KYUNGPOOK Ma h. J. 39(1999), 465-476
A Con ac Ve sion o B.-Y. Chen’s Inequali y and I s Appli-
ca ions o Slan Imme sions
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, Apa ado de Co eos 1160, 41080-Se illa, Spain
e-mail: aca [email protected]
(1991 Ma hema ics Subjec Classi ica ion : 53C15, 53C40)
We es ablish a e sion o B.-Y. Chen’s inequali y o a submani old o a Sasakian-
space- o m, angen o he s uc u e ec o ield o he ambien space. We ob ain some
applica ions and we s udy his inequali y o slan submani olds. We also cha ac e ize
3–dimensional slan submani olds sa is ying he equali y case.
1. In oduc ion
Gi en a Riemannian mani old M, o each poin p∈M, pu
(in K)(p) = in {K(π) : plane sec ions π⊂TpM},
whe e K(π) deno es he sec ional cu a u e o Massocia ed wi h π. Le
δM(p) = τ(p)−in K(p),(1.1)
being τ he scala cu a u e o M. Then, δMis a well–defined Riemannian in a ian ,
which was ecen ly in oduced by B.-Y. Chen [4, 5].
Fo submani olds Min a eal-space- o m e
Rm(c) o cons an sec ional cu a u e
c, Chen ga e he ollowing basic inequali y in ol ing he in insic in a ian δMand
he squa ed mean cu a u e o he imme sion,
δM≤n2(n−2)
2(n−1) |H|2+1
2(n+ 1)(n−2)c, (1.2)
whe e ndeno es he dimension o Mand His he mean cu a u e ec o . On he
o he hand, i was ema ked in [8] ha he exac p oo o (1.2) gi en in [4] yields
he same inequali y o o ally eal submani olds in a complex-space- o m
Mm(4c)
wi h cons an holomo phic sec ional cu a u e 4c.
La e , Chen gene alized he abo e si ua ion by es ablishing an inequali y o an
a bi a y submani old o dimension g ea e han 2 in a complex-space- o m [6]. By
(Recei ed : May 14, 1999)
The au ho is pa ially suppo ed by he PAI p ojec (Jun a de Andaluc´ıa, Spain 1998).
465
466 Al onso Ca iazo
applying his inequali y, he showed ha (1.2) holds o a bi a y submani olds in
he complex hype bolic space CHm(4c) (c < 0) as well. He also s a ed a o mula
o a submani old in he complex p ojec i e space CPm(4c).
In con ac geome y, De e e , Mihai and Ve s aelen ob ained an inequali y si-
mila o (1.2) o C- o ally eal submani olds o a Sasakian-space- o m wi h cons an
ϕ-sec ional cu a u e c[11]:
δM≤n2(n−2)
2(n−1) |H|2+1
2(n+ 1)(n−2)c+ 3
4.(1.3)
Se e al au ho s ha e s udied he equali y cases o he abo e inequali ies (see,
o ins ances, [6, 7, 9, 10, 11, 12, 13]).
C- o ally eal submani olds ha e he s uc u e ec o field ξo he ambien
space as a no mal ec o field (and so, hey a e an i-in a ian submani olds i ha
ambien space is, a leas , a con ac me ic mani old).
The pu pose o he p esen pape is o es ablish a gene al inequali y, simila
o ha o [6], o submani olds angen o he s uc u e ec o field o a Sasakian-
space- o m. We a e specially in e es ed in applying he ob ained esul s o slan
imme sions in con ac geome y (see, o e e ences, [2, 3, 15]).
Thus, in Sec ion 2, we e iew basic o mulas and defini ions o almos con ac
me ic mani olds and hei submani olds, which we shall use la e . In Sec ion 3,
we es ablish he men ioned inequali y and we adap ou p ocedu es o ξ- angen
si ua ion by in oducing a new in a ian δD
M, closely ela ed o δM. Finally, we show
some applica ions in Sec ions 4 and 5, by paying a special a en ion o slan im-
me sions. Fo example, we cha ac e ize 3-dimensional slan submani olds sa is ying
ou equali y case.
When his pape was finished, he au ho lea ned ha in [14], Y.H. Kim and
D.-S. Kim ob ained a basic inequali y o δM o submani olds in a Sasakian-space-
o m. Mo eo e , hey apply i o ge a cha ac e iza ion o an odd-dimensional g ea
sphe e o an odd-dimensional sphe e. On he o he hand, hey do no s udy slan
imme sions and so, hey do no modi y ha inequali y in o de o conside non-
in a ian submani olds sa is ying he equali y case. Hence, e en hough δD
M≤δM,
nei he ou main pinching esul gi en in Theo em 3.5, no he applica ions shown
in Sec ion 5, can be ob ained om Theo em 3.3 o [14].
2. P elimina ies
Le (
M, g) be an odd–dimensional Riemannian mani old and deno e by T
M he
Lie algeb a o ec o fields in
M.
Le ϕbe a (1,1) enso field, ξa global uni ec o field (s uc u e ec o ield),
and ηa 1– o m on
M. I we ha e ϕ2X=−X+η(X)ξ,g(X, ξ) = η(X) and
g(ϕX, ϕY ) = g(X, Y )−η(X)η(Y), o any X, Y ∈T
M, hen
Mis said o ha e an
almos con ac me ic s uc u e (ϕ, ξ, η, g) and i is called an almos con ac me ic
mani old.
A con ac e sion o B.-Y. Chen’s inequali y and i s applica ions 467
Le Φ deno e he undamen al 2– o m in
M, gi en by Φ(X, Y ) = g(X, ϕY )
o all X, Y ∈T
M. I Φ = dη, hen
Mis said o be a con ac me ic mani old.
Mo eo e , i ξis a Killing ec o field wi h espec o g, he con ac me ic s uc u e
is called a K–con ac s uc u e.
The s uc u e o
Mis said o be no mal i [ϕ, ϕ]+2dη⊗ξ= 0, whe e [ϕ, ϕ] is he
Nijenhuis o sion o ϕ. A Sasakian mani old is a no mal con ac me ic mani old.
E e y Sasakian mani old is a K–con ac mani old.
Gi en a Sasakian mani old
M, a plane sec ion πin Tp
Mis called a ϕ–sec ion i
i is spanned by Xand ϕX, whe e Xis a uni angen ec o field o hogonal o ξ.
The sec ional cu a u e e
K(π) o a ϕ–sec ion πis called ϕ–sec ional cu a u e. I a
Sasakian mani old
Mhas cons an ϕ–sec ional cu a u e c,
Mis called a Sasakian-
space- o m and i is deno ed by
M(c). Fo mo e de ails and backg ound, we e e
o he s anda d e e ence [1].
Now, le Mbe a submani old imme sed in (
M, ϕ, ξ, η, g). We also deno e by
g he induced me ic on M. Le TM be he Lie algeb a o ec o fields in M
and T⊥M he se o all ec o fields no mal o M. We deno e by σ he second
undamen al o m o Mand by AV he Weinga en endomo phism associa ed wi h
any V∈T⊥M. We pu σ
ij =g(σ(ei, ej), e ), o any ei, ej∈T M and e ∈T⊥M.
The mean cu a u e ec o His defined by H= (1/dim M) ace σ.Mis
said o be minimal i H anishes iden ically.
F om now on, we deno e by n+ 1 ( esp. m) he dimension o M( esp.
M). We
conside n≥2. We also suppose ha he s uc u e ec o field ξis angen o M.
Hence, i we deno e by D he o hogonal dis ibu ion o ξin T M, we can conside
he o hogonal di ec decomposi ion TM =D ⊕ < ξ >.
Fo any X∈TM, we w i e ϕX =TX +NX, whe e T X ( esp. NX) is he
angen ial ( esp. no mal) componen o ϕX. I
Mis a K-con ac mani old, i is
well-known ha
σ(X, ξ) = −NX, (2.1)
o any X∈TM.
Gi en a local o hono mal ame {e1, . . . , en}o D, we can define he squa ed
no ms o Tand Nby
|T|2=
n
∑
i,j=1
g2(ei, Tej),|N|2=
n
∑
i=1
|Nei|2,(2.2)
espec i ely. I is easy o show ha bo h |T|2and |N|2a e independen o he
choice o he abo e o hono mal ame.
The submani old Mis said o be in a ian i Nis iden ically ze o, ha is,
ϕX ∈TM, o any X∈TM. On he o he hand, Mis said o be an an i–in a ian
submani old i Tis iden ically ze o, ha is, ϕX ∈T⊥M, o any X∈TM.
Fo each nonze o ec o X angen o Ma p, such ha Xis no p opo ional
o ξp, we deno e by θ(X) he angle be ween ϕX and TpM. Then, Mis said o
468 Al onso Ca iazo
be slan [15] i he angle θ(X) is a cons an , which is independen o he choice
o p∈Mand X∈TpM−< ξp>. The angle θo a slan imme sion is called
he slan angle o he imme sion. In a ian and an i–in a ian imme sions a e slan
imme sions wi h slan angle θ= 0 and θ=π/2 espec i ely. A slan imme sion
which is no in a ian no an i–in a ian is called a p ope slan imme sion.
In [3] we ha e p o ed ha a θ-slan submani old Mo an almos con ac me ic
mani old
Msa isfies
g(TX, TY ) = cos2θ(g(X, Y )−η(X)η(Y)),(2.3)
g(NX, NY ) = sin2θ(g(X, Y )−η(X)η(Y)).(2.4)
o any X, Y ∈TM. On he o he hand, Lemma 2.3.8 o [2] implies
n
∑
j=1
g2(ei, ϕej) = cos2θ, (2.5)
o any i= 1, . . . , n, whe e {e1, . . . , en, ξ}is a local o hono mal ame o TM.
I is well-known ha he cu a u e enso Ro a submani old Mo a Sasakian-
space- o m
M(c) sa isfies
R(X, Y ;Z, W ) = g(σ(X, W ), σ(Y, Z)) −g(σ(X, Z), σ(Y, W))+
+c+ 3
4(g(X, W)g(Y, Z)−g(X, Z)g(Y, W)) + c−1
4(η(X)η(Z)g(Y, W)−
−η(Y)η(Z)g(X, W) + η(Y)η(W)g(X, Z)−η(X)η(W)g(Y, Z)+
+g(ϕX, W)g(ϕY, Z)−g(ϕX, Z)g(ϕY, W)+2g(X, ϕY )g(ϕZ, W )),(2.6)
o any X, Y, Z, W ∈TM.
Fo an o hono mal basis {e1, . . . , en+1}o he angen space TpM,p∈M, he
scala cu a u e τa pis defined by
τ=∑
i<j
K(ei∧ej),(2.7)
whe e K(ei∧ej) deno es he sec ional cu a u e o Massocia ed wi h he plane
sec ion spanned by ei, ej. In pa icula , i we pu en+1 =ξp, hen (2.7) implies:
2τ=
n
∑
i=j
K(ei∧ej) + 2
n
∑
i=1
K(ei∧ξ).(2.8)
F om (2.2), (2.6) and (2.8), we ob ain he ollowing ela ion be ween he scala
cu a u e and he mean cu a u e o M,
2τ= (n+ 1)2|H|2− |σ|2+n(n+ 1)c+ 3
4+ 2n+3(c−1)
4|T|2,(2.9)
A con ac e sion o B.-Y. Chen’s inequali y and i s applica ions 469
whe e |σ|deno es he no m o he second undamen al o m σ.
3. Chen’s inequali y in Sasakian-space- o ms
Le Mn+1 be a submani old o
Mm(c), angen o he s uc u e ec o field ξ,
and π⊂ Dpa plane sec ion a p∈M, o hogonal o ξp. Then,
Φ2(π) = g2(e1, ϕe2) (3.1)
is a eal numbe in [0,1] which is independen o he choice o he o hono mal basis
{e1, e2}o π. Deno e by τand K(π) he scala cu a u e o Mand he sec ional
cu a u e o Massocia ed wi h π, espec i ely.
We fi s ecall an algeb aic lemma om [4]:
Lemma 3.1.Le a1, . . . ,ak, c be k+ 1 (k≥2) eal numbe s such ha :
(k
∑
i=1
ai)2
= (k−1) (k
∑
i=1
a2
i+c).
Then 2a1a2≥c, wi h equali y holding i and only i a1+a2=a3=· · · =ak.
Now, we can p o e he ollowing con ac e sion o Theo em 3 o [6]:
Theo em 3.2.Le φ:Mn+1 →
Mm(c)be an isome ic imme sion om a Rie-
mannian (n+ 1)–mani old in o a Sasakian-space- o m
Mm(c), such ha ξ∈TM.
Then, o any poin p∈Mand any plane sec ion π⊂ Dp, we ha e:
τ−K(π)≤(n+ 1)2(n−1)
2n|H|2+1
2(n+ 1)(n−2)c+ 3
4+
+n+3
2|T|2c−1
4−3Φ2(π)c−1
4.(3.2)
Equali y in (3.2) holds a p∈Mi and only i he e exis an o hono mal basis
{e1, . . . , en+1}o TpMand an o hono mal basis {en+2, . . . , em}o T⊥
pMsuch ha
(a) en+1 =ξp,(b) πis spanned by e1, e2and (c) he shape ope a o s A =Ae ,
=n+ 2, . . . , m, ake he ollowing o ms:
An+2 =
a0 0
0−a0
0 0 0n−1
,(3.3)
A =
σ
11 σ
12 0
σ
12 −σ
11 0
0 0 0n−1
, =n+ 3, . . . , m. (3.4)

470 Al onso Ca iazo
P oo . Le Mn+1 be a submani old o
Mm(c). Pu :
ε= 2τ−(n+ 1)2(n−1)
n|H|2−(n+ 1)(n−2)c+ 3
4−2n−3(c−1)
4|T|2.(3.5)
Then, (2.9) and (3.5) yield:
(n+ 1)2|H|2=n|σ|2+n(ε−2(c+ 3)
4).(3.6)
Le π⊂ Dpbe a plane sec ion. We choose an o hono mal ame {e1, . . . , en+1}
o TpMand an o hono mal basis {en+2, . . . , em}o T⊥
pMsuch ha en+1 =ξp,π
is spanned by e1, e2and en+2 is in he di ec ion o he mean cu a u e ec o H.
Hence, (3.6) gi es
(n+1
∑
i=1
σn+2
ii )2
=n


n+1
∑
i=1
(σn+2
ii )2+∑
i=j
(σn+2
ij )2+
m
∑
=n+3 ∑
i,j
(σ
ij)2+ε−2(c+ 3)
4

,
and so, by applying Lemma 3.1, we ob ain:
2σn+2
11 σn+2
22 ≥∑
i=j
(σn+2
ij )2+
m
∑
=n+3 ∑
i,j
(σ
ij)2+ε−2(c+ 3)
4.(3.7)
On he o he hand, om (2.6) we find:
K(π) = σn+2
11 σn+2
22 −(σn+2
12 )2+
m
∑
=n+3
(σ
11σ
22 −(σ
12)2)+
+c+ 3
4+3(c−1)
4g2(e1, ϕe2).(3.8)
Then, om (3.7) and (3.8) we ge :
K(π)≥
m
∑
=n+2 ∑
j>2
{(σ
1j)2+ (σ
2j)2}+1
2∑
i=j>2
(σn+2
ij )2+1
2
m
∑
=n+3 ∑
i,j>2
(σ
ij)2+
+1
2
m
∑
=n+3
(σ
11 +σ
22)2+ε
2+3(c−1)
4g2(e1, ϕe2)≥ε
2+3(c−1)
4g2(e1, ϕe2).(3.9)
Finally, combining (3.1), (3.5) and (3.9), we ob ain (3.2).
I he equali y in (3.2) holds, hen he inequali ies in (3.7) and (3.9) become
equali ies. Thus, we ha e:
σn+2
1j=σn+2
2j=σn+2
ij = 0, i =j > 2;
σ
1j=σ
2j=σ
ij = 0, =n+ 3, . . . , m;i, j = 3, . . . , n + 1;
σn+3
11 +σn+3
22 =· · · =σm
11 +σm
22 = 0.
A con ac e sion o B.-Y. Chen’s inequali y and i s applica ions 471
Fu he mo e, we may choose e1, e2such ha σn+2
12 = 0. Mo eo e , by applying
Lemma 3.1 and (2.1), we also ha e:
σn+2
11 +σn+2
22 =σn+2
33 =···=σn+2
n+1 n+1 = 0.
The e o e, wi h espec o he chosen o hono mal basis {e1, . . . , em}, he shape
ope a o s o M ake he o ms (3.3) and (3.4).
The con e se ollows om a di ec calcula ion.
Now, o each poin p∈M, we define:
(in DK)(p) = in {K(π) : plane sec ions π⊂ Dp}.
Then, in DKis a well–defined unc ion on M. Le δD
Mdeno e he diffe ence
be ween he scala cu a u e and in DK, i.e.:
δD
M(p) = τ(p)−in DK(p).(3.10)
F om (1.1) and (3.10), i is clea ha :
δD
M≤δM.(3.11)
I c= 1, hen we ob ain di ec ly om (3.2) and (3.10) he ollowing esul :
Co olla y 3.3.Le φ:Mn+1 →
Mm(1) be an isome ic imme sion om a Rie-
mannian (n+1)-mani old in o a Sasakian-space- o m wi h cons an ϕ-sec ional cu -
a u e 1, such ha ξ∈T M. Then, we ha e:
δD
M≤(n+ 1)2(n−1)
2n|H|2+1
2(n+ 2)(n−1).(3.12)
No e ha , in ac , (3.12) also ollows om (3.11) and (1.2) wi h c= 1, since
a Sasakian-space- o m wi h cons an ϕ-sec ional cu a u e 1 is a eal-space- o m o
cons an sec ional cu a u e 1. This seems o poin ou ha (3.2) may be a na u al
con ac e sion o (1.2). Ne e heless, by using (2.1), (3.3) and (3.4), we can s a e
he ollowing esul :
Co olla y 3.4.I equali y in (3.2) holds a any p∈M, hen φis an in a ian
imme sion.
Now, we a e going o modi y (3.2) in o de o conside non-in a ian submani-
olds ( o example, p ope slan submani olds) sa is ying a simila equali y. We can
p o e he ollowing heo em:
Theo em 3.5.Le φ:Mn+1 →
Mm(c)be an isome ic imme sion om a Rie-
mannian (n+ 1)–mani old in o a Sasakian-space- o m
Mm(c), such ha ξ∈TM.
Then, o any poin p∈Mand any plane sec ion π⊂ Dp, we ha e:
τ−K(π)≤(n+ 1)2(n−1)
2n|H|2+1
2(n+ 1)(n−2)c+ 3
4+
472 Al onso Ca iazo
+n+3
2|T|2c−1
4−3Φ2(π)c−1
4− |N|2.(3.13)
Equali y in (3.13) holds a p∈Mi and only i he e exis an o hono mal basis
{e1, . . . , en+1}o TpMand an o hono mal basis {en+2, . . . , em}o T⊥
pMsuch ha
(a) en+1 =ξp,(b) πis spanned by e1, e2and (c) he shape ope a o s A =Ae ,
=n+ 2, . . . , m, ake he ollowing o ms:
An+2 =




a0 0 µn+2
1
0−a0.
.
.
0 0 0n−2µn+2
n
µn+2
1· · · µn+2
n0




,(3.14)
A =




σ
11 σ
12 0µ
1
σ
12 −σ
11 0.
.
.
0 0 0n−2µ
n
µ
1· · · µ
n0




, =n+ 3, . . . , m, (3.15)
whe e µ
i=g(ϕei, e ), o any i= 1, . . . , n; =n+ 2, . . . , m.
P oo . We ollow he fi s s eps o he p oo o Theo em 3.2 and we s a e equa ions
(3.5)-(3.9). Then, inequali y (3.9) can now be w i en as:
K(π)≥
m
∑
=n+2
n
∑
j=3
{(σ
1j)2+ (σ
2j)2}+1
2
n
∑
i=j>2
(σn+2
ij )2+1
2
m
∑
=n+3
n
∑
i,j=3
(σ
ij)2+
+1
2
m
∑
=n+3
(σ
11 +σ
22)2+ε
2+3(c−1)
4g2(e1, ϕe2) +
m
∑
=n+2
n
∑
i=1
(σ
i n+1)2≥
≥ε
2+3(c−1)
4g2(e1, ϕe2) +
m
∑
=n+2
n
∑
i=1
(σ
i n+1)2.(3.16)
Bu , om (2.1) and (2.2) we find:
m
∑
=n+2
n
∑
i=1
(σ
i n+1)2=|N|2.(3.17)
Hence, combining (3.1), (3.5), (3.16) and (3.17), we ob ain (3.13).
I he equali y in (3.13) holds, hen he inequali ies in (3.7) and (3.16) become
equali ies. By using his ac , (2.1) and Lemma 3.1, we ha e:
σm+2
1j=σn+2
2j=σn+2
ij = 0,2< i =j < n;
σ
1j=σ
2j=σ
ij = 0, =n+ 3, . . . , m;i, j = 3, . . . , n;
σn+3
11 +σn+3
22 =···=σm
11 +σm
22 = 0;
σn+2
11 +σn+2
22 =σn+2
33 =· · · =σn+2
n+1 n+1 = 0.
A con ac e sion o B.-Y. Chen’s inequali y and i s applica ions 473
Hence, i we also choose e1, e2such ha σn+2
12 = 0, hen we ob ain (3.14) and (3.15).
As in he p oo o Theo em 3.2, he con e se can be e ified by s aigh - o wa d
compu a ion.
Mo eo e , i is ob ious ha (3.2) ollows om (3.13), since |N|2≥0.
On he o he hand, i is also clea ha , i φis an an i-in a ian imme sion, hen
|T|2= 0, |N|2=nand Φ2(π) = 0, o any plane sec ion πo hogonal o ξ. Hence,
om (3.13) we ob ain:
Co olla y 3.6.Le Mn+1 be an an i-in a ian submani old o a Sasakian-space-
o m
Mm(c), such ha ξ∈TM. Then, we ha e:
δD
M≤(n+ 1)2(n−1)
2n|H|2+1
2(n+ 1)(n−2)c+ 3
4.(3.18)
No e ha inequali y (3.18) is he ξ- angen e sion o (1.3), wi h he logical diffe -
ences abou he dimensions.
4. Some applica ions
By using Theo em 3.5, we can find some gene al pinching esul s o δD
Mi ei he
c > 1 o c < 1.
Theo em 4.1.Le φ:Mn+1 →
Mm(c)be an isome ic imme sion om a Rie-
mannian (n+ 1)–mani old (n > 2) in o a Sasakian-space- o m
Mm(c), wi h c > 1,
such ha ξ∈TM. Then:
δD
M≤(n+ 1)2(n−1)
2n|H|2+1
2(n2+ 2n−2)c+ 3
4−n
2.(4.1)
Equali y in (4.1) holds iden ically i and only i nis e en and Mn+1 is imme sed
as an in a ian , o ally geodesic submani old o
Mm(c).
Theo em 4.2.Le φ:Mn+1 →
Mm(c)be an isome ic imme sion om a Rie-
mannian (n+ 1)–mani old (n > 2) in o a Sasakian-space- o m
Mm(c), wi h c < 1,
such ha ξ∈TM. Then:
δD
M≤(n+ 1)2(n−1)
2n|H|2+1
2(n+ 1)(n−2)c+ 3
4+n− |N|2.(4.2)
Equali y in (4.2) holds a a poin po Mi and only i he e exis an o hono mal
basis {e1, . . . , en+1}o TpMand an o hono mal basis {en+2, . . . , em}o T⊥
pMsuch
ha (a) en+1 =ξp,(b) he subspace spanned by e3, . . . , en+1 is an i-in a ian , (c)
K(e1∧e2) = in DKa p, and (d) he shape ope a o s A =Ae , =n+ 2, . . . , m,
ake he o ms (3.14) and (3.15).
Theo ems 4.1 and 4.2 can be p o ed by ollowing he same s eps as in he p oo s
o Theo ems 3 and 4 o [6], espec i ely.