Publ. Ma h. Deb ecen
61 / 3-4 (2002), 523–532
Riemannian subme sions and slan submani olds
By JOS´
E L. CABRERIZO (Se illa), ALFONSO CARRIAZO (Se illa),
LUIS M. FERN´
ANDEZ (Se illa) and MANUEL FERN´
ANDEZ (Se illa)
Abs ac . We s udy he ela ionship be ween slan submani olds in bo h Com-
plex and Con ac Geome y h ough Riemannian subme sions. We p esen some con-
s uc ion p ocedu es o ob ain slan submani olds in he uni sphe e and in a S ie el
mani old. We also gene alize hem by means o he Boo hby–Wang ib a ion. Finally,
we show some cha ac e iza ion heo ems o h ee-dimensional slan submani olds.
0. In oduc ion
The geome y o slan submani olds has been inc easingly s udied
since B.-Y. Chen de ined slan imme sions in complex mani olds as a
na u al gene aliza ion o bo h holomo phic and o ally eal imme sions
(see [7]). La e , a simila no ion o slan submani old was in oduced in
Con ac Geome y, which is specially impo an o submani olds angen
o he s uc u e ec o ield o a con ac me ic mani old. The pu pose o
he p esen pape is o s udy he close ela ionship be ween bo h heo ies
h ough Riemannian subme sions.
In pa icula , we p o e ha , in some condi ions, a submani old o an
almos He mi ian mani old is slan i and only i i s li by a Riemannian
subme sion is a slan submani old o an almos con ac me ic mani old.
Ma hema ics Subjec Classi ica ion: 53C15, 53C40.
Key wo ds and ph ases: Riemannian subme sion, Kaehle ian mani old, Sasakian man-
i old, slan submani old.
The au ho s wish o hank P o . Manuel Ba os o his aluable sugges ions and help ul
ema ks.
The au ho s a e pa ially suppo ed by he PAI p ojec (Jun a de Andaluc´ıa, Spain,
2001).
524 Jos´e L. Cab e izo e al.
We use his esul as a me hod o ind in e es ing examples. Examples o
p ope slan submani olds o a Sasakian-space- o m o cons an φ-sec ional
cu a u e cha e been gi en in [4], [10] (c=−3) and [6] (c < −3), bu , un il
now, he e we e no examples in a Sasakian-space- o m wi h c > −3. In
ac , in his pape we exhibi a cons uc ion p ocedu e o ob ain examples
o slan submani olds in he uni sphe e wi h i s usual Sasakian s uc u e
(c= 1). A e wa ds, we ex end i in o de o ge ample examples in
Sasakian-space- o ms wi h cons an φ-sec ional cu a u e c, o any c > −3.
Mo eo e , we also cons uc examples o slan imme sions in o a S ie el
mani old and we gene alize bo h p ocedu es by using he Boo hby–Wang
ib a ion. Finally, we p esen some classi ica ions o h ee-dimensional slan
submani olds o R5, by a ending o hei second undamen al o m.
1. P elimina ies
In his sec ion, we ecall some basic o mulas and de ini ions abou
slan submani olds in bo h Complex and Con ac Geome y, which we shall
use la e . Fo de ails and backg ound on complex and con ac mani olds,
we e e o he s anda d e e ences [1], [13].
A submani old No an almos He mi ian mani old ( e
N, g, J) is said
o be slan [7] i o each nonze o ec o X angen o Na p, he angle
θ(X), 0 ≤θ(X)≤π/2, be ween JX and TpNis a cons an , called he
slan angle o he submani old. In pa icula , holomo phic and o ally
eal submani olds appea as slan submani olds wi h slan angle 0 and
π/2, espec i ely. A slan submani old is called p ope slan i i is nei he
holomo phic no o ally eal. In he case whe e Nis a Riemann su ace and
e
Nis a Kaehle mani old, S. S. Che n and J. G. Wol son in oduced
he no ion o Kaehle angle, de ined o be he angle be ween J∂/∂x and
∂/∂y, whe e z=x+√−1yis a local complex coo dina e on N[9]. I is
clea ha i Nis a su ace wi h cons an Kaehle angle α, hen i is a slan
submani old wi h slan angle θsa is ying θ=α( esp. θ=π−α) when
α∈[0, π/2] ( esp. α∈(π/2, π]).
Pu JX =PX +FX, o any angen ec o ield X, whe e PX ( esp.
FX) deno es he angen ial ( esp. no mal) componen o JX. Then, θ-
slan submani olds a e cha ac e ized by he o mula:
P2=−cos2θId .
Riemannian subme sions and slan submani olds 525
A special ype o p ope slan submani old is ha o Kaehle ian slan
submani old, i.e., a p ope slan submani old sa is ying ∇0P= 0, whe e
∇0deno es he Le i–Ci i a connec ion on N. I is easy o show ha a
Kaehle ian slan submani old is a Kaehle ian mani old wi h espec o he
induced me ic and wi h he almos complex s uc u e gi en by (sec θ)P.
In a simila way, gi en a submani old M angen o he s uc u e
ec o ield ξo an almos con ac me ic mani old (
M, φ, ξ, η, G), i is
said o be slan [4] i he angle θ(X) be ween φX and TpMis a cons an ,
which is independen o he choice o p∈Mand X∈Tp(M) Span(ξp).
In pa icula , o θ= 0 ( esp. θ=π/2) we ob ain he in a ian ( esp.
an i-in a ian ) submani olds. Now, i we deno e by TX ( esp. NX) he
angen ial ( esp. no mal) componen o φX, he e is an equa ion which
cha ac e izes θ-slan submani olds:
T2=−cos2θ(Id −η⊗ξ).
In con ac geome y, he simila no ion o Kaehle ian slan submani-
olds is gi en by p ope θ-slan submani olds sa is ying
(∇XT)Y= cos2θ(g(X, Y )ξ−η(Y)X),
o any angen ec o ields X, Y , whe e ∇deno es he Le i–Ci i a con-
nec ion on M. This non- i ial ac is shown in [4]. The e o e, by ol-
lowing he complex case no a ion, we call such a submani old a Sasakian
slan submani old. On he o he hand, he possibili y o ob aining an in-
duced con ac me ic s uc u e on a slan submani old o a con ac me ic
mani old is s udied in [5].
2. Main esul s
Le
Mbe a (2m+ 1)-dimensional almos con ac me ic mani old
wi h s uc u e enso s (φ, ξ, η, G) and e
Nbe a eal 2m-dimensional almos
He mi ian mani old wi h s uc u e (J, g). Le suppose ha he e exis s a
Riemannian subme sion π:
M→e
Nsa is ying he condi ions:
i) The e ical subspace Vpo he subme sion a p∈
Mis equal o
he span o ξp,
ii) φX∗= (JX)∗,
526 Jos´e L. Cab e izo e al.
o any ec o ield Xon e
N, whe e ∗deno es he ho izon al li wi h espec
o π. In ac , since πis a Riemannian subme sion, we also ha e
iii) G(X∗, Y ∗) = g(X, Y ),
o any ec o ields X, Y on e
N.
Now, le Mbe an (n+ 1)-dimensional submani old angen o he
s uc u e ec o ield ξo
Mand Nbe an n-dimensional submani old
o e
N. Th oughou in his sec ion we assume ha he ollowing diag am
commu es
(2.1)
M−−−−→
M


y

yπ
N−−−−→ e
N
whe e Mis he se o ib es o e N.
Then, we s a e he ollowing heo em:
Theo em 2.1. In he abo e condi ions, we ha e:
(a) Mis θ-slan in
Mi and only i Nis θ-slan in e
N.
Mo eo e , i
Mis a Sasakian mani old, we also ha e:
(b) Mis Sasakian θ-slan in
Mi and only i Nis Kaehle ian θ-slan
in e
N.
P oo . S a emen (a) ollows di ec ly om i)–iii). In ac , in any
almos con ac me ic mani old, φξ = 0, and hen, he condi ion o M
being a slan submani old is eally ela ed o i s con ac dis ibu ion, which
is he ho izon al subspace o he subme sion a any poin .
Now, suppose ha
Mis a Sasakian mani old and deno e by ∇( esp.
∇0) he Le i–Ci i a connec ion on M( esp. N). I ollows om he well-
known O’Neill equa ions o he subme sion ha
∇X∗Y∗= (∇0
XY)∗+η(∇X∗Y∗)ξ, η(∇X∗Y∗) = −G(X∗, TY ∗),
o any ec o ields X, Y on e
N angen o N. Then, we ha e
(∇X∗T)Y∗= ((∇0
XP)Y)∗−G(X∗, T2Y∗)ξ, (∇ξT)Y∗= 0,
which imply (b). ¤
Riemannian subme sions and slan submani olds 527
No ice ha , in pa icula , s a emen (a) o Theo em 2.1 implies s a e-
men s (3) and (4) o [13, P oposi ion 3.2, p. 459].
By using Theo em 2.1, we can show he ollowing cons uc ion p oce-
du e o gi ing examples o p ope slan submani olds in he uni sphe e.
Le π:S2m+1 →CPm(4) be he well-known Hop ib a ion, whe e
S2m+1 ( esp. CPm(4)) is endowed wi h i s usual Sasakian ( esp. Kaehle-
ian) s uc u e. Gi en any isome ic imme sion :N→CPm(4), hen
M=π−1(N) is a p incipal ci cle bundle o e Nwi h o ally geodesic i-
b es and he li ˆ
:M→S2m+1 o is an isome ic imme sion such ha
he ollowing diag am commu es:
Mˆ
−−−−→ S2m+1


y

yπ
N
−−−−→ CPm(4).
I ollows om Theo em 2.1 ha , in o de o ob ain a θ-slan subman-
i old o S2m+1, i is enough o conside a θ-slan submani old o CPm(4).
Fo example, we could ake he examples gi en in [11]. The abo e p oce-
du e was i s poin ed ou by B.-Y. Chen and Y. Tazawa in [8].
We can also conside he li o he Ve onese sequence in o de o ge
new examples o p ope slan imme sions in o S2m+1. We ecall ha he
Ve onese sequence ψ0, . . . , ψmis de ined, o any p= 0, . . . , m, by
ψp:S2→CPm:ψp[z0, z1] = [gp,0(z0/z1), . . . , gp,m(z0/z1)],
whe e [z0, z1]∈CP1=S2, and
gp,j(z) = p!
(1 + zz)psµm
j¶zj−pX
k
(−1)kµj
p−k¶µm−j
k¶(zz)k,
o any j= 0, . . . , m. I was shown in [2] ha e e y ψpis a con o mal
minimal imme sion wi h cons an cu a u e and cons an Kaehle angle
αpsuch ha
an2αp
2=p(m−p+ 1)
(p+ 1)(m−p).
By combining his p ocedu e and a D-homo he ic de o ma ion, we
may also ob ain he ollowing heo em, simila o [6, Theo em 3.5]:

528 Jos´e L. Cab e izo e al.
Theo em 2.2. Le cbe a cons an wi h c > −3. Then, he e ex-
is p ope slan submani olds in a Sasakian-space- o m wi h cons an φ-
sec ional cu a u e c.
P oo . Fi s , we can choose a p ope slan submani old o S2m+1,
gi en by he abo e cons uc ion p ocedu e. We deno e he usual Sasakian
s uc u e on S2m+1 by (φ, ξ, η, G). Then, o any c > −3, we conside he
cons an a= 4/(c+ 3) >0 and he D-homo he ic de o ma ion:
e
φ=φ, e
ξ=1
aξ, eη=aη, e
G=aG +a(a−1)η⊗η.
I was shown in [1] ha S2m+1 wi h his s uc u e is a Sasakian-space-
o m wi h cons an φ-sec ional cu a u e (4/a)−3 = c.
Finally, i is easy o p o e ha a D-homo he ic de o ma ion maps
slan submani olds in o slan submani olds. ¤
A mo e elabo a e cons uc ion p ocedu e o ob aining slan sub-
mani olds in a ce ain almos con ac me ic mani old can be shown as
ollows. Le Hbe he closed connec ed subg oup in S3×S3gi en by
H={(z, z) : z∈S1}and conside he homogeneous space (S3×S3)/H.
Since his is a compac simply connec ed 5-dimensional spin mani old wi h
H2((S3×S3)/H;Z) = Z, i ollows om a classic esul o Smale [12] ha
i is di eomo phic o S2×S3. On he o he hand, i we deno e by V(2,4)
he S ie el mani old o o hono mal 2- ames in 4-space, i is known ha
V(2,4) is di eomo phic o (S3×S3)/H and hen, he e is a di eomo -
phism :V(2,4) →S2×S3. Le eπ:S3→S2be he Hop ib a ion and
pu F= (id ×eπ)◦ . Hence, F:V(2,4) →Q2is a subme sion, whe e
Q2deno es he complex quad ic S2×S2. Now, pu S2
∗=S2 {(0,0,1)}
and le E:S2
∗→Cbe he co esponding s e eog aphic p ojec ion, which
p ese es he complex s uc u e o C.
Then, V(2,4)∗→Q2∗→C2is a subme sion, whe e Q2∗( esp.
V(2,4)∗) deno es he mani old S2
∗×S2
∗( esp. F−1(Q2∗)). I is clea ha ,
i we conside on C2i s usual Kaehle s uc u e, V(2,4)∗can be endowed
wi h a na u al almos con ac me ic s uc u e such ha (E, E)◦F|V(2,4)∗
is a Riemannian subme sion sa is ying he abo e s a ed condi ions i)–ii).
Hence, we ob ain ample examples o slan su aces in V(2,4)∗by consid-
e ing he li s o slan su aces in C2(see, o ins ance, [7]).
Mo eo e , we can gi e a gene aliza ion o he p e ious cons uc ion
p ocedu es. Le
Mbe a (2m+ 1)-dimensional compac egula con ac
Riemannian subme sions and slan submani olds 529
mani old. Acco ding wi h a classical esul o Boo hby–Wang [3], one
can see
Mas a ci cle bundle o e a 2m-dimensional compac symplec ic
mani old e
N:
π:
M−→ e
N.
Since e
Nca ies a global symplec ic o m Ω, he e exis a Riemannian
me ic gand a enso ield Jo ype (1,1) such ha (g, J) is an almos
Kaehle s uc u e on e
Nwi h Ω as i s undamen al 2- o m. Deno e by η he
con ac o m on
Mwi h d η=π∗Ω and ξi s cha ac e is ic ec o ield and
de ine a enso ield φand a Riemannian me ic Gon
Mby φX = (Jπ∗X)∗
and G=π∗(g)+η⊗η, espec i ely. Then, i can be p o ed ha (φ, ξ, η, G)
is a K-con ac s uc u e on
Mand π: (
M, G)→(e
N, g) is a Riemannian
subme sion. Now, we ha e:
Theo em 2.3. In he abo e condi ions, le Mbe a submani old o
M.
Then, Mis a S1-in a ian θ-slan submani old i and only i M=π−1(N),
whe e Nis a θ-slan submani old o e
N.
P oo . Le Nbe a submani old o e
Nand deno e by M=π−1(N).
Then, Mis a submani old o
Mand he cha ac e is ic ec o ield ξis
angen o M, in pa icula , Mis S1-in a ian .
The con e se o he abo e s a ed ac also holds, ha is, i Mis a
S1-in a ian submani old o
M, hen ξis angen o Mand he e exis s a
submani old Nin e
Nwi h M=π−1(N). Hence, he p oo concludes by
applying Theo em 2.1. ¤
3. Some applica ions
We now p oceed o show some applica ions o he abo e s a ed e-
la ionship be ween slan submani olds and Riemannian subme sions, by
conside ing he di e en ial map gi en by
π:R5−→ C2; (x1, x2, y1, y2, z)7−→ 1
2(y1, y2, x1, x2).
I is easy o see ha , i we ha e on R5( esp. C2) i s usual Sasakian ( esp.
Kaehle ian) s uc u e, hen πis a Riemannian subme sion sa is ying con-
di ions i)–ii). The e o e, by using his subme sion, we can ob ain examples
o slan submani olds in R5by aking he li s o Examples 2.1, 2.3, 2.4
530 Jos´e L. Cab e izo e al.
and 2.5 o [7]. No ice ha hose examples will be simila o Examples 3.7–
3.10 o [4].
Now, suppose ha we ha e a 3-dimensional submani old M angen
o he s uc u e ec o ield on R5and a su ace Nin C2sa is ying dia-
g am (2.1). Then, we ha e he ollowing classi ica ion heo em:
Theo em 3.1. In he abo e condi ions, Mis a 3-dimensional slan
submani old o R5wi h pa allel mean cu a u e ec o i and only i Mis
one o he ollowing submani olds:
(a) a submani old locally isome ic o an open po ion o he p oduc o
a plane ci cle and a ci cula cylinde .
(b) a submani old locally isome ic o an open po ion o he p oduc o
a ci cula cylinde and R.
(c) a minimal slan submani old in R5.
Mo eo e , i ei he case (a) o case (b) occu s, hen Mis an an i-in a ian
submani old.
P oo . Fi s , i is known ha i he mean cu a u e ec o o Mis
pa allel hen he mean cu a u e ec o o Nis also pa allel, and ha M
is minimal i and only i Nis minimal (see, o ins ance, [13, p. 462–463]).
Hence, he p oo o his heo em ollows om Theo em 1.1 o [7, p. 50] and
by aking in o accoun ha , i Mis an an i-in a ian submani old, hen
η(∇X∗Y∗) = 0, o any X,Y angen o N, which means ha , in his
case, Mis locally isome ic o he Riemannian p oduc o Nand R.¤
In he same condi ions, we can also classi y he submani old Ma -
ending o a pa icula beha iou o i s second undamen al o m σ:
Theo em 3.2. Mis a 3-dimensional slan submani old o R5sa is y-
ing
(3.1) (∇Xσ)(Y, Z) = G(Y, TX)NZ +G(Z, TX)NY
o any angen ec o ields X, Y, Z o hogonal o ξ, i and only i Mis
one o he ollowing submani olds:
(a) a submani old locally isome ic o an open po ion o he p oduc o
a plane ci cle and a ci cula cylinde .
(b) a submani old locally isome ic o an open po ion o he p oduc o
a ci cula cylinde and R.
Riemannian subme sions and slan submani olds 531
(c) a li by πo an open po ion o a plane in C2.
Mo eo e , i ei he case (a)o case (b)occu s, hen Mis an an i-in a ian
submani old.
P oo . I ollows om he O’Neill equa ions ha
(∇X∗σ)(Y∗, Z∗) = ((∇Xσ0)(Y, Z))∗+G(Y∗, TX∗)NZ∗+G(Z∗, TX∗)NY ∗,
o any X, Y, Z angen o N, whe e σ0deno es he second undamen al
o m o N, and so, Msa is ies (3.1) i and only i σ0is pa allel.
The e o e, his p oo wo ks as ha o Theo em 3.1, by applying now
Theo em 1.2 o [7, p. 51]. ¤
A su icien condi ion o a submani old M, in he abo e condi ions,
o sa is y equa ion (3.1) is o be o ally con ac geodesic, i.e., such ha
σ(X, Y ) = η(X)σ(Y, ξ) + η(Y)σ(X, ξ),
o any angen ec o ields Xand Y. In ac he e a e examples o o ally
con ac geodesic slan submani olds in R5(see, o ins ance, Example 3.7
o [4]).
Re e ences
[1]
D. E. Blai
, Con ac Mani olds in Riemannian Geome y, Lec u e No es in Ma h-
ema ics, 509, Sp inge -Ve lag,New Yo k, 1976.
[2]
J. Bol on, G. R. Jensen, M. Rigoli
and
L. M. Woodwa d
, On con o mal
minimal imme sions o
S
2in o
CP
n,Ma h. Ann. 279 (1988), 599–620.
[3]
W. M. Boo hby
and
H. C. Wang
, On con ac mani olds, Ann. o Ma h. 68
(1958), 721–734.
[4]
J. L. Cab e izo, A. Ca iazo, L. M. Fe n
andez
and
M. Fe n
andez
, Slan
submani olds in Sasakian mani olds, Glasgow Ma h. J. 42 (2000), 125–138.
[5]
J. L. Cab e izo, A. Ca iazo, L. M. Fe n
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M. Fe n
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, S uc-
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(2000), 857–864.
[6]
J. L. Cab e izo, A. Ca iazo, L. M. Fe n
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M. Fe n
andez
, Exis-
ence and uniqueness heo em o slan imme sions in Sasakian-space- o ms, Publ.
Ma h. Deb ecen 58 no. 4 (2001), 559–574.
[7]
B. Y. Chen
, Geome y o Slan Submani olds, Ka holieke Uni e si ei Leu en,
Leu en, 1990.
[8]
B. Y. Chen
and
Y. Tazawa
, Slan submani olds o complex p ojec i e and com-
plex hype bolic spaces, Glasgow Ma h. J. 42 (2000), 439–454.