Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting structure Jacobi operators

0.21 MB·

by  Hyunjin Lee

REAL HYPERSURFACES IN COMPLEX HYPERBOLIC
TWO-PLANE GRASSMANNIANS WITH COMMUTING
6 STRUCTURE JACOBI OPERATORS
1
0
2 HYUNJINLEE,YOUNGJINSUHANDCHANGHWAWOO

n
a Abstract. In this paper, we introduce anew commuting condition between
J
thestructureJacobioperatorandsymmetric(1,1)-typetensorfieldT,thatis,

5 RξφT =TRξφ,whereT =AorT =S forHopfhypersurfacesincomplexhy-
2 perbolictwo-planeGrassmannians. Byusingsimultaneous diagonalzation for
commuting symmetricoperators, we giveacomplete classification of real hy-

] persurfaces incomplexhyperbolictwo-plane Grassmannianswithcommuting
G
conditionrespectively.
D
.
h
t
a
m Introduction
[

1 Itisoneofthemaintopicsinsubmanifoldgeometrytoinvestigateimmersedreal
v hypersurfacesofhomogeneoustypeinHermitiansymmetricspacesofrank2(HSS2)
8 withcertaingeometricconditions. Understandingandclassifyingrealhypersurfaces
1
in HSS2 is one of importantproblems in differential geometry. One of these spaces
0
is the complex two-plane Grassmannian G (Cm+2) = SU /S(U U ) defined
8 2 2+m 2 m
0 by the set of all complex two-dimensional linear subspaces in Cm+2.·Another one
. is the complex hyperbolic two-plane GrassmannianG (Cm+2)=SU /S(U U )
2 ∗2 2,m 2· m
0 defined by the set of all complex two-dimensional linear subspaces in indefinite
6 complex Euclidean space Cm+2.
2
1 These aretypicalexamplesofHSS2. Characterizingtypicalmodel spacesofreal
:
v hypersurfacesundercertaingeometricconditionsisoneofourmaininterestsinthe
Xi classification theory in G2(Cm+2) or SU2,m/S(U2·Um) (see [13] and [14]).
Our recent interest is the study by applying geometric conditions used in sub-
r manifolds in G (Cm+2) to submanifolds in SU /S(U U ).
a 2 2,m 2 m
G (Cm+2) = SU /S(U U ) has compact transiti·ve group SU , however
2 2+m 2 m 2+m
·
SU /S(U U ) has noncompact indefinite transitive group SU . This distinc-
2,m 2 m 2,m
·
tion gives various remarkable results.
Thecomplexhyperbolictwo-planeGrassmannianSU /S(U U )istheunique
2,m 2 m
·
noncompact, irreducible, K¨ahler and quaternionic K¨ahler manifold which is not a
hyperk¨ahler manifold.
12010 Mathematics Subject Classification: Primary53C40; Secondary53C15.
2Key words : Real hypersurfaces; complex hyperbolictwo-plane Grassmannians, Hopfhyper-
surface,shapeoperator,Riccitensor,structureJacobioperator,commutingcondition.
*This work was supported by Grant Proj. No. NRF-2015-R1A2A1A-01002459 and the third
authorissupportedbyNRFGrantfundedbytheKoreanGovernment(NRF-2013-FosteringCore
LeadersofFutureBasicScienceProgram).
1
2 HYUNJINLEE,YOUNGJINSUHANDCHANGHWAWOO
Let M be a real hypersurface in complex hyperbolic two-plane Grassmannian
SU /S(U U ). Let N be a local unit normalvector field on M. Since the com-
2,m 2 m
·
plex hyperbolic two-plane Grassmannians SU /S(U U ) has the K¨ahler struc-
2,m 2 m
·
tureJ,wemaydefineaReebvector fieldξ = JN anda1-dimensionaldistribution
−
=Span ξ .
⊥
C { }
Let be the orthogonal complement of distribution in T M at p M. It is
⊥ p
C C ∈
the complex maximal subbundle of T M. Thus the tangent space of M consists of
p
the direct sum of and as follows: T M = . The real hypersurface M
⊥ p ⊥
C C C ⊕C
is said to be Hopf if A , or equivalently, the Reeb vector field ξ is principal
C ⊂ C
withprincipalcurvatureα=g(Aξ,ξ),whereg denotesthemetric. Inthiscase,the
principal curvature α is said to be a Reeb curvature of M.
FromthequaternionicK¨ahlerstructureJ=Span J ,J ,J ofSU /S(U U ),
1 2 3 2,m 2 m
{ } ·
therenaturallyexistalmostcontact3-structurevectorfieldsξ = J N,ν =1,2,3.
ν ν
−
Let = Span ξ ,ξ ,ξ . It is a 3-dimensional distribution in the tangent space
⊥ 1 2 3
Q { }
T M of M at p M. In addition, stands for the orthogonalcomplement of
p ⊥
∈ Q Q
inT M. ItisthequaternionicmaximalsubbundleofT M. Thusthetangentspace
p p
of M can be splitted into and as follows: T M = .
⊥ p ⊥
Q Q Q⊕Q
Thus,wehaveconsideredtwonaturalgeometricconditionsforrealhypersurfaces
in SU /S(U U ) such that the subbundles and of TM are both invariant
2,m 2 m
· C Q
under the shape operator. By using these geometric conditions, we will use the
results in Suh [13, Theorem 1].
Ontheotherhand,aJacobifieldalonggeodesicsofagivenRiemannianmanifold
(M¯,g¯) plays an important role in the study of differential geometry. It satisfies a
well-known differential equation which inspires Jacobi operators. It is defined by
(R¯ (Y))(p) = (R¯(Y,X)X)(p), where R¯ denotes the curvature tensor of M¯ and
X
X,Y denoteanyvectorfieldsonM¯. Itisknowntobeaself-adjointendomorphism
on the tangent space T M¯, p M¯. Clearly, each tangent vector field X to M¯
p
∈
provides a Jacobi operator with respect to X. Thus the Jacobi operator on a real
hypersurface M of M¯ with respect to ξ is said to be a structure Jacobi operator
and will be denoted by R . The Riemannian curvature tensor of M (resp., M¯) is
ξ
denoted by R (resp., R¯).
For a commuting problem concernedwith the structure Jacobioperator R and
ξ
thestructuretensorφofHopfhypersurfaceM inG (Cm+2),thatis,R φA=AR φ,
2 ξ ξ
Lee,Suh andWoo [3]provedthata Hopf hypersurfaceM withR φA=AR φ and
ξ ξ
ξα = 0 is locally congruent to an open part of a tube around a totally geodesic
G (Cm+1) in G (Cm+2). Motivated by this result, we consider the same condition
2 2
in the different ambient space, that is,
(C-1) R φAX =AR φX
ξ ξ
foranytangentvectorfieldX onM inSU /S(U U ). Thegeometricmeaningof
2,m 2 m
·
R φAX =AR φX canbeexplainedinsuchawaythatanyeigenspaceofR onthe
ξ ξ ξ
distribution = X T M X ξ ,p M,isinvariantundertheshapeoperator
p
C { ∈ | ⊥ } ∈
A of M in SU /S(U U ). Then by using [13, Theorem 1], we give a complete
2,m 2 m
·
classification of Hopf hypersurfaces in SU /S(U U ) with R φAX = AR φX
2,m 2 m ξ ξ
·
as follows:
Theorem 1. LetM beaHopf hypersurface in complex hyperbolic two-planeGrass-
mannians SU /S(U U ), m 3 with R φA = AR φ. If the Reeb curvature
2,m 2 m ξ ξ
· ≥
COMMUTING STRUCTURE JACOBI OPERATORS 3
α = g(Aξ,ξ) is constant along the Reeb direction of the structure vector field ξ,
then M is locally congruent to one of the following:
(i) a tube over a totally geodesic SU /S(U U ) in SU /S(U U ) or
2,m 1 2 m 1 2,m 2 m
− · − ·
(ii) a horosphere in SU /S(U U ) whose center at infinity is singular and of
2,m 2 m
·
type JX JX.
∈
FromtheRiemanniancurvaturetensorRofM inSU /S(U U )wecandefine
2,m 2 m
·
the Ricci tensor S of M in such a way that
4m 1
g(SX,Y)= − g(R(e ,X)Y,e ),
i i
i=1
X
where e , ,e denotes a basis of the tangent space T M of M, p M, in
1 4m 1 p
{ ··· − } ∈
SU /S(U U ) (see [15]). Then we can consider another new commuting condi-
2,m 2 m
·
tion
(C-2) R φSX =SR φX
ξ ξ
for any tangent vector field X on M. That is, the operator R φ commutes with
ξ
the Ricci tensor S.
Then by [13, Theorem 1], we also give another classificationrelated to the Ricci
tensor S of M in SU /S(U U ) as follows:
2,m 2 m
·
Theorem 2. LetM beaHopf hypersurface in complex hyperbolic two-planeGrass-
mannians SU /S(U U ), m 3 with R φS = SR φ. If the smooth function
2,m 2 m ξ ξ
· ≥
α = g(Aξ,ξ) is constant along the direction of ξ, then M is locally congruent to
one of the following:
(i) a tube over a totally geodesic SU /S(U U ) in SU /S(U U ) or
2,m 1 2 m 1 2,m 2 m
− · − ·
(ii) a horosphere in SU /S(U U ) whose center at infinity is singular and of
2,m 2 m
·
type JX JX.
∈
Inthispaper,werefer[10],[13],[14]and[15]forRiemanniangeometricstructures
of complex hyperboilc two-plane Grassmannians SU /S(U U ), m 3.
2,m 2 m
· ≥
1. The complex hyperbolic two-plane Grassmannian SU /S(U U )
2,m 2 m
·
Inthissectionwesummarizebasicmaterialaboutcomplexhyperbolictwo-plane
Grassmann manifolds SU /S(U U ), for details we refer to [9], [11], [13] and
2,m 2 m
·
[15]. The Riemannian symmetric space SU /S(U U ), which consists of all
2,m 2 m
·
complex two-dimensional linear subspaces in indefinite complex Euclidean space
Cm+2 is a connected, simply connected, irreducible Riemannian symmetric space
2
of noncompact type and with rank two. Let G = SU and K = S(U U ),
2,m 2 m
·
and denote by g and k the corresponding Lie algebra of the Lie group G and
K respectively. Let B be the Killing form of g and denote by p the orthogonal
complement of k in g with respect to B. The resulting decomposition g = k p is
⊕
a Cartan decomposition of g. The Cartan involution θ Aut(g) on su is given
2,m
∈
by θ(A)=I AI , where
2,m 2,m
I 0
I2,m =(cid:18)0−m,22 I2m,m(cid:19),
I and I denote the identity 2 2-matrix and m m-matrix respectively. Then
2 m
× ×
<X,Y >= B(X,θY) becomes a positive definite Ad(K)-invariantinner product
−
4 HYUNJINLEE,YOUNGJINSUHANDCHANGHWAWOO
ong. ItsrestrictiontopinducesametricgonSU /S(U U ),whichisalsoknown
2,m 2 m
·
as the Killing metric on SU /S(U U ). Throughout this paper we consider
2,m 2 m
·
SU /S(U U ) together with this particular Riemannian metric g.
2,m 2 m
·
The Lie algebra k decomposes orthogonally into k = su su u , where u
2 m 1 1
⊕ ⊕
is the one-dimensional center of k. The adjoint action of su on p induces the
2
quaternionic K¨ahler structure J on SU /S(U U ), and the adjoint action of
2,m 2 m
·
mi I 0
Z = m+2 2 2,m u
(cid:18) 0m,2 m−+2i2Im(cid:19)∈ 1
inducestheK¨ahlerstructureJ onSU /S(U U ). Byconstruction,J commutes
2,m 2 m
·
witheachalmostHermitianstructureJ inJforν =1,2,3. Recallthatacanonical
ν
localbasis J ,J ,J ofaquaternionicK¨ahlerstructureJconsistsofthreealmost
1 2 3
{ }
Hermitian structures J ,J ,J in J such that J J = J = J J , where
1 2 3 ν ν+1 ν+2 ν+1 ν
−
the index ν is to be taken modulo 3. The tensor field JJ , which is locally defined
ν
on SU /S(U U ), is self-adjoint and satisfies (JJ )2 = I and tr(JJ ) = 0,
2,m 2 m ν ν
·
where I is the identity transformation. For a nonzero tangent vector X, we define
RX = λX λ R , CX =RX RJX, and HX =RX JX.
{ | ∈ } ⊕ ⊕
We identify the tangent space T SU /S(U U ) of SU /S(U U ) at o
o 2,m 2 m 2,m 2 m
· ·
with p in the usual way. Let a be a maximal abelian subspace of p. Since
SU /S(U U ) has rank two, the dimension of any such subspace is two. Every
2,m 2 m
·
nonzerotangentvectorX ∈ToSU2,m/S(U2·Um)∼=p is containedin some maximal
abelian subspace of p. Generically this subspace is uniquely determined by X, in
which case X is called regular. If there exist more than one maximal abelian sub-
spaces of p containing X, then X is called singular. There is a simple and useful
characterization of the singular tangent vectors: A nonzero tangent vector X p
∈
is singular if and only if JX JX or JX JX.
∈ ⊥
Up to scaling there exists a unique SU -invariant Riemannian metric g on
2,m
SU /S(U U ). Equipped with this metric, SU /S(U U ) is a Riemannian
2,m 2 m 2,m 2 m
· ·
symmetric space of rank two which is both K¨ahler and quaternionic K¨ahler. For
computational reasons we normalize g such that the minimal sectional curvature
of(SU /S(U U ),g) is 4. The sectionalcurvature K of the noncompactsym-
2,m 2 m
· −
metric space SU /S(U U ) equipped with the Killing metric g is bounded by
2,m 2 m
4 K 0. The sectional c·urvature 4 is obtained for all two-planes CX when X
− ≤ ≤ −
is a non-zero vector with JX JX.
∈
When m = 1, G (C3) = SU /S(U U ) is isometric to the two-dimensional
complex hyperbolic s∗2pace CH2 w1i,t2h con1st·an2t holomorphic sectional curvature 4.
−
When m = 2, we note that the isomorphism SO(4,2) SU yields an isom-
2,2
etry between G (C4) = SU /S(U U ) and the indefinite≃real Grassmann mani-
foldG (R6)ofo∗2rientedtwo-2d,2imensi2o·na2llinearsubspacesofanindefiniteEuclidean
∗2 2
space R6. For this reason we assume m 3 from now on, although many of the
2 ≥
subsequent results also hold for m=1,2.
From now on, hereafter X,Y and Z always stand for any tangent vector fields
on M.
COMMUTING STRUCTURE JACOBI OPERATORS 5
The Riemannian curvature tensor R¯ of SU /S(U U ) is locally given by
2,m 2 m
·
2R¯(X,Y)Z =g(Y,Z)X g(X,Z)Y +g(JY,Z)JX
− −
g(JX,Z)JY 2g(JX,Y)JZ
− −
3
+ g(J Y,Z)J X g(J X,Z)J Y 2g(J X,Y)J Z
ν ν ν ν ν ν
{ − − }
νX=1
3
+ g(J JY,Z)J JX g(J JX,Z)J JY ,
ν ν ν ν
{ − }
νX=1
where J ,J ,J is any canonical local basis of J.
1 2 3
{ }
2. Fundamental formulas in SU /S(U U )
2,m 2 m
·
In this section, we derive some basic formulas and the Codazzi equation for a
real hypersurface in SU /S(U U ) (see [13], [14] and [15]).
2,m 2 m
·
Let M be a real hypersurface in complex hyperbolic two-plane Grassmannian
SU /S(U U ),thatis,ahypersurfaceinSU /S(U U )withrealcodimension
2,m 2 m 2,m 2 m
· ·
one. TheinducedRiemannianmetriconM willalsobedenotedbyg,and denotes
∇
the LeviCivita covariantderivative of(M,g). We denote by and the maximal
C Q
complexandquaternionicsubbundle ofthe tangentbundle TM ofM,respectively.
Now let us put
(2.1) JX =φX +η(X)N, J X =φ X +η (X)N
ν ν ν
foranytangentvectorfieldX ofarealhypersurfaceM inSU /S(U U ),where
2,m 2 m
·
φX denotes the tangential component of JX and N a unit normal vector field of
M in SU /S(U U ).
2,m 2 m
·
Fromthe K¨ahlerstructure J ofSU /S(U U ) there exists analmostcontact
2,m 2 m
·
metric structure (φ,ξ,η,g) induced on M in such a way that
(2.2) φ2X = X +η(X)ξ, η(ξ)=1, φξ =0, η(X)=g(X,ξ)
−
foranyvectorfieldX onM. Furthermore,let J ,J ,J beacanonicallocalbasis
1 2 3
{ }
ofJ. ThenthequaternionicK¨ahlerstructureJ ofSU /S(U U ),togetherwith
ν 2,m 2 m
·
the condition J J = J = J J in section 1, induces an almost contact
ν ν+1 ν+2 ν+1 ν
−
metric 3-structure (φ ,ξ ,η ,g) on M as follows:
ν ν ν
φ2X = X +η (X)ξ , η (ξ )=1, φ ξ =0,
ν − ν ν ν ν ν ν
φ ξ = ξ , φ ξ =ξ ,
ν+1 ν ν+2 ν ν+1 ν+2
(2.3) −
φ φ X =φ X +η (X)ξ ,
ν ν+1 ν+2 ν+1 ν
φ φ X = φ X+η (X)ξ
ν+1 ν ν+2 ν ν+1
−
for any vector field X tangent to M. Moreover, from the commuting property of
J J =JJ ,ν =1,2,3insection1and(2.1),therelationbetweenthesetwocontact
ν ν
metric structures (φ,ξ,η,g) and (φ ,ξ ,η ,g), ν =1,2,3, can be given by
ν ν ν
φφ X =φ φX +η (X)ξ η(X)ξ ,
ν ν ν ν
(2.4) −
η (φX)=η(φ X), φξ =φ ξ.
ν ν ν ν
6 HYUNJINLEE,YOUNGJINSUHANDCHANGHWAWOO
Onthe otherhand,fromtheparallelismofK¨ahlerstructureJ,thatis, J =0and
∇
thequaternionicK¨ahlerstructureJ,togetherwithGaussandWeingartenformulas,
e
it follows that
(2.5) ( φ)Y =η(Y)AX g(AX,Y)ξ, ξ =φAX,
X X
∇ − ∇
(2.6) ξ =q (X)ξ q (X)ξ +φ AX,
X ν ν+2 ν+1 ν+1 ν+2 ν
∇ −
( φ )Y = q (X)φ Y +q (X)φ Y +η (Y)AX
X ν ν+1 ν+2 ν+2 ν+1 ν
(2.7) ∇ −
g(AX,Y)ξ .
ν
−
Combining these formulas, we find the following:
(φ ξ)= (φξ )
X ν X ν
∇ ∇
=( φ)ξ +φ( ξ )
X ν X ν
(2.8) ∇ ∇
=q (X)φ ξ q (X)φ ξ+φ φAX
ν+2 ν+1 ν+1 ν+2 ν
−
g(AX,ξ)ξ +η(ξ )AX.
ν ν
−
Finally, using the explicit expression for the Riemannian curvature tensor R¯ of
SU /S(U U ) in [14], the Codazzi equation takes the form
2,m 2 m
·
2( A)Y +2( A)X =η(X)φY η(Y)φX 2g(φX,Y)ξ
X Y
− ∇ ∇ − −
3
+ η (X)φ Y η (Y)φ X 2g(φ X,Y)ξ
ν ν ν ν ν ν
− −
νX=1(cid:8) (cid:9)
(2.9) 3
+ η (φX)φ φY η (φY)φ φX
ν ν ν ν
−
νX=1(cid:8) (cid:9)
3
+ η(X)η (φY) η(Y)η (φX) ξ ,
ν ν ν
−
νX=1(cid:8) (cid:9)
for any vector fields X and Y on M.
On the other hand, by differentiating Aξ = αξ and using (2.9), we get the
following
3
g(φX,Y) η (X)η (φY) η (Y)η (φX) g(φ X,Y)η (ξ)
ν ν ν ν ν ν
− { − − }
Xν=1
(2.10) =g(( A)Y ( A)X,ξ)
X Y
∇ − ∇
=g(( A)ξ,Y) g(( A)ξ,X)
X Y
∇ − ∇
=(Xα)η(Y) (Yα)η(X)+αg((Aφ+φA)X,Y) 2g(AφAX,Y).
− −
Putting X =ξ gives
3
(2.11) Yα=(ξα)η(Y)+2 η (ξ)η (φY).
ν ν
νX=1
Then, substituting (2.11) into (2.10) the above equation, we have the following
3
α
AφAY = (Aφ+φA)Y + η(Y)η (ξ)φξ +η (ξ)η (φY)ξ
ν ν ν ν
2
νX=1(cid:8) (cid:9)
(2.12)
3
1 1
φY η (Y)φξ +η (φY)ξ +η (ξ)φ Y .
ν ν ν ν ν ν
− 2 − 2
Xν=1(cid:8) (cid:9)
COMMUTING STRUCTURE JACOBI OPERATORS 7
By differentiating and using (2.4), (2.5) and (2.6), we have
(grad α)=X(ξα)ξ+(ξα)φAX
X
∇
3
2 q (X)η (ξ) q (X)η (ξ)+2η (φAX) φξ
ν+2 ν+1 ν+1 ν+2 ν ν
− νX=1n − o
3
2 η (ξ) q (X)φ ξ+q (X)φ ξ+η (ξ)AX
ν ν+1 ν+2 ν+2 ν+1 ν
− νX=1 n−
g(AX,ξ)ξ +φ φAX
ν ν
− o
3
=X(ξα)ξ+(ξα)φAX 4 η (φAX)φξ
ν ν
−
νX=1
3
2 η (ξ) η (ξ)AX g(AX,ξ)ξ +φ φAX .
ν ν ν ν
− νX=1 n − o
By taking the skew-symmetric part to the above equation, we have
0=X(ξα)η(Y) Y(ξα)η(X)+(ξα)g (Aφ+φA)X,Y
−
3 (cid:0) (cid:1)
4 η (φAX)g(φξ ,Y) η (φAY)g(φξ ,X)
ν ν ν ν
− νX=1n − o
3
+2α η (ξ) η(X)η (Y) η(Y)η (X)
ν ν ν
νX=1 n − o
3
2 η (ξ) g(φ φAX,Y) g(φ φAY,X) .
ν ν ν
− νX=1 n − o
From this, by putting X =ξ we have the following
3 3
(2.13) Y(ξα)=ξ(ξα)η(Y)+2α η (ξ)η (Y) 2 η (ξ)η (AY).
ν ν ν ν
−
νX=1 νX=1
From this, if we assume that ξα=0, then it follows that
3 3
η (ξ)η (AX)=α η (ξ)η (X).
ν ν ν ν
νX=1 νX=1
Lemma 2.1. Let M be a Hopf real hypersurface in SU /S(U U ). If the prin-
2,m 2 m
·
cipal curvature α is constant along the direction of ξ, then the distribution or
Q
component of the structure vector field ξ is invariant by the shape operator.
⊥
Q
3. Proof of Theorem 1
Let M be a Hopf hypersurface in SU /S(U U ) with
2,m 2 m
·
(C-1) R φAX =AR φX.
ξ ξ
The structure Jacobi operator R of M is defined by R X = R(X,ξ)ξ for any
ξ ξ
tangent vector X T M, p M (see [1] and [7]). Then for any tangent vector
p
∈ ∈
8 HYUNJINLEE,YOUNGJINSUHANDCHANGHWAWOO
field X on M in SU /S(U U ), we calculate the structure Jacobi operator R
2,m 2 m ξ
·
2R (X)=2R(X,ξ)ξ
ξ
3
(3.1) = X+η(X)ξ+ η (X)ξ η(X)η (ξ)ξ
ν ν ν ν
− −
Xν=1(cid:8)
+3η (φX)φ ξ+η (ξ)φ φX +2αAX 2η(AX)Aξ,
ν ν ν ν
−
(cid:9)
where α denotes the Reeb curvature defined by g(Aξ,ξ).
Lemma 3.1. Let M be a Hopf hypersurface in SU /S(U U ) with the commut-
2,m 2 m
·
ing condition R φAX = AR φX. If the smooth function α is constant along the
ξ ξ
direction of ξ on M, then the Reeb vector field ξ belongs to either the distribution
or the distribution .
⊥
Q Q
Proof. To prove this lemma, without loss of generality, ξ may be written as
(*) ξ =η(X )X +η(ξ )ξ
0 0 1 1
where X (resp., ξ ) is a unit vector in (resp., ) and η(X )η(ξ )=0.
0 1 ⊥ 0 1
Q Q 6
From (*) and φξ =0, we have
φX = η(ξ )φ X ,
0 1 1 0
−
(3.2) φξ1 =φ1ξ =η(X0)φ1X0,

φ φX =η (ξ)X .
1 0 1 0
LetU= p M α(p)=0 beanopensubsetofM. Fromnowon,wediscussour
{ ∈ | 6 }
argumentsonU. ByvirtueofLemma2.1,ξα=0givesAX =αX andAξ =αξ .
0 0 1 1
The equation (2.12) yields αAφX =(α2 2η2(X ))φX by substituting X =X .
0 0 0 0
−
Since α is non-vanishing on U, it becomes
(3.3) AφX =σφX ,
0 0
where σ = α2−2η2(X0).
α
From (3.2) and (3.3), we have
R (X )=α2X α2η(X )ξ,
ξ 0 0 0
−
(3.4) R (ξ )=α2ξ α2η(ξ )ξ,
ξ 1 1 1
 −
R (φX )= α2 4η2(X ) φX .
ξ 0 0 0
−
On U, substituting Xby φX into ((cid:0)C-1), we have(cid:1)
0
(3.5) X η(X )ξ =0,
0 0
−
which is a contradiction. Therefore, U = , and thus it must be p M U. Since
∅ ∈ −
the setM U=Int(M U) ∂(M U),we considerthe followingtwocases. Here
− − ∪ −
Int (resp., ∂) denotes an interior (resp., the boundary) of (M U).
−
Case 1. p Int(M U).
• ∈ −
Ifp Int(M U),thenα=0. Forthiscase,itwasprovedbytheequation(2.11).
∈ −
Case 2. p ∂(M U).
• ∈ −
Since p ∂M U, there exists a sequence of points p such that p p with
n n
∈ − →
α(p) = 0 and α(p )= 0. Such a sequence will have an infinite subsequence where
n
6
η(ξ ) = 0 (in which case ξ at p, by the continuity) or an infinite subsequence
1
∈ Q
where η(X )=0 (in which case ξ at p).
0 ⊥
∈Q
COMMUTING STRUCTURE JACOBI OPERATORS 9
Accordingly, we get a complete proof of our lemma. (cid:3)
FromLemma3.1,weconsiderthecasethatξ belongstothedistribution . Thus
⊥
Q
without loss of generality, we may put ξ = ξ . Differentiating ξ = ξ along any
1 1
direction X TM and using (2.5) and (2.6), it gives us
∈
(3.6) 2η (AX)ξ 2η (AX)ξ +φ AX φAX =0.
3 2 2 3 1
− −
Then, by using the symmetric (resp., skew-symmetric)property ofthe shape oper-
ator A (resp., the structure tensor field φ), we also obtain
(3.7) 2η (X)Aξ 2η (X)Aξ +Aφ X AφX =0.
3 2 2 3 1
− −
Applying φ to (3.6), it implies
1
(3.8) 2η3(AX)ξ3+2η2(AX)ξ2 AX +αη(X)ξ φ1φAX =0.
− −
On the other hand, replacing X =φX into (3.6), we have
(3.9) 2η2(X)Aξ2 2η3(X)Aξ3+Aφ1φX AX αη(X)ξ =0.
− − − −
Lemma 3.2. Let M be a Hopf hypersurface in SU /S(U U ), m 3, with
2,m 2 m
· ≥
R φA = AR φ. If the Reeb vector field ξ belongs to the distribution , then the
ξ ξ ⊥
Q
shape operator A commutes with the structure tensor field φ.
Proof. Applying ξ =ξ into right hand side (resp., left hand side) of (C-1), we get
1
2R φAX = AφX +2αA2φX 2η (X)Aξ +2η (X)Aξ Aφ X,
ξ 3 2 2 3 1
− − −
2AR φX = φAX +2αAφAX 2η (AX)ξ +2η (AX)ξ φ AX.
ξ 3 2 2 3 1
− − −
Combining (3.6) and (3.7), the above equations become
R φAX = AφX +αA2φX,
ξ
−
AR φX = φAX +αAφAX.
ξ
−
Hence, (C-1) is equivalent to
(3.10) Aφ φA=αA(Aφ φA)
− −
Taking the symmetric part of (3.10), we have
(3.11) Aφ φA=α(Aφ φA)A.
− −
From this, we can divide into the following three cases:
First, let us consider an open subset U = p M α(p) = 0 of M. Naturally
{ ∈ | 6 }
we can apply (3.10) and (3.11) on the open subset U.
(Aφ φA)AX =A(Aφ φA)X.
− −
Since the shape operator A and the tensor Aφ φA are both symmetric oper-
−
ators and commute with each other, there exists a common orthonormal basis
E which gives a simultaneous diagonalization. Specifically, we have
i i=1,...,4m 1
{ } −
(3.12) AE =λ E ,
i i i
(3.13) (Aφ φA)E =β E ,
i i i
−
where λ and β are scalars for all i=1,2,...,4m 1.
i i
−
Taking the inner product with E into (3.13), we have
i
(3.14) β g(E ,E )=g (Aφ φA)E ,E =2λ g(φE ,E )=0.
i i i i i i i i
−
(cid:0) (cid:1)
10 HYUNJINLEE,YOUNGJINSUHANDCHANGHWAWOO
Since g(E ,E )=1, β =0 for all i=1,2,...,4m 1. Hence AφX =φAX for any
i i i
−
tangent vector field X on U.
Next, if p Int(M U), then α(p) = 0. From this, the equation (3.11) gives
∈ −
(Aφ φA)X(p)=0.
−
Finally, let us assume that p ∂(M U), where ∂(M U) is the boundary
∈ − −
of M U. Then there exists a subsequence p U such that p p. Since
n n
− { } ⊂ →
(Aφ φA)X(p ) = 0 on the open subset U in M, by the continuity we also get
n
−
(Aφ φA)X(p)=0.
−
Summing up these observations, it is natural that the shape operator A com-
mutes with the structure tensor field φ under our assumption. (cid:3)
By[11]weassertM withtheassumptionsgiveninlemma3.2islocallycongruent
to one of the following hypersurfaces:
( ) a tube over a totally geodesic SU /S(U U ) in SU /S(U U )
A 2,m 1 2 m 1 2,m 2 m
T − · − ·
or,
( ) a horosphere in SU /S(U U ) whose center at infinity is singular and
A 2,m 2 m
H ·
of type JX JX.
∈
Inapaperdueto[11],SuhgavesomeinformationrelatedtotheshapeoperatorA
of and as follows:
A A
T H
PropositionA. LetM beaconnectedrealhypersurface incomplexhyperbolic two-
plane Grassamannian SU /S(U U ), m 3. Assume that the maximal complex
2,m 2 m
≥
subbundle of TM and the maximal quaternionic subbundle of TM are both
C Q
invariant under the shape operator of M. If JN JN, then one of the following
∈
statements holds:
( ) M has exactly four distinct constant principal curvatures
A
T
α=2coth(2r), β =coth(r), λ =tanh(r), λ =0,
1 2
and the corresponding principal curvature spaces are
T =TM , T = , T =E , T =E .
α ⊖C β C⊖Q λ1 −1 λ2 +1
The principal curvature spaces T and T are complex (with respect to J)
λ1 λ2
and totally complex (with respect to J).
( ) M has exactly three distinct constant principal curvatures
A
H
α=2, β =1, λ=0
with corresponding principal curvature spaces
T =TM , T =( ) E , T =E .
α β 1 λ +1
⊖C C⊖Q ⊕ −
Here, E and E are the eigenbundles of φφ with respect to the
+1 1 1
− |Q
eigenvaleus +1 and 1, respectively.
−
Since the symmetric tensor Aφ φA vanishes identically on (resp. ), it
A A
− T H
triviallysatisfies(3.10). Henceweassertthat (resp., )incomplexhyperbolic
A A
T H
two-planeGrassmanniansSU /S(U U )hastheourcommutingcondition(C-1)
2,m 2 m
·
(see [11]).
Next, due to Lemma 3.1, let us suppose that ξ (i.e., JN JN).
∈Q ⊥
By virtue of the result in [13], we assert that a Hopf hypersurface M in complex
hyperbolictwo-planeGrassmanniansSU /S(U U )satisfyingthehypothesesin
2,m 2 m
·
Theorem 1 is locally congruent to