No e di Ma ema ica 20, n. 2, 2000/2001, 125–133.
The con ac Whi ney sphe e
Da id E. Blai
Depa men o Ma hema ics, Michigan S a e Uni e si y,
Eas Lansing, Michigan 48824-1027, USA
[email p o ec ed]
Al onso Ca iazo∗
Depa amen o de Geome ´ıa y Topolog´ıa, Facul ad de Ma em´a icas, Uni e sidad de Se illa,
Apdo. Co eos 1160, 41080–Se illa, Spain
[email p o ec ed]
Recei ed: 2 Ma ch 2001; accep ed: 27 Ma ch 2001.
Abs ac . In his pape , we in oduce he con ac Whi ney sphe e as an imbedding o he
n–dimensional uni sphe e as an in eg al submani old o he s anda d con ac s uc u e on
R2n+1. We ob ain a gene al inequali y o in eg al submani olds in R2n+1, in ol ing bo h he
scala cu a u e and he mean cu a u e, and we use he equali y case in o de o cha ac e ize
he con ac Whi ney sphe e. We also s udy a simila p oblem o an i-in a ian submani olds
o R2n+1, angen o he s uc u e ec o field.
Keywo ds: In eg al submani olds, Whi ney sphe es, scala cu a u e, mean cu a u e
MSC 2000 classi ica ion: 53C15, 53C40, 53D10.
In oduc ion
One o he mos in e es ing opics in con ac geome y is he s udy o in eg al
submani olds, i.e., submani olds imme sed in a con ac mani old, such ha he
con ac o m es ic ed o he submani old anishes. In pa icula , 1–dimensional
in eg al submani olds a e called Legend e cu es.
A well-known p ope y o Legend e cu es o he s anda d con ac s uc u e
dz−ydxon R3is ha he p ojec ion ¯γo a closed Legend e cu e γin R3
o he xy-plane mus ha e sel -in e sec ions and algeb aic (signed) a ea ze o.
On he o he hand, we can hink o he pai o γand i s p ojec ion ¯γin he
ollowing e ms. Suppose ha γi sel does no ha e sel -in e sec ions and ega d
¯γas a Lag angian submani old in C∼
=R2wi h sel -in e sec ions; hen hink o
going om ¯γ o γas a way o emo ing he singula i y bu p ese ing he
“Lag angian-Legend e” p ope y.
A gene aliza ion o his can be done wi h he Whi ney sphe es. The Whi ney
sphe es a e usually defined as a amily o Lag angian imme sions o he uni
∗Pa ially suppo ed by he PAI p ojec (Jun a de Andaluc´ıa, Spain 1999)
126 D. E. Blai , A. Ca iazo
sphe e Sn, cen e ed a he o igin o Rn+1,in oCn∼
=R2ngi en by
(u0,u
1,...,u
n)→
1+u2
0
(u1,...,u
n,u
0u1,...,u
0un)+B, (1)
whe e is a posi i e numbe and Bis a ec o o Cn. The numbe and he
ec o Ba e called he adius and he cen e o he Whi ney sphe e, espec i ely.
F om a opological poin o iew, i is well-known ha he sphe e can no be
imbedded in Cnas a Lag angian submani old. The Whi ney sphe es ha e he
bes possible beha iou , because hey ha e only one double poin a he poles
o Sn.
In he con ac mani old R2n+1 wi h i s usual con ac me ic s uc u e, we
ha e he ollowing p esen a ion o he Whi ney sphe es as a amily o imbedded
sphe es and in eg al submani olds o he con ac s uc u e
(u0,u
1,...,u
n)→
1+u2
0
(u0u1,...,u
0un,u
1,...,u
n, u0
1+u2
0
+C(1 + u2
0)) + B,
whe e is a posi i e numbe , Bis a ec o o R2n+1 and Cis a eal cons an .
We e e o hese sphe es as he con ac Whi ney sphe es. The e o e, wi h his
p esen a ion we ha e emo ed he p e ious singula i y a he poles o Sn.
On he o he hand, he Whi ney sphe es in Cnha e an in e es ing geome -
ic p ope y. I was p o en by Bo elli, Chen and Mo an [3] and independen ly
by Ros and U bano [6] ha i Mnis a Lag angian submani old o Cn,wi h
mean cu a u e ec o Hand scala cu a u e τ, hen |H|2≥2(n+2)
n2(n−1)τ,wi h
equali y i and only i Mis ei he o ally geodesic o a (piece o a) Whi ney
sphe e. Mo eo e , in [4], Cas o cons uc s a one-pa ame e amily o Lag angian
sphe es including he Whi ney sphe e, such ha hey sa is y a geome ic equal-
i y o ype τ=µ|H|2,wi hµ>0.
In his pape , we es ablish an analogue o he abo e esul o in eg al sub-
mani olds in R2n+1 wi h i s s anda d Sasakian s uc u e, such ha he equali y
case holds o he con ac Whi ney sphe es. We also gi e a cha ac e iza ion by
he second undamen al o m, simila o ha o Ros and U bano in [6].
Finally, we s udy he same p oblem o an i-in a ian submani olds angen
o he s uc u e ec o field on R2n+1 and ob ain he co esponding esul s.
1 P elimina ies.
Le (R2n+1,φ,ξ,η,g) deno e he mani old R2n+1 wi h i s usual Sasakian
s uc u e gi en by
η=1
2(dz −
n

i=1
yidxi),ξ=2
∂
∂z,
The con ac Whi ney sphe e 127
g=η⊗η+1
4
n

i=1
(dxi⊗dxi+dyi⊗dyi),
φ(
n

i=1
(Xi
∂
∂xi
+Yi
∂
∂yi
)+Z∂
∂z)=
n

i=1
(Yi
∂
∂xi
−Xi
∂
∂yi
)+
n

i=1
Yiyi
∂
∂z,
whe e (xi,y
i,z),i=1...n a e he ca esian coo dina es.
I is well-known ha (R2n+1,φ,ξ,η,g) is a Sasakian-space- o m, wi h con-
s an φ–sec ional cu a u e equal o −3. Hence, i s cu a u e enso 
Ris gi en
by 
R(X,Y )Z=−η(X)η(Z)Y+η(Y)η(Z)X−g(X,Z)η(Y)ξ+
+g(Y,Z)η(X)ξ−g(Z, φY )φX +g(Z, φX)φY −2g(X,φY )φZ, (2)
o any ec o fields X,Y, Z. Fo mo e de ails and backg ound, we e e o [2].
Le Mbe an n-dimensional Riemannian mani old isome ically imme sed in
he Sasakian-space- o m R2n+1. We also deno e by g he me ic on M.
Le ∇( esp. 
∇) be he Le i–Ci i a connec ion o M( esp. R2n+1). Then,
he Gauss–Weinga en o mulas a e gi en by

∇XY=∇XY+σ(X,Y ),
∇XV=−AVX+DXV,
o any angen ec o fields Xand Yand any no mal ec o field V, whe e
Dis he connec ion in he no mal bundle, σis he second undamen al o m
o Mand Ais he shape ope a o . The mean cu a u e ec o His defined by
H=(1/n) ace σ.
A submani old Mis called an in eg al submani old i η es ic ed o M
anishes. I is well-known ha he con ac subbundle {η=0}admi s in eg al
submani olds up o and including dimension n, bu o no highe dimension, see
e.g. [2]. A di ec consequence o his defini ion is ha φX is a no mal ec o
field, o any angen ec o field X, i.e., Mis an an i-in a ian submani old o
R2n+1. Hence, in a neighbo hood o e e y poin p∈M,wecanconside alocal
o hono mal ame {e1,...,e
n,e
1∗,...,e
n∗,ξ}, such ha e1,...,e
na e angen
o Mand ei∗=φei, o any i=1,...,n. Such a ame is called a Legend e
ame. I we pu σk
ij =g(σ(ei,e
j),e
k∗), i can be p o ed ha
σi
jk =σk
ji =σj
ik,(3)
o any i, j, k =1,...,n.
On he o he hand, we ha e [2, p. 128]
Aξ=0,(4)
equi alen ly σ(X, Y ) is pe pendicula o ξ, o any angen ec o fields Xand
Y.
128 D. E. Blai , A. Ca iazo
2 Cha ac e izing he Con ac Whi ney Sphe e
Le Mbe an n–dimensional submani old o R2n+1. Gi en a angen o hono -
mal ame {e1,...,e
n}, he scala cu a u e τo Mis defined by
τ=
i<j
K(ei∧ej),
whe e K(ei∧ej) is he sec ional cu a u e o he plane sec ion spanned by ei
and ej.
By ollowing he same s eps as in he p oo o Lemma 1 o [3], and by i ue
o (2), (3) and (4), we ob ain he ollowing inequali y o in eg al submani olds,
in ol ing bo h cu a u es |H|and τ:
P oposi ion 1. Le Mnbe an in eg al submani old o R2n+1. Then, he
squa e o i s mean cu a u e |H|2and i s scala cu a u e τsa is y a each
poin he ollowing inequali y:
|H|2≥2(n+2)
n2(n−1)τ. (5)
The equali y holds i and only i he e exis s a eal unc ion λdefined on M,
such ha he second undamen al o m σo Msa isfies
σ(e1,e
1)=3λe1∗,σ(e2,e
2)=... =σ(en,e
n)=λe1∗,
σ(e1,e
j)=λej∗,σ(ej,e
k)=0,2≤j=k≤n, (6)
whe e {e1,...,e
n,e
1∗,...,e
n∗,ξ}is a Legend e ame such ha e1∗is pa allel o
H.
On he o he hand, we can also cha ac e ize he equali y case o he abo e
inequali y by he beha io o he second undamen al o m o he submani old.
We ob ain a o mula simila o ha o [6]:
P oposi ion 2. Le Mnbe an in eg al submani old o R2n+1. Then, M
sa isfies he equali y case o (5) a e e y poin , i and only i
σ(X,Y )= n
n+2{g(X, Y )H+g(φX, H)φY +g(φY, H)φX},(7)
o any angen ec o fields Xand Y.
P oo . I he equali y case o (5) holds o e e y p∈M, hen, in a neigh-
bo hood o e e y poin , we can find a Legend e ame wi h e1∗pa allel o Hand
such ha i sa isfies (6), wi h λ=n
n+2|H|. Hence, i ollows ha (7) holds o
any angen ec o fields Xand Y. The con e se can be e ified di ec ly. QED
The con ac Whi ney sphe e 129
We now p oceed o show ha he only non- i ial example o an in eg al sub-
mani old sa is ying he equali y in (5), is he con ac Whi ney sphe e. Fi s , we
s a e he ollowing wo lemmas. The fi s one can be easily p o ed by s aigh -
o wa d compu a ion.
Lemma 1. Le π:R2n+1 →Cnbe he diffe en ial map gi en by:
π(x1,...,x
n,y
1,...,y
n,z)=1
2(y1,...,y
n,x
1,...,x
n).(8)
Then, π:(R2n+1,φ,ξ,η,g)→(Cn,J,G)is a Riemannian subme sion, whe e we
deno e by (Cn,J,G) he usual Kaehle ian s uc u e on Cn. Mo eo e , i sa isfies
he ollowing condi ions:
i) The e ical subspace Vpo he subme sion a p∈R2n+1 is equal o he
span o ξp;
ii) g=π∗G+η⊗η;
iii) φX =(Jπ∗X)∗, o any ec o field Xon R2n+1, whe e ∗deno es he
ho izon al li wi h espec o η.
Lemma 2 (Uniqueness o he li o he Whi ney sphe e). Le π:
R2n+1 →Cnbe he Riemannian subme sion gi en by (8),andw:Sn→Cna
Whi ney imme sion gi en by (1).I ψ:Sn→R2n+1 is an in eg al imme sion,
such ha π◦ψ=w, hen ψis he con ac Whi ney imme sion gi en by
ψ(u0,u
1,...,u
n)= 2
1+u2
0
(u0u1,...,u
0un,u
1,...,u
n,2 u0
1+u2
0
+C(1+u2
0))+ ˜
B,
whe e Cis a cons an and ˜
Bisa ec o o R2n+1 such ha π(˜
B)=B.
P oo . Since π◦ψ=wand n
i=0 u2
i= 1, we can w i e ψas
ψ(u0,u
1,...,u
n)= 2
1+u2
0
(u0u1,...,u
0un,u
1,...,u
n, (u1,...,u
n))+ ˜
B, (9)
o any cons an ec o ˜
Bsuch ha π(˜
B)=B.
Then, diffe en ia ing (9), we ob ain, o any i=1,...,n:
∂
∂ui
=2 
u2
0(1 + u2
0)−u2
i(1 −u2
0)
u0(1 + u2
0)2
∂
∂xi
−
j=i
uiuj(1 −u2
0)
u0(1 + u2
0)2
∂
∂xj
+
+1+u2
0+2u2
i
(1 + u2
0)2
∂
∂yi
+2
j=i
uiuj
(1 + u2
0)2
∂
∂yj
+
2ui +(1+u2
0)∂
∂ui
(1 + u2
0)2
∂
∂z



.(10)

130 D. E. Blai , A. Ca iazo
Since ψis an in eg al imme sion in R2n+1,weha eη(∂
∂ui)=0, o any
i=1,...,n, whe e ηis he con ac o m on R2n+1. Hence, i ollows om (10)
ha he unc ion mus sa is y he sys em o pa ial diffe en ial equa ions
∂
∂ui
(
1+u2
0
)=2 ui(3u2
0−1)
u0(1 + u2
0)3,i=1,...,n,
which implies ha
=2 u0
1+u2
0
+C(1 + u2
0),
whe e Cis a eal cons an . QED
We can now s a e he main heo em:
Theo em 1. Le Mnbe an in eg al submani old o he s anda d con ac
s uc u e on R2n+1. Then, he equali y case o (5) holds a e e y poin p∈M,
i and only i ei he Mis a o ally geodesic submani old o i is a po ion o a
con ac Whi ney sphe e.
P oo . Le ψ:M→R2n+1 be an in eg al imme sion and pu 
ψ=π◦ψ.
Then, i ollows om Lemma 1 ha 
ψ:M→Cnis a Lag angian imme sion,
and ha he me ics induced on Mby bo h ψand 
ψag ee.
Deno e by σand ∇ he second undamen al o m o 
ψand he Le i–Ci i a
connec ion o Cn, espec i ely. Then, i ollows om he well-known O’Neill
equa ions [5], ha

∇X∗Y∗=(∇XY)∗+1
2η([X∗,Y∗])ξ
o any ec o fields X,Y on Cn angen o M.SinceMis no mal o ξ,weha e
η([X∗,Y∗]) = 0, and he e o e
σ(X,Y )=(σ(X,Y ))∗and 
H∗=H, (11)
whe e 
His he mean cu a u e ec o o 
ψ.
Suppose ha he equali y case o (5) holds o any p∈M. Then, P oposi-
ion 2 implies ha σsa isfies (7). Hence, by i ue o (11), we ha e ha
σ(X,Y )= n
n+2G(X, Y )
H+G(JX, 
H)JY +G(JY, 
H)JX
o any ec o fields X, Y on Cn angen o M. The e o e, i ollows om [6,
Theo em 2], ha ei he 
ψis o ally geodesic o 
ψ(M) is an open po ion o a
Whi ney sphe e.
I he fi s case holds, (11) implies ha ψis also o ally geodesic. On he
o he hand, i 
ψ(M) is a po ion o a Whi ney sphe e, Lemma 2 implies ha
ψ(M) mus be a po ion o a co esponding con ac Whi ney sphe e.
The con e se can be p o ed by s aigh o wa d compu a ion. QED
The con ac Whi ney sphe e 131
3 An i-in a ian Submani olds.
We now conside an (n+ 1)–dimensional an i-in a ian submani old Mn+1
o R2n+1; such a submani old is angen o he s uc u e ec o field ξ. In pa -
icula , om he Sasakian condi ion on R2n+1,
∇Xξ=−φX, and he e o e
σ(X,ξ)=−φX, (12)
o any ec o field X angen o M.
In his case, i is possible o s a e an inequali y simila o (5):
P oposi ion 3. Le Mn+1 be an (n+1)–dimensional an i-in a ian sub-
mani old o R2n+1. Then, he squa e o i s mean cu a u e |H|2and i s scala
cu a u e τsa is y a each poin he ollowing inequali y:
|H|2≥2(n+2)
(n+1)
2(n−1)τ. (13)
The equali y holds i and only i he e exis s a eal unc ion λdefined on M,
such ha he second undamen al o m σo Msa isfies
σ(e1,e
1)=3λe1∗,σ(e2,e
2)=... =σ(en,e
n)=λe1∗,
σ(e1,e
j)=λej∗,σ(ej,e
k)=0,2≤j=k≤n,
whe e {e1,...,e
n,ξ,e
1∗,...,e
n∗}is a local o hono mal ame such ha e1,...,
en,ξa e angen o M,ei∗=φei, o any i=1,...,n,ande1∗is pa allel o H.
We can also desc ibe he second undamen al o m o he submani olds sa -
is ying he equali y case o (13) a e e y poin :
P oposi ion 4. Le Mn+1 be an (n+1)–dimensional an i-in a ian sub-
mani old o R2n+1. Then, Msa isfies he equali y case o (13) a e e y poin ,
i andonlyi
σ(X,Y )=n+1
n+2{(g(X, Y )−η(X)η(Y))H+
+(g(φX, H)−n+2
n+1η(X))φY +(g(φY, H)−n+2
n+1η(Y))φX},(14)
o any angen ec o fields Xand Y.
A submani old Mo R2n+1, angen o ξ,issaid obe o ally con ac
geodesic [1, p.110] i
σ(X,Y )=η(X)σ(Y,ξ)+η(Y)σ(X,ξ),(15)
o any angen ec o fields Xand Y. I is clea ha e e y o ally con ac
geodesic submani old is minimal. Then, i ollows om (12), (14) and (15) ha
132 D. E. Blai , A. Ca iazo
e e y an i-in a ian o ally con ac geodesic submani old sa isfies he equali y
case o (13). Wi h an addi ional condi ion, we can cha ac e ize he non- i ial
an i-in a ian submani olds sa is ying ha equali y.
Le π:R2n+1 →Cnbe he Riemannian subme sion gi en by (8), and
suppose ha he e exis s a Lag angian submani old No Cn, such ha he
ollowing diag am commu es
M−→ R2n+1
↓↓π
N−→ Cn,
whe e Mis he se o fib es o e N. Now, we s a e he ollowing heo em:
Theo em 2. Unde he abo e condi ions, he equali y case o (13) holds o
any p∈M, i and only i ei he Mis a o ally con ac geodesic submani old o
is locally isome ic o he Riemannian p oduc o a po ion o a Whi ney sphe e
and R.
P oo . As in he p oo o Theo em 1, i ollows om he O’Neill equa ions
o he Riemannian subme sion π ha
σ(X∗,Y∗)=(σ(X, Y ))∗and 
H∗=n+1
nH, (16)
whe e σand 
Ha e he second undamen al o m and he mean cu a u e ec o
o N, espec i ely. On he o he hand, we also know ha
η(∇X∗Y∗)=−g(∇X∗ξ,Y ∗)=g(φX∗,Y∗)=0,
since Mis an i-in a ian . This implies ha Mis locally isome ic o he Rie-
mannian p oduc o Nand R.
Suppose now ha he equali y case o (13) holds o any p∈M. Then,
P oposi ion 4 implies ha σsa isfies (14). Hence, i ollows om (11) ha σ
sa isfies he equa ion o [6, Theo em 2], and so, ei he Nis o ally geodesic o
is a po ion o a Whi ney sphe e.
Finally, i only emains o ema k ha , i Nis o ally geodesic, hen, we
ob ain om (16) ha σ(X,Y ) = 0, o any angen ec o fields Xand Y,
pe pendicula o ξ, and he e o e, Mis o ally con ac geodesic.
The con e se can be p o ed by s aigh o wa d compu a ion. QED
Acknowledgemen s. This a icle was w i en while he second au ho was
isi ing Michigan S a e Uni e si y in 1999. He would like o exp ess his deep
app ecia ion o p o esso s D.E. Blai and B.-Y. Chen o he wa m hospi ali y
he ecei ed du ing his isi .
The con ac Whi ney sphe e 133
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[3] V.Bo elli,B.Y.Chen,J.M.Mo anUne ca ac ´e isa ion g´eom´e ique de la sph`e e
de Whi ney,C.R.Acad.Sci.Pa is,S´e ie I 321 (1995), 1485–1490.
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equali y, Geom. Dedica a, 70 n. 2 (1998), 197–208.
[5] B. O’Neill:The undamen al equa ions o a subme sion, Michigan Ma h. J., 13 (1966),
459–469.
[6] A. Ros, F. U bano:Lag angian submani olds o Cnwi h con o mal Maslo o m and
he Whi ney sphe e, J. Ma h. Soc. Japan, 50 n. 1 (1998), 203–226.