,··· ,em+n, and let θ1,··· ,θm+n be the dual frame such that ds2N =
m+n(θ )2. Thenwehavethefirstandsecondstructural equations:
PA=1 A
dθ = θ θ ,θ +θ = 0,
 A AB B AB BA
 dθ = θ ∧ θ Θ ,
AB AC CB AB
 ∧ −
where θ are the connection 1-forms, and Θ = 1K θ θ are the curvature 2-forms
AB AB 2 ABCD C ∧ D
with K thecomponents ofthecurvaturetensor.
ABCD
In this section, we will consider the geometry of vector bundles and sphere bundles. Let
(M,ds2 )beam-dimensional closed Riemannianmanifold, andξ aRiemannian vectorbundle
M
of rank n over M with total space E and projection π : E M. For our purpose, we fix a
→
metric , onξ andchooseaconnection Dcompatible with , .
h· ·i h· ·i
Let U be an open neighborhood in M, and ds2 = m (ω)2, where ω ,...,ω are 1-
M|U Pi=1 i 1 m
formsonU. Thenthestructural equations of(M,ds2 )aregivenby
M
dω = ω ω ,ω +ω = 0,
 i ij j ij ji
 dω = ω ∧ ω Ω ,
ij ik kj ij
 ∧ −
whereω areconnection 1-forms,andΩ = 1R ω ω arethecurvature2-formswithR
ij ij 2 ijkl k∧ l ijkl
thecomponents ofthecurvaturetensor.
Assume U issmall enough such that ξ istrivial, wecan choose orthonormal cross sec-
U
|
tions em+1,...,em+n on U for ξ, i.e. eα,eβ = δαβ, and ΨU : U Rn π−1(U) is the local
h i × →
trivialization given by Ψ (p,v ,...,v ) = m+n v e (p). Then De = ω e , where (ω )
U 1 n Pα=m+1 α α α αβ β αβ
is the matrix expression of the connection D. Note that D is compatible with , means that
h· ·i
(ω ) is skew-symmetric. The curvature 2-forms Ω of the connection D are determined by
αβ αβ
dω = ω ω Ω ,whereΩ = 1R ω ω .
αβ αγ ∧ γβ− αβ αβ 2 αβij i∧ j
RICCICURVATUREOFDOUBLEMANIFOLDSVIAISOPARAMETRICFOLIATIONS 5
Define θ := π ω and θ := dv + v ω . Then ds2 = (θ)2 + (θ )2 is a well-defined
i ∗ i α α β βα ξ i α
Riemannian metric on E, i.e., the so-called connection metric(cf. [Je73], [Na79]). For con-
venience, we give the formulae of connection 1-forms θ and curvature 2-forms Θ of the
AB AB
connection metricinthefollowingbymakinguseofthemovingframemethod.
Proposition 2.1. Theconnection 1-formsθ aredetermined by
AB
θ = π ω +A θ ,
ij ∗ ij ijα α
θ = A θ ,
iα ijα j
θ = ω ,
αβ αβ
andthecurvature 2-formsΘ aregivenby
AB
1 1
Θ = Ω A A θ θ A A θ θ R v θ θ + R θ θ
ij ij ikα jlα k l ijα klα k l αβij,k β k α αβij α β
− ∧ − ∧ − 2 ∧ 2 ∧
+A A θ θ ,
ikα kjβ α β
∧
1 1
Θ = R v θ θ + R θ θ A A θ θ ,
iα αβij,k β j k αβij j β ikβ kjα j β
2 ∧ 2 ∧ − ∧
Θ = A A θ θ +Ω ,
αβ ijα ikβ j k αβ
− ∧
where A = 1R v , and R is defined by R ω := dR +R ω +R ω +
ijα 2 αβij β αβij,k αβij,k k αβij γβij γα αγij γβ
R ω +R ω .
αβlj li αβil lj
Proof. Byusingthefirstandsecondstructural equations andadirectcomputation. (cid:3)
As a consequence, we can obtain the components of curvature tensor, which are defined
byΘ := 1Θ θ θ .
AB 2 ABCD C ∧ D
Proposition 2.2. Thecomponents ofthecurvature 2-formsΘ aregivenby
AB
Θ = R A A +A A 2A A ,
ijkl ijkl ikα jlα ilα jkα ijα klα
− −
1
Θ = R v ,
ijkα αβij,k β
−2
Θ = R +A A A A ,
ijαβ αβij ikα kjβ ikβ kjα
−
1
Θ = R A A ,
iαjβ αβij ikβ kjα
2 −
Θ = 0, Θ = 0.
iαβγ αβγσ
Define S (ξ) := (p,v) E v π 1(p), v,v = r2 , the associated sphere bundle of
r −
{ ∈ | ∈ h i }
radiusr > 0. Atpresent, weareintheposition toproveTheorem1.2.
ProofofTheorem1.2:
Proof. At first, observe that the function f : E R,(p,v) v,v is a transnormal func-
→ 7→ h i
tion(cf. Theorem 2.2 in [QT15]). Next, we need to compute the principal curvatures of
S (ξ) E for any r > 0. On π 1(U), r2 = v2. Define u := vα. For r > 0, η = u ∂ is
r ⊂ − Pα α α r α∂vα
6 C.K.PENGANDC.QIAN
a natural global normal vector field of S (ξ) E. In order to study the extrinsic geometry of
r
⊂
S (ξ),weintroduce theorthogonal transformation P :Rn Rn definedby
r
→
I xTx xT
(Pαβ)=  n−1 −x 1−xn xn ,
where x = (um+1,...,um+n 1) and xn = um+n. Choose the moving frame ϕi,ϕa,ϕm+n by
− { }
ϕi = θi and ϕα = Pαβθβ. Particularly, ϕm+n = uαθα = dr. Therefore, ϕi,ϕa,ϕm+n is exactly
{ }
theadaptedmovingframewithrespecttoS (ξ) E. Theassociatedconnection1-forms(ϕ )
r AB
⊂
aregivenby
ϕ ϕ θ θ P
(ϕ )=  ij iα  =  ij iβ βα .
AB  ϕαi ϕαβ   Pαβθβi dPαγPγβ+PαγθγδPδβ 
Itfollowsthatϕim+n = 0andϕam+n = −ϕra. Therefore, foreachr > 0,Sr(ξ)isanisoparametric
hypersurface in E with constant principal curvatures 0 and 1 with multiplicities m and n,
−r
respectively.
Now,undertheassumption(M,ds2 )haspositiveRiccicurvatureandn 3,wewillprove
M ≥
theinduced metricofS (ξ) (E,ds2)haspositive Riccicurvature forr > 0smallenough. To
r ⊂ ξ
computetheRiccicurvatures ofSr(ξ),choosethemovingframe{ϕi,ϕa,ϕm+n}of(E,ds2ξ),and
define Φ := 1K ϕ ϕ for 1 A,B,C,D m+n, where Φ are curvature 2-forms
AB 2 ABCD C ∧ D ≤ ≤ AB
of ϕi,ϕa,ϕm+n . Then
{ }
Φ Φ Θ Θ P
(Φ ) =  ij iα =  ij iβ βα .
AB  Φαi Φαβ   PαβΘβi PαγΘγδPδβ 
Defineψ = ϕ ,ψ = ϕ ,then ψ,ψ isamovingframeforS (ξ). Letψ ,ψ ,ψ be
i i|Sr(ξ) a a|Sr(ξ) { i a} r ij ia ab
the associated connection 1-forms, and Ψ ,Ψ ,Ψ the associated curvature 2-forms respec-
ij ia ab
tively. BytheGaussequations ofS (ξ) E,itfollowsthat
r
⊂
Ψij = Φij|Sr(ξ)+ϕim+n∧ϕjm+n|Sr(ξ),
Ψia = Φia|Sr(ξ)+ϕim+n∧ϕam+n|Sr(ξ),
Ψab = Φab|Sr(ξ)+ϕam+n∧ϕbm+n|Sr(ξ).
AssumeΨ = 1Ψ ψ ψ +Ψ ψ ψ + 1Ψ ψ ψ ,Ψ = 1Ψ ψ ψ +Ψ ψ
ij 2 ijkl k∧ l ijka k∧ a 2 ijab a∧ b ia 2 iakl k∧ l iakb k∧
ψ + 1Ψ ψ ψ ,andΨ = 1Ψ ψ ψ +Ψ ψ ψ + 1Ψ ψ ψ .According to
b 2 iabc b ∧ c ab 2 abkl k ∧ l abkc k ∧ c 2 abcd c ∧ d
Proposition 2.1,theRiccicurvature tensorRicSr ofS (ξ)isgivenby
r
RicSr = Ψ +Ψ ,
ij ikjk iaja
RicSr = Ψ +Ψ ,
ia ikak ibab
RicSr = Ψ +Ψ ,
ab akbk acbc
RICCICURVATUREOFDOUBLEMANIFOLDSVIAISOPARAMETRICFOLIATIONS 7
where
Ψ = R 3A A ,
ikjk ikjk ikα jkα
−
Ψ = A A P P ,
iaja ikα jkβ αa βa
1
Ψ = R v P ,
ikak αβik,k β αa
2
Ψ = 0,
ibab
Ψ = A A P P ,
akbk kmα kmβ αa βb
1
Ψ = δ (n 2).
acbc r2 ab −
Observe that A = 1R v = 1rR u , and the norms of R , R are bounded on M,
ijα 2 αβij β 2 αβij β αβij αβij,k
since MisaclosedRiemannianmanifold. Bytheassumption MhaspositiveRiccicurvature,it
follows that, forr > 0smallenough, theRiccicurvature tensor RicSr ispositive definite. And
theproofiscomplete. (cid:3)
3. Riccicurvatureandisoparametricfoliation ondoublemanifold
In this section, we will prove Theorem 1.3. The key observation is the following fact in
[TXY12].
Proposition 3.1. ([TXY12]) Let ξ be a vector bundle of rank n over M of dimension m and
D(ξ,M)thedoublemanifoldbygluingtwocopiesofdiscbundle ofξ. ThenD(ξ,M)isdiffeo-
morphic toS(ξ 1),whereS(ξ 1)isthesphere bundleoftheWhitneysumbetweenξ anda
⊕ ⊕
triviallinebundle 1.
For convenience, denote the total space of ξ 1 by E(ξ 1) and the projection also by
⊕ ⊕
π : E(ξ 1) M. As in Section 2, we fix a metric , on ξ and choose a connection D
⊕ → h· ·i
compatible with , . Atpresent, forthevector bundle ξ 1, itisnatural toextend themetric
h· ·i ⊕
, onξtoξ 1suchthatξ 1isanorthogonalWhitneysum,denotedalsoby , . Moreover,
h· ·i ⊕ ⊕ h· ·i
chooseaunitsectionem+n+1 forthetriviallinebundle1. Bydemandingthatem+n+1 isparallel,
weextendnaturally theconnection Donξ toaconnection onξ 1,denotedalsobyD.
⊕
Assume U issmall enough such that ξ istrivial, wecan choose orthonormal cross sec-
U
|
tions em+1,...,em+n onU forξ. Thenem+1,...,em+n,em+n+1 isanorthonormal cross sections on
U forξ 1. Moreover,form+1 α,β m+n,Deα = ωαβeβ,andDem+n+1 = 0,where(ωαβ)
⊕ ≤ ≤
isthematrixexpression oftheconnection D. Consequently, ωm+n+1α = ωαm+n+1 = 0.
−
ProofofTheorem1.3:
Proof. By the assumption, choose the Riemannian metric ds2 on M with positive Ricci cur-
M
vature. AccordingtoTheorem1.2,fortheconnection metricon E(ξ 1),andforr > 0small
0
⊕
enough, the sphere bundle S (ξ 1) of radius r with the induced metric has positive Ricci
r0 ⊕ 0
curvature.
8 C.K.PENGANDC.QIAN
Next, wewill construct the natural isoparametric foliation on S (ξ 1). Let f : S (ξ
r0 ⊕ r0 ⊕
1) R be a smooth function such that f(p,v) is the em+n+1-component of v for each point
→
(p,v) S (ξ 1). Wewillprovethat f isanisoparametric function.
∈ r0 ⊕
Chooseanopenneighborhood U in M suchthatds2 = m (ω)2,whereω ,...,ω are
M|U Pi=1 i 1 m
1-forms onU. Letω and Ω betheconnection 1-forms and curvature 2-forms, respectively.
ij ij
Moreover, assume U is small enough such that ξ is trivial, and choose orthonormal cross
U
|
sections em+1,...,em+n for ξU, with connection 1-forms ωαβ and curvature 2-forms Ωαβ. For
|
thevectorbundleξ 1,weassumeem+n+1 isparallelasbefore,i.e.,ωm+n+1α = ωαm+n+1 = 0.
⊕ −
Under this local trivialization of (ξ 1) , for each point (p,v) (ξ 1) , we have
U U
⊕ | ∈ ⊕ |
v = vαeα +vm+n+1em+n+1. Define θi := π∗ωi, θα := dvα +vβωβα and θm+n+1 := dvm+n+1. The
connectionmetricds2(ξ 1) canbeexpressedasds2(ξ 1) = (θi)2+(θα)2+(θm+n+1)2. Writeuα = vrα
andum+n+1 = vm+rn+1,w⊕herer2 = (vα)2+(vm+n+1)2⊕. Then,forthesphere bundle Sr0(ξ⊕1), the
induced metricds2 metriccanbeexpressedas
Sr0(ξ⊕1)
ds2Sr0(ξ⊕1) = (θi)2+r02(duα +uβωβα)2+r02(dum+n+1)2.
Under this local coordinate system, f(p,v) = r0um+n+1 for (p,v) ∈ Sr0(ξ ⊕1). It follows that
f 2 = 1 f2.
|∇ | − r2
0
Tocompletetheproof,itissufficienttoshowtheregularhypersurfacesof f haveconstant
principal curvatures. Writew = uα ,then
α
√1 (um+n+1)2
−
ds2Sr0(ξ⊕1) = (θi)2+r021(du(mu+mn++n1+)12)2 +r02(1−(um+n+1)2)(dwα +wβωβα)2.
−
Usingthesubstitution t = arccosum+n+1 for 1< um+n+1 < 1,weobtainthat
−
ds2 = (θ)2+r2(dt)2 +r2sin2t(dw +w ω )2.
Sr0(ξ⊕1) i 0 0 α β βα
Now,wecanchooseϕ = θ,ϕ = r dt,andϕ = r sintψ suchthat
i i 0 0 a 0 a
m+n 1
−
(ψ )2 = (dw +w ω )2.
X a α β βα
a=m+1
Then ϕ ,ϕ,ϕ is a moving frame for S (ξ 1) with ϕ = 0 for r < c < r . Let
{ 0 i a} r0 ⊕ 0|f−1(c) − 0 0
ϕ ,ϕ ,ϕ ,ϕ ,ϕ bethe connection 1-forms. Toobtain the principal curvatures of f 1(c)
0i 0a ij ia ab −
⊂
S (ξ 1),itissufficient todeterminetheformsϕ andϕ . Define
r0 ⊕ 0i 0a
ϕ = h ϕ +h ϕ ,
0i ij j ia a
ϕ = h ϕ +h ϕ ,
0a ai i ab b
ϕ = λ ϕ mod ϕ,ϕ ,
ij ij 0 i a
{ }
ϕ = λ ϕ mod ϕ,ϕ ,
ia ia 0 i a
{ }
ϕ = λ ϕ mod ϕ,ϕ .
ab ab 0 i a
{ }
RICCICURVATUREOFDOUBLEMANIFOLDSVIAISOPARAMETRICFOLIATIONS 9
Fromthefirststructural equations, wehave
dϕ = ϕ ϕ +ϕ ϕ ,
0 0i i 0a a
∧ ∧
dϕ = ϕ ϕ +ϕ ϕ +ϕ ϕ ,
i i0 0 ij j ia a
∧ ∧ ∧
dϕ = ϕ ϕ +ϕ ϕ +ϕ ϕ .
a a0 0 ai i ab b
∧ ∧ ∧
Byadirectcomputation, wehave
dϕ = 0,
0
dϕ = π ω ϕ ,
i ∗ ij j
∧
cott
dϕ = ϕ ϕ mod ϕ ϕ ,ϕ ϕ ,ϕ ϕ .
a 0 a i j i a a b
r ∧ { ∧ ∧ ∧ }
0
Itfollowsthat
h = h ,h = h ,h = h ;
 ij ji ia ai ab ba
 hij = −λij,hia = −λia;
 hai = λia,hab = −λab+ cro0ttδab.
Therefore, ϕ = 0 and ϕ = cottϕ . That is to say, the hypersurface f 1(c) has constant
0i 0a r0 a −
principal curvatures 0and cott witht = arccos c for the unit normal vector field determined
− r0 r0
byϕ . Theproofiscomplete. (cid:3)
0
4. Applicationstoisoparametricfoliationofunitspheres
Inthissection,basedonTheorem1.3,wewillstudytheRiccicurvatureandisoparametric
foliation ondoublemanifolds D(Sn+1),andproveTheorem1.4.
±
ProofofTheorem1.4:
Proof. Let Yn be a closed isoparametric hypersurface with 4 distinct principal curvatures in
Sn+1(1)withtwofocalsubmanifolds M+and M ofcodimensionm1+1andm2+1inSn+1(1)
−
respectively(cf. [CR85]). And n = 2(m + m ). Without loss of generality, we can assume
1 2
m m .
1 2
≤
Case (1). (m ,m ) = (1,k): According to[Ta76], the isoparametric hypersurfaces inthis case
1 2
mustbehomogeneous. Moreprecisely, theyaretheprincipal orbits ofisotropy representation
ofthesymmetricpair(SO(k+4),SO(2) SO(k+2)). Forthesymmetricpair(SO(k+4),SO(2)
× ×
SO(k+2)),wehave theCartan decomposition o(k+4) = (o(2)+o(k+2)) p,wherepisthe
⊕
orthogonal complement of (o(2) + o(k + 2)) with respect to the Killing form B of o(k + 4),
and p has the canonical induced inner product. Let S(p) be the unit sphere in p. Then G =
SO(2) SO(k +2) acts on p by conjugation and it induces the cohomogeneity one action on
×
S(p), the principal orbits of which are isoparametric hypersurfaces in this case. Let K+ =
∆SO(2) SO(k), K = Z SO(k + 1) and H = Z SO(k) be the closed subgroups of G.
2 2
× − · ·
Under the notations of [GZ02], (S(p),G) is a chomogeneity one manifold determined by the
group diagram H K+,K G. Thetwofocal submanifolds(singular orbits) M+ and M are
⊂ − ⊂ −
10 C.K.PENGANDC.QIAN
determined by M+ = G/K+ (cid:27) Vk+2,2, the Stiefel manifold of 2-orthonormal frames in Rk+2,
and
M =G/K (cid:27) (S1 Sk+1)/Z = (S1 Sk+1)/(θ,x) (θ+π, x).
2
− − × × ∼ −
Thus,thedoublemanifoldD(Sn+1)hasaninducedcohomogeneityoneactionbyGwithgroup
+
diagram H K+,K+ G. By Van Kampen’s theorem, π1D(S+n+1) (cid:27) π1M+ = Z2 if k = 1,
⊂ ⊂
and π1D(S+n+1) (cid:27) π1M+ = 0 if k > 1. It follows from Theorem A in [GZ02] that D(S+n+1)
{ }
admits an invariant metric with positive Ricci curvature. Meanwhile, the principal orbits are
homogeneous isoparametric hypersurfaces. Forthecaseof D(Sn+1),ithasthecohomogeneity
−
one action by G with group diagram H K ,K G. By Van Kampen’s theorem again,
π D(Sn+1) (cid:27) π M = Z. Thus, by Theor⊂em −A in−[G⊂Z02], D(Sn+1) only admits an invariant
1 1
− − −
metricwithnonnegativeRiccicurvature, andahomogeneous isoparametric foliation.
Case (2). 2 m m : According to [Mu¨80], both of the focal submanifolds are minimal
1 2
≤ ≤
submanifolds in Sn+1(1). Moreover, for any unit normal vector η of M+( respectively M ),
−
the principal curvatures of the shape operator A are 1, 1,0. To complete the proof, it is
η
−
sufficient to only consider M+ and prove the induced metric of M+ in Sn+1(1) has positive
Riccicurvature. Choosealocaladaptedmovingframe ei,eα of M+for1 i m1+2m2 and
{ } ≤ ≤
m1 +2m2+1 α n+1, whereei’saretangent to M+ and eα’sare normalto M+. Then, by
≤ ≤
the Gauss equation, for each unit tangent vector X of M+, the Ricci curvature RicM+ is given
by
RicM+(X,X)= 2m +m 1 A X,A X .
2 1 X α α
− − h i
α
For any unit normal vector η of M+, the principal curvatures of the shape operator Aη are
1, 1,0. It follows that A X,A X 1. Therefore, RicM+(X,X) 2(m 1) > 0 if m 2.
α α 2 2
− h i ≤ ≥ − ≥
Now,theproofiscomplete. (cid:3)
Acknowledgements . The authors are very grateful to Professor Zizhou Tang for valuable
discussions. Thesecond author isalso very grateful to Professor Yoshihiro Ohnita for helpful
conversations when he attended the 8th OCAMI-KNUGRG Joint International Workshop on
Submanifold Geometry and RelatedTopics atOsakaCityUniversity in2014. Thefirstauthor
wassupportedinpartbytheNSFC(No. 11331002,No. 11471299),andthesecondauthorwas
partially supported bytheNSFC(No. 11401560 andNo. 11571339).
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