NON-HOPF REAL HYPERSURFACES WITH CONSTANT
PRINCIPAL CURVATURES IN COMPLEX SPACE FORMS
JOS´
E CARLOS D´
IAZ-RAMOS AND MIGUEL DOM´
INGUEZ-V´
AZQUEZ
Abs ac . We classi y eal hype su aces in complex space o ms wi h con-
s an p incipal cu a u es and whose Hop ec o ield has wo non i ial p o-
jec ions on o he p incipal cu a u e spaces.
In complex p ojec i e spaces such eal hype su aces do no exis . In com-
plex hype bolic spaces hese a e holomo phically cong uen o open pa s o
ubes a ound he uled minimal submani olds wi h o ally eal no mal bun-
dle in oduced by Be nd and B ¨uck. In pa icula , hey a e open pa s o
homogenous ones.
1. In oduc ion
A homogeneous submani old o a Riemannian mani old is an o bi o he ac ion o
a closed subg oup o he isome y g oup o he ambien mani old. One o he aims o
submani old geome y is o classi y homogeneous submani olds o a gi en mani old
and o cha ac e ize hem in e ms o geome ic da a. O pa icula in e es a e
homogeneous hype su aces, which a ise as p incipal o bi s o cohomogenei y one
ac ions. Ob iously, homogeneous hype su aces ha e cons an p incipal cu a u es,
ha is, he eigen alues o hei shape ope a o a e cons an . I is an ou s anding
p oblem o de e mine unde which condi ions hype su aces wi h cons an p incipal
cu a u es a e open pa s o homogeneous ones.
In spaces o cons an cu a u e, a hype su ace has cons an p incipal cu a u es
i and only i i is isopa ame ic. The classi ica ion o isopa ame ic hype su aces
was achie ed by Seg e [20] in Euclidean spaces and by Ca an [9] in eal hype bolic
spaces. They all a e open pa s o homogeneous ones. The si ua ion is mo e in ol ed
in sphe es. Ca an classi ied hype su aces wi h g∈ {1,2,3}cons an p incipal cu -
a u es in sphe es. Subsequen ly, Hsiang and Lawson [12] classi ied homogeneous
hype su aces in sphe es; hey ha e g∈ {1,2,3,4,6}p incipal cu a u es. Then,
M¨unzne [18] showed ha g∈ {1,2,3,4,6} o isopa ame ic hype su aces in gen-
e al. Su p isingly, o g= 4 he e a e isopa ame ic hype su aces ha a e no
homogeneous [13]. Recen ly, Cecil, Chi and Jensen [10], and Imme oll [14] showed
ha , wi h a ew possible excep ions, hype su aces wi h g= 4 cons an p incipal
cu a u es a e among he known homogeneous and inhomogeneous examples. Some
p og ess has been made o g= 6 by Ab esch [1] and Do meis e and Nehe [11],
bu he p oblem emains open in ull gene ali y. See [24] o a su ey.
1991 Ma hema ics Subjec Classi ica ion. P ima y 53C40, Seconda y 53C55, 53C35.
Key wo ds and ph ases. Hop hype su aces, homogeneous hype su aces, cons an p incipal
cu a u es.
The i s au ho has been suppo ed by a Ma ie-Cu ie Eu opean Rein eg a ion G an
(PERG04-GA-2008-239162). The second au ho has been suppo ed by he FPU p og amme o
he Spanish Go e nmen . Bo h au ho s ha e been suppo ed by p ojec MTM2009-07756 (Spain).
1
2 J.C. D´
IAZ-RAMOS AND M. DOM´
INGUEZ-V´
AZQUEZ
The p oblem is e en mo e di icul in complex space o ms. See [19] o a su ey
on his and ela ed opics. By c6= 0 we deno e he cons an holomo phic sec ional
cu a u e o a complex space o m; hus, i c > 0 ( esp. c < 0) we ha e a complex
p ojec i e ( esp. hype bolic) space CPn(c) ( esp. CHn(c)). We deno e by Ji s
K¨ahle s uc u e. Le Mbe a eal hype su ace o a complex space o m and ξ
a (local) uni no mal ec o ield. Then, Jξ is angen o Mand is called he
Hop ec o ield o M. The hype su ace Mis said o be Hop i Jξ is a p incipal
cu a u e ec o ield. The mo i a ion o ou wo k is o add ess he classi ica ion
o eal hype su aces wi h cons an p incipal cu a u es in complex space o ms.
We b ie ly summa ize he cu en s a e o he p oblem.
Assume Mis a eal hype su ace o a complex space o m wi h gdis inc con-
s an p incipal cu a u es. Fo p∈Mdeno e by h(p) he numbe o non i ial
p ojec ions o Jξpon o he p incipal cu a u e spaces o M. Clea ly, his unc-
ion is in ege - alued and Mis Hop i and only i h= 1. The classi ica ion o
homogeneous eal hype su aces in complex p ojec i e spaces CPn(c) was de i ed
by Takagi [21]. I ollows om his classi ica ion ha g∈ {2,3,5}. A ema kable
ea u e o homogeneous eal hype su aces in CPn(c) is ha hey a e Hop . Sub-
sequen ly, Takagi classi ied eal hype su aces wi h g∈ {2,3}cons an p incipal
cu a u es [22], [23] ([25] o n= 2, g= 3). I ollows om his wo k ha hey
all a e Hop and open pa s o homogeneous ones. Kimu a [15] classi ied Hop eal
hype su aces wi h cons an p incipal cu a u es in CPnand showed ha hese a e
open pa s o homogeneous ones. No examples a e known o eal hype su aces wi h
cons an p incipal cu a u es in CPn(c) wi h h > 1. Su p isingly, in CHn(c) he e
a e non-Hop homogeneous eal hype su aces. The i s example was disco e ed
by Lohnhe [16] and u he examples we e gi en by Be nd and B ¨uck [3], [4]. We
e e o §2.2 o a b ie in oduc ion and o [7] o a deepe s udy o hei geome y.
Be nd and Tama u ob ained in [8] he classi ica ion o cohomogenei y one ac ions
on CHn(c). The numbe o p incipal cu a u es o he homogeneous examples is
g∈ {2,3,4,5}. Mon iel [17] classi ied eal hype su aces wi h g= 2 cons an p in-
cipal cu a u es in CHn(c) (n≥3). Be nd and he i s au ho sol ed he case
g= 3 and g= 2, n= 2 [5], [6]. I ollows om [17] ha h= 1 when g= 2, and om
[5] and [6] we ge h≤2 i g= 3. Hop eal hype su aces wi h cons an p incipal
cu a u es in CHn(c) we e classi ied by Be nd [2] and hey all a e open pa s o
homogeneous ones. To ou knowledge, [5] and [6] a e he i s classi ica ions o his
kind in ol ing non-Hop eal hype su aces. No hing is known abou hi g≥4.
Ou aim in his pape is o ca y ou he nex na u al s ep a e Be nd and
Kimu a’s classi ica ion o Hop eal hype su aces wi h cons an p incipal cu a u es
in CPn(c) and CHn(c). Thus, we classi y eal hype su aces wi h cons an p incipal
cu a u es whose Hop ec o ield Jξ has h= 2 non i ial p ojec ions on o he
p incipal cu a u e spaces.
Main Theo em. We ha e:
(a) The e a e no eal hype su aces wi h cons an p incipal cu a u es in CPn(c),
n≥2, whose Hop ec o ield has h= 2 non i ial p ojec ions on o he p in-
cipal cu a u e spaces.
(b) Le Mbe a connec ed eal hype su ace in CHn(c),n≥2, wi h cons an p in-
cipal cu a u es and whose Hop ec o ield has h= 2 non i ial p ojec ions
on o he p incipal cu a u e spaces o M. Then, Mhas g∈ {3,4}p incipal
cu a u es and is holomo phically cong uen o an open pa o :
NON-HOPF HYPERSURFACES WITH CONSTANT PRINCIPAL CURVATURES 3
(i) a uled minimal eal hype su ace W2n−1⊂CHn(c)o one o he equidis-
an hype su aces o W2n−1, o
(ii) a ube a ound a uled minimal Be nd -B ¨uck submani old wi h o ally eal
no mal bundle W2n−k⊂CHn(c), o some k∈ {2, . . . , n −1}.
In pa icula , Mis an open pa o a homogeneous eal hype su ace o CHn(c).
The uled minimal submani olds W2n−k⊂CHn(c) a e homogeneous and ha e
o ally eal no mal bundle o ank k∈ {1, . . . , n −1}. Ac ually, W2n−1was dis-
co e ed by Lohnhe [16]. Then, Be nd s udied he geome y o he equidis an
hype su aces o W2n−1[3]. This cons uc ion was gene alized by Be nd and
B ¨uck in [4]. Bo h W2n−1and any o i s equidis an hype su aces ha e g= 3
p incipal cu a u es. The ubes a ound W2n−k,k∈ {2, . . . , n −1}ha e g= 4
p incipal cu a u es i 6= (1/√−c) log(2 + √3) and g= 3 p incipal cu a u es i
= (1/√−c) log(2 + √3). See [7] o a de ailed desc ip ion.
The p oo is as ollows. Fi s we use he Gauss and Codazzi equa ions o de i e
some algeb aic p ope ies o he eigen alue s uc u e o he shape ope a o . The
me hods used o his a e simila o hose o [5], al hough a bi mo e gene al. We
would like o emphasize ha whene e we use a me hod simila o one in [5] we
explici ly poin i ou and skip he de ails as much as possible. On he o he hand,
we ocus on he new echniques and esul s, especially on Subsec ion 3.4. The
mos c ucial s ep o he p oo is o show ha he numbe go cons an p incipal
cu a u es sa is ies g≤4. Fo his we use a no el app oach based on he s udy o
some inequali ies sa is ied by he p incipal cu a u es. Using s anda d Jacobi ield
heo y one can deduce he geome y o he ocal submani olds o hese hype su aces
and hen he esul ollows om a igidi y esul in [7].
The pape is o ganized as ollows. In Sec ion 2 we in oduce he basic elemen s
o ou pape . Subsec ion 2.1 is de o ed o p esen he equa ions o submani old
geome y ha we will use in he es o he pape . In §2.2 we b ie ly desc ibe he
uled minimal Be nd -B ¨uck submani olds W2n−k. We p o e ou Main Theo em in
Sec ion 3. The p oo is di ided in se e al s eps. Some ec o ields and unc ions
a ise na u ally in ou p oo (§3.1 and §3.2). We ge some o hei p ope ies in
Subsec ion 3.3. In §3.4 we show ha he numbe go p incipal cu a u es sa is ies
g∈ {3,4}. We summa ize all he eigen alue s uc u e in §3.5. In Subsec ion 3.6
we use s anda d Jacobi ield heo y o inish he p oo o he Main Theo em.
2. P elimina ies
In his sec ion we in oduce he basic no a ion o his pape . We w i e down he
Gauss and Codazzi equa ions o a hype su ace in a complex space o m and de i e
some basic consequences. Then, we b ie ly men ion how he examples o he Main
Theo em a e cons uc ed.
2.1. The equa ions o a hype su ace. Le ¯
M(c) be a complex space o m
o cons an holomo phic sec ional cu a u e c6= 0 and complex dimension n. I
c > 0 hen ¯
M(c) is a complex p ojec i e space CPn(c) o cons an holomo phic
sec ional cu a u e c. Analogously, i c < 0 hen ¯
M(c) is a complex hype bolic
space CHn(c). We deno e by h·,·i i s inne p oduc , by Ji s K¨ahle s uc u e,
and by ¯
∇i s Le i-Ci i a connec ion. The cu a u e enso is de ined by ¯
R(X, Y ) =
4 J.C. D´
IAZ-RAMOS AND M. DOM´
INGUEZ-V´
AZQUEZ
[¯
∇X,¯
∇Y]−¯
∇[X,Y ], so in his case we ha e
¯
R(X, Y )Z=c
4(hY, ZiX−hX, ZiY+hJY, ZiJX −hJX, ZiJY −2hJX, Y iJZ).
Le Mbe a connec ed submani old o ¯
M(c). We deno e by ∇and Ri s Le i-
Ci i a connec ion and i s cu a u e enso espec i ely. By TM and νM we deno e
he angen and no mal bundles o M. We use he symbol Γ(·) o e e o he
smoo h sec ions o any ec o bundle. Le X, Y, Z, W ∈Γ(TM) and ξ∈Γ(νM).
The second undamen al o m II o Mis de ined by he Gauss o mula as ¯
∇XY=
∇XY+II(X, Y ). The Weinga en o mula is hen w i en as ¯
∇Xξ=−SξX+∇⊥
Xξ,
whe e Sξis he shape ope a o wi h espec o ξand ∇⊥is he induced no mal
connec ion on νM. The second undamen al o m and he shape ope a o a e
ela ed by hSξX, Y i=hII(X, Y ), ξi.
Now le Mbe a connec ed eal hype su ace o ¯
M(c). The wo d ‘ eal’ emphasizes
he ac ha he eal codimension is one. Fix ξ∈Γ(νM) a (local) uni no mal
ec o ield. We w i e Sins ead o Sξ. The Gauss o mula can be ew i en as
¯
∇XY=∇XY+hSX, Y iξ,
and hence, he Weinga en o mula is SX =−¯
∇Xξ. Mo eo e , he Gauss and
Codazzi equa ions o a hype su ace a e
h¯
R(X, Y )Z, W i=hR(X, Y )Z, Wi−hSY, ZihSX, Wi+hSX, ZihSY, Wi,and
h¯
R(X, Y )Z, ξi=h(∇XS)Y−(∇YS)X, Zi.
We assume om now on ha Mhas cons an p incipal cu a u es, ha is, he
eigen alues o he shape ope a o Sa e cons an . Fo each p incipal cu a u e λo
Mwe deno e by Tλ he dis ibu ion on M o med by he p incipal cu a u e spaces
o λalong M.
The Codazzi equa ion implies (see [5, Sec ion 2] o a p oo )
Lemma 2.1.
(i) Le p∈M. I he o hogonal p ojec ion o Jξpon o Tα(p)is nonze o, hen
Tα(p)is a eal subspace o Tp¯
M(c), ha is, JTα(p)is o hogonal o Tα(p).
(ii) Le X, Y ∈Γ(Tα)and Z∈Γ(Tβ)wi h α6=β. Then
h∇XY, Zi=c
4(α−β)(hJY, ZihX, Jξi+hJX, Y ihZ, Jξi+ 2hJX, ZihY, Jξi).
(iii) Le X∈Γ(Tα),Y∈Γ(Tβ)and Z∈Γ(Tγ). Then
h¯
R(X, Y )Z, ξi= (β−γ)h∇XY, Zi−(α−γ)h∇YX, Zi.
The Gauss equa ion implies (again, see [5, Lemma 4] o a p oo )
Lemma 2.2. Le X∈Γ(Tα)and Y∈Γ(Tβ), wi h α6=β, be uni ec o ields.
Then
0=(β−α)(−c−4αβ −2chJX, Y i2+ 8h∇XY, ∇YXi−4h∇XX, ∇YYi)
−4chJX, Y i(XhY, Jξi+YhX, Jξi)
−chX, Jξi(3YhJX, Y i+h∇YX, JY i−2h∇XY, JY i)
−chY, Jξi(3XhJX, Y i−h∇XY, JXi+ 2h∇YX, JXi).
NON-HOPF HYPERSURFACES WITH CONSTANT PRINCIPAL CURVATURES 5
2.2. Discussion o examples. Pa (a) o he Main Theo em s a es ha he e
a e no examples o eal hype su aces wi h cons an p incipal cu a u es in CPn(c)
whose Hop ec o ield has h= 2 non i ial p ojec ions on o he p incipal cu a u e
spaces o M. Thus, we will ocus on desc ibing b ie ly he examples o pa (b) o he
Main Theo em. These examples whe e i s cons uc ed in [4] and hei geome y
was s udied in [7].
The connec ed simple Lie g oup G=SU(1, n) ac s ansi i ely on CHn(c). Fix a
poin o∈CHn(c) and le Kbe he iso opy g oup o Ga o. The subg oup Ko Gis
isomo phic o S(U(1)U(n)). Fu he mo e, (G, K) is a symme ic pai and CHn(c)
may be iden i ied wi h he quo ien G/K. W i e g o he Lie algeb a o Gand k o
he Lie algeb a o K. Le g=k⊕pbe he Ca an decomposi ion o gwi h espec
o o∈CHn(c). We choose a maximal abelian subspace ao p; hen, dim a= 1 since
CHn(c) has ank one. Le g=g−2α⊕g−α⊕g0⊕gα⊕g2αbe he es ic ed oo space
decomposi ion o gwi h espec o aand assume ha αis a posi i e oo . Then, n=
gα⊕g2αis a 2-s ep nilpo en subalgeb a o gisomo phic o he (2n−1)-dimensional
Heisenbe g algeb a. Fu he mo e, g=k⊕a⊕nis an Iwasawa decomposi ion o g. I
Aand Ndeno e he connec ed subg oups o Gwhose Lie algeb as a e aand n, hen
G=KAN is an Iwasawa decomposi ion o G. The sol able g oup AN is simply
connec ed and ac s simply ansi i ely on CHn(c). Thus, we can iden i y a⊕n
wi h ToCHn(c). The Riemannian me ic o CHn(c) induces a le -in a ian me ic
on AN which makes AN isome ic o CHn(c). Simila ly, he complex s uc u e
Jon ToCHn(c) induces a complex s uc u e on a⊕nwhich we also deno e by J.
We ha e Ja=g2α, and gαis J-in a ian . Le B∈abe a uni ec o and de ine
Z=JB ∈g2α.
Le wbe a linea subspace o gαsuch ha he o hogonal complemen w⊥=
gαªwo win gαhas cons an K¨ahle angle ϕ∈(0, π/2], ha is, he angle be ween
J and w⊥is ϕ o all nonze o ∈w⊥. Then, ϕ=π/2 i and only i w⊥is eal,
o equi alen ly, i and only i Jw⊥is o hogonal o w⊥. Le kbe he dimension o
w⊥. Then, s=a⊕w⊕g2αis a subalgeb a o a⊕n. Le Sbe he connec ed simply
connec ed subg oup o AN whose Lie algeb a is s. We de ine he Be nd -B ¨uck
submani olds as [4] (see [16] o k= 1)
W2n−k
ϕ=S·o, and W2n−k=W2n−k
π/2.
The Be nd -B ¨uck submani olds W2n−k
ϕa e homogeneous, ha e no mal bundle
o ank kand cons an K¨ahle angle ϕ∈(0, π/2], and hei second undamen-
al o m II is gi en by he i ial symme ic bilinea ex ension o II(Z, P ξ) =
(sin2(ϕ)√−c/2)ξ o all ξ∈w⊥, whe e Pξ is he o hogonal p ojec ion o Jξ on o
TW2n−k
ϕ. In pa icula , he submani olds W2n−k
ϕa e minimal, and uled by he
o ally geodesic complex hype bolic subspaces de e mined by hei maximal holo-
mo phic dis ibu ion. I ϕ=π/2 hen P=Jand he Be nd -B ¨uck submani olds
ha e o ally eal no mal bundle. Con e sely [7, Theo em 1],
Theo em 2.3. Le Mbe a (2n−k)-dimensional connec ed submani old in CHn(c),
n≥2, wi h no mal bundle νM o cons an K¨ahle angle ϕ∈(0, π/2]. Assume ha
he e exis s a uni ec o ield Z angen o he maximal holomo phic subbundle o
TM such ha he second undamen al o m II o Mis gi en by he i ial symme ic
bilinea ex ension o
II(Z, Pξ) = sin2(ϕ)√−c
2ξ
6 J.C. D´
IAZ-RAMOS AND M. DOM´
INGUEZ-V´
AZQUEZ
o all ξ∈Γ(νM). Then Mis holomo phically cong uen o an open pa o he
uled minimal submani old W2n−k
ϕ.
In pa icula , he Be nd -B ¨uck submani olds W2n−ka e de e mined by he
equa ion II(Z, Jξ) = (√−c/2)ξand he ac ha hei no mal bundle is o ally
eal. Geome ically, hey a e cons uc ed in he ollowing way. Fix a ho osphe e
Hin a o ally geodesic eal hype bolic space RHk+1(c)⊂CHn(c). A ach a each
poin he o ally geodesic CHn−k(c) which is angen o he o hogonal complemen
o he complex span o he angen space o Ha p. The esul ing submani old is
cong uen o W2n−k.
Le N0
K(S) deno e he connec ed componen o he iden i y ans o ma ion o
he no malize o Sin K. Then, N0
K(S)Sac s on CHn(c) wi h cohomogenei y one
and W2n−k
ϕ=N0
K(S)S·o. I k > 1, hen he p incipal o bi s o N0
K(S)Sa e ubes
a ound W2n−k
ϕ. I k= 1, hen ϕ=π/2, he ac ion o N0
K(S)Sis o bi equi alen
o he ac ion o S, and i s o bi s o m a homogeneous olia ion on CHn(c) ha was
i s s udied in [3].
Le Mbe a p incipal o bi o N0
K(S)S. I ϕ∈(0, π/2) hen he Hop ec o ield
o Mhas h= 3 non i ial p ojec ions on o he p incipal cu a u e spaces o M. I
ϕ=π/2, hen he Hop ec o ield o Mhas h= 2 non i ial p ojec ions on o he
p incipal cu a u e spaces o M. The objec i e o pa (b) o he Main Theo em is
o gi e a geome ic cha ac e iza ion o he ubes a ound W2n−k,k∈ {2, . . . , n−1},
and he equidis an hype su aces o W2n−1.
3. P oo o he Main Theo em
In his sec ion we p o e he Main Theo em. Ou main goal is o desc ibe accu-
a ely he eigen alue s uc u e o a eal hype su ace in he condi ions o he Main
Theo em (Theo em 3.12). Then we inish he p oo using s anda d Jacobi ield
heo y (§3.6).
3.1. No a ion and se up. Le Mbe a connec ed eal hype su ace wi h g > 1
dis inc cons an p incipal cu a u es in a complex space o m ¯
M(c). Since he
calcula ions ha ollow a e local we may assume ha we ha e a globally de ined
uni no mal ec o ield ξ. We deno e by λ1, . . . , λg he p incipal cu a u es o M.
By assump ion, he numbe o non i ial p ojec ions o Jξ on o he p incipal
cu a u e dis ibu ions Tλi,i∈ {1, . . . , g}, is h= 2. By elabeling he indices
we may also assume ha Jξ has non i ial p ojec ion on o Tλ1and Tλ2. Hence,
he e exis uni ec o s ields Ui∈Γ(Tλi), i∈ {1,2}, and posi i e smoo h unc ions
bi:M→R,i∈ {1,2}, such ha
Jξ =b1U1+b2U2.
Ob iously, b2
1+b2
2= 1. Mo eo e ,
Lemma 3.1. We ha e g≥3,hJU1, U2i= 0 and he e exis s a uni ec o ield
A∈Γ(⊕g
k=3Tλk)such ha
JUi= (−1)ibjA−biξ, (i, j ∈ {1,2}, i 6=j),
JA =b2U1−b1U2.
P oo . The p oo is simila o ha o [5, Lemma 7], so we jus ske ch i . We will
assume in wha ollows i, j ∈ {1,2},i6=j, and k∈ {3, . . . , g}.
NON-HOPF HYPERSURFACES WITH CONSTANT PRINCIPAL CURVATURES 7
Since Tλi,i∈ {1,2}, is eal by Lemma 2.1 (i) we can w i e JUi=hJUi, UjiUj+
Wij +Pg
k=3 Wik −biξ, whe e Wij ∈Γ(TλjªRUj) and Wik ∈Γ(Tλk). (He e
and hence o h, he symbol ªis used o deno e o hogonal complemen .) F om
Jξ =b1U1+b2U2we ge
−ξ=J2ξ=b2(hJU2, U1iU1+W21)+b1(hJU1, U2iU2+W12)+
g
X
k=3
(b1W1k+b2W2k)−ξ.
Thus, g≥3, hJU1, U2i= 0, W12 =W21 = 0, and b1W1k+b2W2k= 0 o all k. I
we de ine A∈Γ(⊕g
k=3Tλk) by Pg
k=3 Wik = (−1)ibjA, hen he las equali y implies
Pg
k=3 Wjk = (−1)jbiA( ecall i, j ∈ {1,2},i6=j). This gi es he desi ed exp ession
o JUi,i∈ {1,2}. Finally, om b2
1+b2
2= 1 and −U1=J(JU1) = −b2JA−b1Jξ =
−b2JA −U1+b2
2U1−b1b2U2we ob ain JA =b2U1−b1U2.¤
3.2. The ec o ield A.In iew o Lemma 3.1 we may w i e
A=
g
X
k=3
Ak,wi h Ak∈Γ(Tλk), k ∈ {3, . . . g}.
The aim o his subsec ion is o show ha all bu one Aka e ze o and hence we can
assume o example ha A∈Γ(Tλ3) (P oposi ion 3.3). The main di icul y he e is
he ac ha gis no known. We s a wi h he ollowing
Lemma 3.2. Le i, j ∈ {1,2}wi h i6=j. Then we ha e
∇UiUi=
g
X
k=3
(−1)j3cb1b2
4(λk−λi)Ak,∇UiUj=
g
X
k=3
(−1)jµλi−3cb2
i
4(λk−λi)¶Ak.
P oo . Again, his is qui e simila o [5, Lemma 8]. We assume i, j ∈ {1,2},i6=j,
and k∈ {3, . . . , g}. Le Wi∈Γ(TλiªRUi) and Wk∈Γ(TλkªRAk).
Since Uihas uni leng h, h∇UiUi, Uii= 0. Lemma 2.1 (ii) yields h∇UiUi, Uji=
h∇UiUi, Wji=h∇UiUi, Wki= 0, and h∇UiUi, Aki= 3(−1)jcb1b2/(4(λk−λi)).
F om ¯
∇J= 0, he Weinga en o mula, and Lemma 3.1, we ob ain hWi,¯
∇UiJξi=
−λihWi, JUii= 0. Hence, using Jξ =b1U1+b2U2, and Lemma 2.1 (ii), we ge
0 = UihWi, Jξi=h∇UiWi, Jξi+hWi,¯
∇UiJξi=−bih∇UiUi, Wii.
Since bi6= 0 he exp ession o ∇UiUi ollows.
As Ujhas uni leng h, h∇UiUj, Uji= 0. F om Lemma 2.1 (ii) we ob ain
h∇UiUj, Uii=h∇UiUj, Wii= 0. Now, he Weinga en o mula and Lemma 3.1
imply hWj,¯
∇UiJξi=−λihWj, JUii= 0, and hus, Lemma 2.1 (ii), yields
0 = UihWj, Jξi=h∇UiWj, Jξi+hWj,¯
∇UiJξi=bjh∇UiWj, Uji.
This implies h∇UiWj, Uji= 0. A simila calcula ion gi es h∇UiWk, Uji= 0. Fi-
nally, by Lemma 2.1 (ii), and Lemma 3.1 we ha e
0 = UihAk, Jξi=h∇UiAk, Jξi+hAk,¯
∇UiJξi
= (−1)i3cb2
ibj
4(λk−λi)−bjh∇UiUj, Aki−(−1)iλibj,
om whe e we ge h∇UiUj, Aki. Al oge he his yields he o mula o ∇UiUj.¤
Now we can p o e he main esul o his sec ion.
P oposi ion 3.3. A∈Γ(Tλk) o some k∈ {3, . . . , g}.
8 J.C. D´
IAZ-RAMOS AND M. DOM´
INGUEZ-V´
AZQUEZ
P oo . On he con a y, assume ha he e exis s a poin p∈Mand wo dis inc
in ege s , s ∈ {3, . . . , g}such ha (A )p,(As)p6= 0. Hence, in a neighbo hood o
pwe ha e A , As6= 0 as well. We will wo k in ha neighbo hood om now on.
Applying Lemma 2.1 (iii) o he ec o ields U1,U2, and Ak,k∈ { , s}, and
using Lemma 3.2 we easily ge
(1) 3c(λ2−λk)
4(λ1−λk)b2
1+3c(λ1−λk)
4(λ2−λk)b2
2=−c
4−λ1(λ2−λk)−λ2(λ1−λk), k ∈ { , s}.
Toge he wi h b2
1+b2
2= 1, his yields a linea sys em o h ee equa ions wi h
unknowns b2
1and b2
2. This sys em mus be compa ible. We show i is de e mined
( ha is, i has a unique solu ion). I i we e no , he ank o he sys em would, a
mos , be one. In pa icula ,
¯¯¯¯¯
3c(λ2−λk)
4(λ1−λk)
3c(λ1−λk)
4(λ2−λk)
1 1 ¯¯¯¯¯
= 3c(λ2−λ1)(λ1+λ2−2λk)
4(λ1−λk)(λ2−λk)= 0, k ∈ { , s},
which implies λ1+λ2−2λk= 0, k∈ { , s}, and hence λ =λs, con adic ion. We
conclude ha he abo e sys em is de e mined. The e o e, we can ind an exp ession
o b2
1and b2
2in e ms o he p incipal cu a u es and c. Since hese a e cons an ,
i ollows ha b1and b2a e cons an .
We ake i, j ∈ {1,2},i6=j, and k∈ { , s}. Since biis cons an and Uihas uni
leng h, using Jξ =b1U1+b2U2, he Weinga en o mula, and Lemma 3.1 we ge
0 = Ak(bi) = AkhUi, Jξi=h∇AkUi, Jξi+hUi,¯
∇AkJξi=bjh∇AkUi, Uji−(−1)jbjλk,
and hus, h∇AkUi, Uji= (−1)jλk. Taking his, Lemma 3.1, and Lemma 3.2 in o
accoun , Lemma 2.1 (iii) o Ak,U1and U2yields
c
4(2b2
2−b2
1) = h¯
R(Ak, U1)U2, ξi= (λ1−λ2)λk+ (λk−λ2)µλ1−3cb2
1
4(λk−λ1)¶,
o k∈ { , s}. We can ea ange his as:
(2) µc
4−3c(λk−λ2)
4(λk−λ1)¶b2
1−c
2b2
2= (λ2−λ1)λk+λ1(λ2−λk), k ∈ { , s}.
Hence, (1), (2), and b2
1+b2
2= 1 gi e a linea sys em o i e equa ions wi h unknowns
b2
1and b2
2. This sys em is compa ible by assump ion, so i has ank wo. Then, all
mino s o o de h ee o he augmen ed ma ix o he sys em anish. This implies
( ake (1), (2), and b2
1+b2
2= 1, wi h k∈ { , s}, and hen bo h equa ions in (2) and
b2
1+b2
2= 1):
3c(λ1−λ2)2(−12λ2
k+ 8λ1λk+ 8λ2λk+c−4λ1λ2)
16(λ1−λk)(λk−λ2)= 0, k ∈ { , s},(3)
3c(λ2−λ1)(λ −λs)(4λ2
1−4λ λ1−4λsλ1+c+ 2λ2λ + 2λ2λs)
8(λ1−λ )(λ1−λs)= 0.(4)
In pa icula , (3) implies −12λ2
k+ 8λ1λk+ 8λ2λk+c−4λ1λ2= 0. Pu ing k=
and k=s, and sub ac ing, we ge 4(2λ1+ 2λ2−3λ −3λs)(λ −λs) = 0, om
whe e we ob ain λ +λs= 2(λ1+λ2)/3. Taking his in o accoun , (4) gi es
(4λ2
1−4λ1λ2+ 4λ2
2+ 3c)/3 = 0. The disc iminan o −12λ2
k+ 8λ1λk+ 8λ2λk+c−
4λ1λ2= 0 as a quad a ic equa ion in λkis p ecisely 16(4λ2
1−4λ1λ2+ 4λ2
2+ 3c), so
his disc iminan anishes. As a consequence, his quad a ic equa ion has a unique
NON-HOPF HYPERSURFACES WITH CONSTANT PRINCIPAL CURVATURES 9
solu ion and hence λ =λs. This is a con adic ion. The e o e, all bu one Ak,
k∈ {3, . . . , g}, a e ze o o each p. The esul ollows by con inui y. ¤
3.3. Some p ope ies o he p incipal cu a u e spaces. In iew o P oposi-
ion 3.3, we may assume om now on ha A∈Γ(Tλ3). Mo eo e , we can choose
an o ien a ion on Mand a elabeling o he indices so ha
λ1< λ2,and λ3≥0.
We will ollow his con en ion om now on.
Fi s we calcula e some co a ian de i a i es.
Lemma 3.4. Le i, j ∈ {1,2}wi h i6=j. Then we ha e
∇UiUi= (−1)j3cb1b2
4(λ3−λi)A,(5)
∇UiUj= (−1)jµλi−3cb2
i
4(λ3−λi)¶A,(6)
∇UiA= (−1)i3cb1b2
4(λ3−λi)Ui+ (−1)iµλi−3cb2
i
4(λ3−λi)¶Uj,(7)
∇AUi=(−1)j
λi−λjÃc(2b2
j−b2
i)
4+ (λj−λ3)µλi−3cb2
i
4(λ3−λi)¶!Uj,(8)
∇AA= 0.(9)
P oo . The p oo is simila o ha o [5, Lemma 8]. Equa ions (5) and (6) a e a
di ec consequence o Lemma 3.2 and P oposi ion 3.3. Assume i, j ∈ {1,2},i6=j,
and k∈ {4, . . . , g}. Le Wi∈Γ(TλiªRUi), W3∈Γ(Tλ3ªRA) and Wk∈Γ(Tλk).
Acco ding o (5) and (6), in o de o p o e (7) we ha e o show h∇UiA, Ai= 0
(ob ious because Ais a uni ec o ield), and h∇UiA, Wli= 0 o all l∈ {1, . . . , g}.
The la e ollows om ¯
∇J= 0, he Weinga en o mula, Lemma 3.1, and (5), wi h
0 = UihJUi, Wli=h¯
∇UiJUi, Wli+hJUi,¯
∇UiWli
=−h∇UiUi, JWli+ (−1)ibjhA, ∇UiWli−bihξ, ¯
∇UiWli= (−1)jbjh∇UiA, Wli.
We now p o e (8). Ob iously, h∇AUi, Uii= 0, and by Lemma 2.1 (ii) we ge
h∇AUi, Ai= 0. Applying Lemma 2.1 (iii) o A,Uiand Uj, using Lemma 3.1 and
(6), gi es
c
4(−1)i(b2
i−2b2
j) = (λi−λj)h∇AUi, Uji−(λ3−λj)(−1)iµλi−3cb2
i
4(λ3−λi)¶,
om whe e we ge h∇AUi, Uji. Fo l∈ {j, 3, . . . , g}, a simila a gumen wi h
Lemma 2.1 (iii) applied o A,Ui, and Wl, aking Lemma 3.1 and (7) in o accoun ,
yields h∇AUi, Wli= 0. Finally, he p e ious equali y (in e changing iand jand
pu ing l=i) gi es
0 = AhWi, Jξi=h∇AWi, Jξi+hWi,¯
∇AJξi
=bih∇AWi, Uii+bjh∇AWi, Uji−λ3hWi, JAi=−bih∇AUi, Wii.
Al oge he his p o es (8).
16 J.C. D´
IAZ-RAMOS AND M. DOM´
INGUEZ-V´
AZQUEZ
I we examine he p oo o ou heo em, so a we ha e ac ually shown ha
o any poin p∈M he e exis s a neighbo hood o pwhe e he conclusion o
Theo em 3.12 is sa is ied. Howe e , by he connec edness o Mand a con inui y
a gumen , i can be easily shown ha Mis o ien able and ha he conclusion o
Theo em 3.12 is sa is ied globally.
3.6. Jacobi ield heo y and igidi y o ocal submani olds. In his las sec-
ion we inish he p oo o pa (b) o he Main Theo em. Since we use s anda d
Jacobi ield heo y, we p o ide he eade jus wi h he undamen al de ails and
skip he long calcula ions. Acco ding o [5] we jus ha e o ake ca e o he case
g= 4. Howe e , i is no much o e load o deal wi h he wo cases simul aneously,
so o he sake o comple eness we will do so in wha ollows.
Le Mbe a eal hype su ace o CHn(c) in he condi ions o Theo em 3.12 (b).
Fo ∈Rwe de ine he map Φ :M→CHn(c), p7→ expp( ξp), whe e exppis he
Riemannian exponen ial map o CHn(c) a p. Then, Φ (M) is ob ained by mo ing
Ma dis ance along i s no mal di ec ion. The singula i ies o Φ a e he ocal
poin s o M. We will ind a pa icula dis ance o which Φ
∗has cons an ank,
whe e Φ
∗deno es he di e en ial o Φ . Then we will apply Theo em 2.3 o Φ (M)
o his choice o . This way, Φ (M) will be an open pa o he uled minimal
Be nd -B ¨uck submani old W2n−k,k∈ {1, . . . , n −1}, and hence Mwill be an
open pa o a ube a ound his uled minimal submani old W2n−k. (I k= 1 hen
Mwill be an equidis an hype su ace o he uled minimal hype su ace W2n−1
a dis ance .)
Le p∈Mand deno e by γp he geodesic de e mined by he ini ial condi ions
γp(0) = pand ˙γp(0) = ξp. Fo any ∈TpMle B be he pa allel ec o ield
along he geodesic γpsuch ha B (0) = , and le ζ be he Jacobi ield along
γpwi h ini ial condi ions ζ (0) = and ζ0(0) = −Sp . He e 0deno es co a ian
de i a i e along γp. Since ζ is a solu ion o he di e en ial equa ion 4ζ00
+cζ +
3chζ , J ˙γpiJ˙γp= 0, i ∈Tλi(p) hen
ζ ( ) = i( )B ( ) + h , Jξigi( )J˙γp( ),
whe e
i( ) = cosh µ √−c
2¶−2λi
√−csinh µ √−c
2¶,
gi( ) = µcosh µ √−c
2¶−1¶µ1 + 2 cosh µ √−c
2¶−2λi
√−csinh µ √−c
2¶¶.
We also de ine he smoo h ec o ield η along Φ by η
p= ˙γp( ). I is known ha
ζ ( ) = Φ
∗ and ζ0
( ) = ¯
∇Φ
∗ η .
We now de e mine he alue o . Since 0 ≤λ3<√−c/2 we can ind a eal
numbe ≥0 such ha
λ3=√−c
2 anh µ √−c
2¶.
Le p∈M. We de ine ui= (Ui)p,i∈ {1,2}. Le 2∈Tλ2(p)ªRu2and
k∈Tλk(p) o 3 ≤k≤g(whene e hese spaces a e non i ial). The explici
solu ion o he Jacobi equa ion abo e implies
(Φ
∗u1,Φ
∗u2) = (Bu1( ), Bu2( ))D( ),
NON-HOPF HYPERSURFACES WITH CONSTANT PRINCIPAL CURVATURES 17
Φ
∗ 2= 0,Φ
∗ 3= sech µ √−c
2¶B 3( ),Φ
∗ 4= 0,
whe e
D( ) = µ 1( ) + b2
1g1( )b1b2g2( )
b1b2g1( ) 2( ) + b2
2g2( )¶.
Since de (D( )) = sech3¡ √−c/2¢we conclude ha Φ
∗has cons an ank 2n−k
(see Theo em 3.12 (b )-(b i) o he de ini ion o k). Then, o each poin p∈M
he e exis s an open neighbo hood Vo psuch ha W= Φ (V) is an embedded
submani old o CHn(c) and Φ :V → W is a subme sion. (I k= 1, hen Φ is
ac ually a local di eomo phism.)
Le q= Φ (p)∈ W. The exp ession abo e o Φ
∗shows ha he angen space
TqWo Wa qis ob ained by pa allel ansla ion o Ru1⊕Ru2⊕Tλ3(p) along he
geodesic γp om p=γp(0) o q=γp( ). The e o e, he no mal space νqWo W
a qis ob ained by pa allel ansla ion o (ke Φ
∗p)⊕Rξpalong γp om p=γp(0)
o q=γp( ). The la e is (Tλ2ªRu2)⊕Rξpi g= 3 (see Theo em 3.12 (b i)), o
Tλ4(p)⊕Rξpi g= 4 (see Theo em 3.12 (b )). In any case, by Theo em 3.12 (b )-
(b i) i ollows ha Whas o ally eal no mal bundle o ank k.
We ha e ha η
p=Bξp( ) is a uni no mal ec o o Wa q. I S deno es
he shape ope a o o W, hen i is known ha S
η
pΦ
∗ =−(ζ0
( ))>, whe e (·)>
deno es o hogonal p ojec ion on o he angen space o W. Using he explici
exp ession o ζ abo e, we ge
(S
η
pBu1( ), S
η
pBu2( )) = (Bu1( ), Bu2( ))C( ),and
S
η
pB 3( ) = 0 o all 3∈Tλ3(p),
whe e C( ) = −D0( )D( )−1. A leng hy and edious calcula ion shows ha
C( ) = √−c
2µ−2b1b2b2
1−b2
2
b2
1−b2
22b1b2¶.
Since Jη
p=BJξp( ) = b1Bu1( ) + b2Bu2( ), and BJAp( ) = b2Bu1( )−b1Bu2( ),
he abo e exp ession o C( ) implies
S
η
pBJAp( ) = −√−c
2Jη
p, S
η
pJη
p=−√−c
2BJAp( ),
and S
η
p anishes on he o hogonal complemen o RJη
p⊕RBJAp( ) in TqW.
We ha e ha J(νqW ªRη
p) is con ained in he pa allel ansla ion along γpo
Tλ3(p). This ollows om Theo em 3.12 (b )-(b i) and he ac ha νqW ª Rη
p
is he pa allel ansla ion along γp om γp(0) = p o γp( ) = qo Tλ2(p)ªRu2i
g= 3, and o Tλ4(p) i g= 4. The linea i y o S
ηp
implies
(13) S
η
pJ˜η=−√−c
2hη
p,˜ηiBJAp( ), o all ˜η∈νqW.
I ollows om he Gauss o mula and ¯
∇J= 0 ha S
˜ηJη
p=S
η
pJ˜η, and hence,
S
˜ηJη
p= 0 o all ˜η∈νqW ª Rη
p. Le αbe a cu e in (Φ )−1({q})∩ V wi h
α(0) = p. Since η
pand η
α( )−hη
α( ), η
piη
pa e pe pendicula , S
˜ηJη
p= 0, and he
linea i y o η7→ S
ηimply
0 = S
η
α( )−hη
α( ),η
piη
pJη
p=S
η
α( )Jη
p+√−c
2hη
α( ), η
piBJAp( ),
18 J.C. D´
IAZ-RAMOS AND M. DOM´
INGUEZ-V´
AZQUEZ
which oge he wi h (13) (wi h α( ) ins ead o p) yields
−√−c
2hη
α( ), η
piBJAp( ) = S
η
α( )Jη
p=−√−c
2hη
α( ), η
piBJAα( )( ).
Since αis a bi a y we ge ha he map ˜p7→ BJA˜p( ) is cons an in he connec ed
componen V0o (Φ )−1({q})∩V con aining p. Thus i makes sense o de ine he
uni ec o z=−BJA˜p( )∈TqW o any ˜p∈ V0.
We may conside η as a map om V0 o he uni sphe e o νqW. The angen
space o V0a pis gi en by he ke nel o Φ
∗p. I ∈ke Φ
∗p, hen η
∗p =ζ0
( ). I
g= 3, hen ∈ke Φ
∗p=Tλ2(p)ªRu2, and η
∗p =−p−c/2B ( ). I g= 4, hen
∈ke Φ
∗p=Tλ4(p), and η
∗p =−csch( √−c/2)B ( ). In any case, we ge ha
η is a local di eomo phism om V0in o he uni sphe e o νqW(no e ha his is
i ial i g= 3 and k= 1). Hence, η (V0) is an open subse o he uni sphe e o
νqW. Bu since η7→ S
ηdepends analy ically on ηwe conclude
S
ηJη =√−c
2z, S
ηz=√−c
2Jη, S
η = 0,
o all uni η∈νqW, and ∈TqWª(RJη⊕Rz). The e o e, he second undamen al
o m II o Wa qis gi en by he i ial symme ic bilinea ex ension o II (z, Jη) =
(√−c/2)η o all η∈νqW. By cons uc ion, zdepends smoo hly on he poin
q∈ W and hence gi es ise o a ec o ield Zwhich is angen o he maximal
holomo phic dis ibu ion o W. The ela ion S
ηJη = (√−c/2)Zensu es ha Zcan
ac ually be de ined on Φ (M), and hence, he second undamen al o m o Φ (M)
is gi en by he i ial symme ic bilinea ex ension o II (Z, Jη) = (√−c/2)η o all
η∈Γ(νΦ (M)). Since Φ (M) has o ally eal no mal bundle o ank kwe conclude
om Theo em 2.3, and he ema k ha ollows, ha Φ (M) is holomo phically
cong uen o an open pa o he uled minimal Be nd -B ¨uck submani old W2n−k.
This eadily implies ha Mis an open pa o a ube (an equidis an hype su ace
i g= 3 and k= 1) o adius a ound he uled minimal Be nd -B ¨uck submani old
W2n−k.
Finally, le us poin ou ha i g= 3 and λ3= 0, hen = 0 and Mis an open
pa o he uled minimal hype su ace W2n−1. Also, i g= 3 and k > 1 hen λ3=
√−c/(2√3) acco ding o Theo em 3.12 b( i)B, and hence = (1/√−c) log(2+√3).
The ube a ound he uled minimal submani old W2n−k,k > 1, o adius =
(1/√−c) log(2+√3) has g= 3 p incipal cu a u es whe eas i 6= (1/√−c) log(2+
√3) he ube o adius a ound he uled minimal submani old W2n−k,k > 1, has
g= 4 p incipal cu a u es. This inishes he p oo o he Main Theo em.
Re e ences
1. U. Ab esch, Isopa ame ic hype su aces wi h ou o six dis inc p incipal cu a u es. Neces-
sa y condi ions on he mul iplici ies. Ma h. Ann. 264 (1983), 283–302.
2. J. Be nd , Real hype su aces wi h cons an p incipal cu a u es in complex hype bolic space.
J. Reine Angew. Ma h. 395 (1989), 132–141.
3. J. Be nd , Homogeneous hype su aces in hype bolic spaces. Ma h. Z. 229 (1998), 589–600.
4. J. Be nd , M. B ¨uck, Cohomogenei y one ac ions on hype bolic spaces. J. Reine Angew. Ma h.
541 (2001), 209–235.
5. J. Be nd , J.C. D´ıaz-Ramos, Real hype su aces wi h cons an p incipal cu a u es in complex
hype bolic spaces. J. London Ma h. Soc. 74 (2006), 778–798.
6. J. Be nd , J.C. D´ıaz-Ramos, Real hype su aces wi h cons an p incipal cu a u es in he
complex hype bolic plane. P oc. Ame . Ma h. Soc. 135 (2007), 3349–3357.
NON-HOPF HYPERSURFACES WITH CONSTANT PRINCIPAL CURVATURES 19
7. J. Be nd , J.C. D´ıaz-Ramos, Homogeneous hype su aces in complex hype bolic spaces. Geom.
Dedica a 138 (2009), 129-150.
8. J. Be nd , H. Tama u, Cohomogenei y one ac ions on noncompac symme ic spaces o ank
one. T ans. Ame . Ma h. Soc. 359 (2007), 3425–3438.
9. ´
E. Ca an, Familles de su aces isopa am´e iques dans les espaces `a cou bu e cons an e. Ann.
Ma . Pu a Appl., IV. Se . 17 (1938), 177–191.
10. T. Cecil, Q.-S. Chi, G. Jensen, Isopa ame ic hype su aces wi h ou p incipal cu a u es.
Ann. o Ma h. (2) 166 (2007), 1–76.
11. J. Do meis e , E. Nehe , Isopa ame ic hype su aces, case g= 6, m= 1. Commun. Algeb a
13 (1985), 2299–2368.
12. W.-Y. Hsiang, H.B. Lawson J ., Minimal submani olds o low cohomogenei y. J. Di e en ial
Geom. 5(1971), 1–38.
13. D. Fe us, H. Ka che , H.-F. M¨unzne , Cli o dalgeb en und neue isopa ame ische Hy-
pe l¨achen. Ma h. Z. 177 (1981), 479–502.
14. S. Imme oll, On he classi ica ion o isopa ame ic hype su aces wi h ou p incipal cu a-
u es in sphe es. Ann. o Ma h. (2) 168 (2008), 1011–1024.
15. M. Kimu a, Real hype su aces and complex submani olds in complex p ojec i e space. T ans.
Ame . Ma h. Soc. 296 (1986), 137–149.
16. M. Lohnhe , H. Reckziegel, On uled eal hype su aces in complex space o ms. Geom.
Dedica a 74 (1999), 267–286.
17. S. Mon iel, Real hype su aces o a complex hype bolic space. J. Ma h. Soc. Japan 37 (1985),
515–535.
18. H.F. M¨unzne , Isopa ame ische Hype l¨achen in Sph¨a en. Ma h. Ann. 251 (1980), 57–71.
19. R. Niebe gall, P.J. Ryan, Real Hype su aces in Complex Space Fo ms. Tigh and Tau Sub-
mani olds, MSRI Publica ions, Volume 32, 1997.
20. B. Seg e, Famiglie di ipe supe icie isopa ame iche negli spazi euclidei ad un qualunque nu-
me o di dimensioni. A i Accad. Naz. Lincei Rend. Cl. Sci. Fis. Ma . Na u . (6) 27 (1938),
203–207.
21. R. Takagi, On homogeneous eal hype su aces in a complex p ojec i e space. Osaka J. Ma h.
10 (1973), 495–506.
22. R. Takagi, Real hype su aces in a complex p ojec i e space wi h cons an p incipal cu a-
u es. J. Ma h. Soc. Japan 27 (1975), 43–53.
23. R. Takagi, Real hype su aces in a complex p ojec i e space wi h cons an p incipal cu a-
u es, II. J. Ma h. Soc. Japan 27 (1975), 507–516.
24. G. Tho be gsson, A su ey on isopa ame ic hype su aces and hei gene aliza ions. Hand-
book o di e en ial geome y, Vol. I, 963–995 (No h-Holland, Ams e dam, 2000).
25. Q.M. Wang, Real hype su aces wi h cons an p incipal cu a u es in complex p ojec i e
spaces I. Sci. Sinica Se . A 26 (1983), 1017–1024.
Depa men o Geome y and Topology, Uni e si y o San iago de Compos ela,
Spain.
E-mail add ess:[email p o ec ed]
Depa men o Geome y and Topology, Uni e si y o San iago de Compos ela,
Spain.
E-mail add ess:[email p o ec ed]
