New Rela ionships in ol ing he Mean
Cu a u e o Slan Submani olds in
S-space- o ms
Luis M. Fe n´andez, Ma ´ıa B. Hans-Ube 1
(2000 Ma hema ics Subjec Classi ica ion: 53C25, 53C40).
(Keywo ds and sen ences: S-space- o m, Slan Imme sion, Mean Cu a u e Vec-
o , Ricci Cu a u e, Shape Ope a o ).
Abs ac . Rela ionships be ween he Ricci cu a u e and he squa ed mean
cu a u e and be ween he shape ope a o associa ed wi h he mean cu a-
u e ec o and he sec ional cu a u e unc ion o slan submani olds o an
S-space- o m a e p o ed, pa icula izing hem o in a ian and an i-in a ian
submani olds angen o he s uc u e ec o ields.
1 In oduc ion.
In wo ds o B.-Y. Chen, o “ ind simple ela ionships be ween he main ex insic
in a ian and he main in insic in a ian s o a submani old” is one o he basic
p oblems in he heo y o submani olds ([7]). In his way, he es ablished a
ela ionship be ween sec ional cu a u e unc ion and he shape ope a o o
submani olds in eal space- o ms [7] and ano he ela ionship be ween he Ricci
cu a u e and he squa ed mean cu a u e [8]. Co esponding ela ionships ha e
been p o ed in [13] o slan submani olds o Sasakian space- o ms.
Slan imme sions in complex geome y we e de ined by B.-Y. Chen as a na u-
al gene aliza ion o bo h holomo phic and o ally eal imme sions [9]. Recen ly,
A. Lo a has in oduced he no ion o slan imme sion o a Riemannian mani old
in o an almos con ac me ic mani old [11] and slan submani olds o Sasakian
mani olds ha e been s udied in [2]. Fo a gene al iew abou slan submani olds,
he su ey w i en by A. Ca iazo ([3]) can be consul ed. On he o he hand,
o mani olds wi h an -s uc u e, D.E. Blai has in oduced S-mani olds as he
analogue o he Kaehle s uc u e in he almos complex case and o Sasakian
1The au ho s a e pa ially suppo ed by he PAI p ojec (Jun a de Andaluc´ıa, Spain, 2004).
1
s uc u e in he almos con ac case [1] and we ha e de ined and begun wi h he
s udy o slan submani olds in such S-mani olds [4,5,6].
The pu pose o his pape is o ob ain simila ela ionships o Chen’s ones
men ioned abo e, gene alizing and imp o ing in some sense he ones p o ed in
[13], o slan submani olds in S-space- o ms. To his end, a e e iewing, o
la e use, necessa y de ails abou S-mani olds and slan submani olds in Sec ion
2, we de o e Sec ion 3 o ge an inequali y be ween Ricci cu a u e and squa ed
mean cu a u e ec o and discuss he equali y case. Finally, in Sec ion 4 we
es ablish an inequali y be ween he shape ope a o associa ed wi h he mean
cu a u e ec o and he sec ional cu a u e unc ion o slan submani olds o
S-space o ms. In bo h cases, we pa icula ize hese inequali ies o in a ian
and an i-in a ian submani olds angen o he s uc u e ec o ields.
2 P elimina ies.
Le (
M, g) be a Riemannian mani old and deno e by T
M he Lie algeb a o
ec o ields in
M.
Mis said o be a me ic -mani old i he e exis a (1,1)
enso ield ,sglobal uni ec o ields ξ1, . . . , ξs(called s uc u e ec o ields)
and s1- o ms η1, . . . , ηson
Msuch ha
2X=−X+
s
X
α=1
ηα(X)ξα, g(X, ξα) = ηα(X),
ξα= 0, ηα◦ = 0
and
g( X, Y ) = g(X, Y )−
s
X
α=1
ηα(X)ηα(Y),
o any X, Y ∈T
Mand α= 1, . . . , s. Le Fdeno e he undamen al 2- o m in
Mgi en by F(X, Y ) = g(X, Y ), o any X, Y ∈T
M. The -s uc u e is said
o be no mal i [ , ] + 2 Pαξα⊗dηα= 0, whe e [ , ] is he Nijenhuis o sion
o .
Mis called an S-mani old i he s uc u e is no mal and F= dηα, o
any α= 1, . . . , s.
Gi en an S-mani old
M, a plane sec ion πin Tp
Mis called an -sec ion i i is
spanned by Xand X, whe e Xis a uni angen ec o ield o hogonal o he
dis ibu ion Mspanned by he s uc u e ec o ields. The sec ional cu a u e
K(π) o an -sec ion πis called -sec ional cu a u e. An S-mani olds is said
o be an S-space- o m i i has cons an -sec ional cu a u e cand hen, i
is deno ed by
M(c).
2
Now, le Mbe a submani old isome ically imme sed in an S-mani old
M.
Le TM be he Lie algeb a o ec o ields in Mand T⊥M he se o all ec o
ields no mal o M. We deno e by σ he second undamen al o m o Mand by
AV he shape ope a o associa ed wi h any V∈T⊥M. They a e ela ed by he
equa ion g(σ(X, Y ), V ) = g(AVX, Y ), o any X, Y ∈TM and any V∈T⊥M.
The mean cu a u e ec o His de ined by H= (1/dim(M)) ace(σ). The
submani old is said o be minimal i H anishes iden ically and i is said o be
o ally geodesic i σ(X, Y ) = 0, o any X, Y ∈TM. Mo eo e , he ela i e
null space o Mis de ined by:
N={X∈TM :σ(X, Y ) = 0, o all Y∈T M}.
Fo any X∈TM, we pu X =T X +NX, whe e T X ( esp. NX) is he
angen ial ( esp. no mal) componen o X. The submani old Mis said o be
in a ian i Nis iden ically ze o, ha is, i X ∈TM, o any X∈TM and
i is said o be an i-in a ian i Tis iden ically ze o, ha is, i X ∈T⊥M, o
any X∈TM.
F om now on, we suppose ha he s uc u e ec o ields a e angen o M
and we deno e by n+s( esp. 2m+s) he dimension o M( esp.
M). Hence,
i we deno e by L he o hogonal dis ibu ion o Min TM, we can w i e he
o hogonal di ec decomposi ion TM =L⊕M.
I is well-known ha
σ(X, ξα) = −NX, (2.1)
o any X∈TM and any α= 1, . . . , s. In pa icula , σ(ξα, ξβ) = 0, o any
α, β = 1, . . . , s.
Now, gi en a local o hono mal basis
{e1, . . . , en, en+1, . . . , e2m, ξ1, . . . , ξs}
o T
M, such ha {e1, . . . , en}is a local o hono mal basis o L, we can w i e
he mean cu a u e ec o Hand he squa ed no ms o Tand σby:
H=1
n+s
n
X
i=1
σ(ei, ei),(2.2)
kTk2=
n
X
i,j=1
g2(ei, Tej),(2.3)
kσk2=
n
X
i=1
kσ(ei, ei)k2+ 2 X
1≤i<j≤n
kσ(ei, ej)k2+ 2
n
X
i=1
s
X
α=1
kσ(ei, ξα)k2.(2.4)
3
The cu a u e enso ield Ro a submani old Mo an S-space- o m
M(c)
sa is ies
R(X, Y, Z, W) = g(σ(X, W), σ(Y, Z)) −g(σ(X, Z), σ(Y, W ))+
+X
α,β
(g( X, W)ηα(Y)ηβ(Z)−g( X, Z)ηα(Y)ηβ(W)+
+g( Y, Z)ηα(X)ηβ(W)−g( Y, W)ηα(X)ηβ(Z))+
+c+ 3s
4(g( X, W)g( Y, Z)−g( X, Z)g( Y, W))+
+c−s
4(F(X, W)F(Y, Z)−F(X, Z)F(Y, W)−2F(X, Y )F(Z, W)),(2.5)
o any X, Y, Z, W ∈T
M(see, o e e ences, [1,10]).
The scala cu a u e τo Mis de ined by
τ=1
2X
i6=j
K(ei∧ej) +
n
X
i=1
s
X
α=1
K(ei∧ξα),(2.6)
whe e K(X∧Y) deno es he sec ional cu a u e o Massocia ed wi h he plane
sec ion spanned by X, Y ∈TM.
F om (2.1)-(2.6), we ob ain he ollowing ela ion be ween he scala cu a u e
and he mean cu a u e o M[4]:
2τ= (n+s)2kHk2− kσk2+n(n−1)c+ 3s
4+ 2ns +3(c−s)
4kTk2.(2.7)
I o each nonze o ec o X∈TpM−Mp, we conside he angle θ(X) be ween
X and TpM, hen he submani old is said o be θ-slan [6] i such angle is a
cons an , which is independen on he choice o p∈Mand X∈TpM− Mp.
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 submani olds angen o he s uc u e ec o ields
a e slan submani olds 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 [6], we ha e p o ed ha a θ-slan submani old Mo a me ic -mani old
Msa is ies
g(TX, TY ) = cos2θg( X, Y ),
o any X, Y ∈TM. Mo eo e , i is easy o show ha , using a local o hono mal
basis {e1, . . . , en+s}o TM,
n+s
X
j=1
g2(ei, ej) = cos2θ(1 −
s
X
α=1
η2
α(ei)),(2.8)
o any i= 1, . . . , s.
4
3 The Ricci cu a u e o slan imme sions.
Le Mbe an (n+s)-dimensional isome ically imme sed submani old in a (2m+
s)-dimensional S-space- o m
M(c), angen o he s uc u e ec o ields. In his
sec ion, we wan o s udy he Ricci cu a u e o uni ec o ields in M, no mal
o he s uc u e ec o ields, when Mis a slan submani old and o ela e hem
wi h he mean cu a u e ec o . Fo his eason, we shall assume ha n≥2,
because i is known ha he e a e no p ope slan submani olds o dimension
1 + s([5]). Th oughou he sec ion, we conside local o hono mal basis o
T
M(c),
{e1, . . . , en, en+1, . . . , e2m, e2m+1 =ξ1, . . . , e2m+s=ξs},(3.1)
such ha {e1, . . . , en}is a local o hono mal basis o L. Fi s , we ha e he
ollowing gene al esul :
Theo em 3.1. Le Mbe an (n+s)-dimensional submani old
M(c), angen o
he s uc u e ec o ields. Then,
4Ric(U) ≤(n+s)2kHk2+ (n−1)(c+ 3s) + kTUk2(3c+s),(3.2)
o any uni ec o ield U∈ L.
P oo . We choose a local o hono mal basis o T
M(c) as in (3.1) and such ha
e1=U. Then:
kσk2=1
2(n+s)2kHk2+ 2 X
1≤i<j≤n
kσ(ei, ej)k2+ 2
n
X
i=1
s
X
α=1
kσ(ei, ξα)k2+
+1
2
n
X
i=1
kσ(ei, ei)k2−X
1≤i<j≤n
g(σ(ei, ei), σ(ej, ej)).(3.3)
Thus, om (2.7) and (3.3), we ha e:
1
4(n+s)2kHk2=τ−1
8n(n−1)(c+ 3s)−3
8kTk2(c−s)−ns+
+X
1≤i<j≤n
kσ(ei, ej)k2+
n
X
i=1
s
X
α=1
kσ(ei, ξα)k2+
+1
4
n
X
i=1
kσ(ei, ei)k2−1
2X
1≤i<j≤n
g(σ(ei, ei), σ(ej, ej)).(3.4)
5

On he o he hand, since om (2.6),
τ= Ric(U) + X
2≤i<j≤n
K(ei∧ej) +
n
X
i=2
s
X
α=1
K(ei∧ξα)
and, by using (2.5), we ge
X
2≤i<j≤n
K(ei∧ej) = 1
8n(n−1)(c+ 3s) + 3
4µkTk2
2− kTUk2¶(c−s)+
+X
2≤i<j≤n¡g(σ(ei, ei), σ(ej, ej)) − kσ(ei, ej)k2¢
and n
X
i=2
s
X
α=1
K(ei∧ξα) = (n−1)s−
n
X
i=2
s
X
α=1
kσ(ei, ξαk2,
hen, subs i u ing in o (3.4) and aking in o accoun (2.1), we ob ain:
Ric(U) = 1
4(n+s)2kHk2+1
4(n−1)(c+ 3s) + 1
4kTUk2(3c+s)−
−1
2X
2≤i<j≤n
g(σ(ei, ei), σ(ej, ej)) + 1
2
n
X
i=2
g(σ(U, U), σ(ei, ei))−
−
n
X
i=2
kσ(U, ei)k2−1
4
n
X
i=1
kσ(ei, ei)k2.(3.5)
Now, i we pu σ
ij =g(σ(ei, ej), e ), o any i, j = 1, . . . , n and =n+1, . . . , 2m,
(3.5) becomes
Ric(U) = 1
4(n+s)2kHk2+1
4(n−1)(c+ 3s) + 1
4kTUk2(3c+s)−
−
2m
X
=n+1 
1
4Ãσ
11 −
n
X
i=2
σ
ii!2
+
n
X
i=2
(σ
1i)2
,(3.6)
which comple es he p oo .
Obse e ha , i we also pu σ
1(2m+α)=g(σ(U, ξα), e ) = −g(NU, e ), o any
α= 1, . . . , s and =n+ 1, . . . , 2m, we ha e
Ric(U) = 1
4(n+s)2kHk2+1
4n(c+ 3s) + 1
4(3kTUk2−1)(c−s)−
−
2m
X
=n+1 
1
4Ãσ
11 −
n
X
i=2
σ
ii!2
+
n
X
i=2
(σ
1i)2+
s
X
α=1
(σ
1(2m+α))2
(3.7)
6
and, consequen ly:
4Ric(U)≤(n+s)2kHk2+n(c+ 3s) + (3kTUk2−1)(c−s) (3.8).
When s= 1, his uppe bound is he one ob ained in [13] o submani olds o
Sasakian space- o ms angen o he s uc u e ec o ield. Howe e , i is easy
o show ha i is wo se (in he sense o highe ) han he one o (3.2). Mo eo e ,
bo h uppe bounds a e equal i and only i kNUk2= 0, ha is, i and only i ,
om (2.1), σ(U, ξα) = 0, o any α= 1, . . . , s. I i is he case, hei common
alue is (n+s)2kHk2+n(c+ 3s) + 2(c−s). Then, we can p o e he ollowing
heo em.
Theo em 3.2. Le Mbe an (n+s)-dimensional minimal submani old o
M(c),
angen o he s uc u e ec o ields. Then, a uni ec o ield Uin Lsa is ies
he equali y case o (3.8) i and only i Ulies in he ela i e null space o M.
Mo eo e , in his case:
4Ric(U) = n(c+ 3s) + 2(c−s).
P oo . I U∈ L is a uni ec o ield sa is ying he equali y case o (3.8), hen i
also sa is ies he equali y case o (3.2) and so, σ(U, ξα) = 0, o any α= 1, . . . , s.
Fu he mo e, choosing a local o hono mal basis o T
M(c) as in (3.1) such ha
e1=U, om (3.7) we ge σ
1i= 0, o any i= 2, . . . , n, =n+ 1, . . . , 2mand
σ
11 =
n
X
i=2
σ
ii,
o any =n+ 1, . . . , 2m. Bu , since H= 0,
σ
11 =−
n
X
i=2
σ
ii,
o any =n+ 1, . . . , 2m, ha is, σ
11 = 0. Thus, U∈ N.
Con e sely, i U∈ N , choosing a local o hono mal basis o T
M(c) as in
(3.1) wi h e1=U, we ha e ha σ
1i= 0 and σ
1(2m+α)= 0, o any i=
1, . . . , n,α= 1, . . . , s, =n+ 1, . . . , 2m. Again, since H= 0, we ob ain ha
σ
22 +· · · +σ
nn = 0, o any =n+ 1, . . . , 2m. Then, om (3.7) we ge he
equali y case o (3.8). Finally, om (2.1) we comple e he p oo .
Now, obse e ha i he equali y case o (3.8) holds o all uni ec o ields
U∈ L, he equali y case o (3.2) is ue o hese ec o ields oo. Thus,
7
om (2.1), NU = 0 o any U∈ L and Mis an in a ian submani old. Then,
i is easy o show ha i is minimal. Consequen ly, making use o Theo em
3.2, U∈ N, o any U∈ L and Mis o ally geodesic. The con e se esul is
a s aigh o wa d compu a ion. So, we ha e p o ed he ollowing co olla y o
Theo em 3.2.
Co olla y 3.1. Le Mbe an (n+s)-dimensional minimal submani old o
M(c),
angen o he s uc u e ec o ields. Then, he equali y case o (3.8) holds o
all uni ec o ield in Li and only i Mis a o ally geodesic submani old.
Nex , we a e going o s udy he equali y case o (3.2). To his end, we ecall
ha an (n+s)-dimensional submani old o an S-mani old, angen o he s uc-
u e ec o ields, is said o be a o ally -geodesic submani old ( esp., o ally
-umbilical) i he dis ibu ion Lis o ally geodesic ( esp., o ally umbilical),
ha is, i σ(X, Y ) = 0 ( esp., σ(X, Y ) = g(X, Y )V, being
V=n+s
nH),
o any X, Y ∈ L ([12]). Then, we can p o e he ollowing heo em.
Theo em 3.3. Le Mbe an (n+s)-dimensional (n≥2) submani old o
M(c),
angen o he s uc u e ec o ields. Then, he equali y case o (3.2) holds
o all uni ec o ield in Li and only i ei he Mis a o ally -geodesic
submani old o n= 2 and Mis a o ally –umbilical submani old.
P oo . I he equali y case o (3.2) is ue o any uni ec o ield U∈ L, hen,
by choosing local o hono mal basis o T
M(c) as in (3.1) and since e1can be
chosen o be any a bi a y uni ec o ields in L, om (3.6) we ge
2σ
ii =σ
11 +· · · +σ
nn, i = 1, . . . , n,
σ
ij = 0, i 6=j,
o any =n+ 1, . . . , 2m. Thus, we ha e wo cases, namely ei he n= 2 o
n > 2. In he i s case, σ
11 =σ
22, o any and Mis a o ally -umbilical
submani old, while in he second case σ
ii = 0, i= 1, . . . n and Mis a o ally -
geodesic submani old. The con e se pa is a s aigh o wa d compu a ion.
The abo e esul s co espond o ha one p o ed by B.-Y. Chen in [8] o
submani olds in eal space- o ms. Mo eo e , hey imply he ollowing heo em
o a slan submani old isome ically imme sed in an S-space- o m.
8
Theo em 3.4. Le Mbe an (n+s)-dimensional (n≥2) θ-slan submani old
o an S-space- o m
M(c). Then:
(i) Fo each uni ec o ield U∈ L, we ha e:
4Ric(U) ≤(n+s)2kHk2+ (n−1)(c+ 3s) + cos2θ(3c+s).(3.9)
(ii) The equali y case o (3.9) holds o all uni ec o ield in Li and only i
ei he Mis a o ally -geodesic submani old o n= 2 and Mis a o ally
–umbilical submani old.
P oo . Fo any uni ec o ield U∈ L, by using a local o hono mal basis o
T
M(c) as in (3.1), such ha e1=U, we ge om (2.8) ha
kTUk2= cos2θ
and so, om (3.2) we ha e (3.9). Res o he p oo is simila o ha o Theo em
3.3.
In pa icula , i Mis an in a ian submani old ( hen, i is a minimal mani old
oo), we ha e:
Theo em 3.5. Le Mbe an (n+s)-dimensional (n≥2) in a ian submani old
o an S-space- o m
M(c) angen o he s uc u e ec o ields. Then:
(i) Fo each uni ec o ield U∈ L, we ha e:
4Ric(U) ≤n(c+ 3s) + 2(c−s).(3.10)
(ii) The equali y case o (3.10) holds o all uni ec o ield in Li and only i
ei he Mis a o ally -geodesic submani old o n= 2 and Mis a o ally
–umbilical submani old.
Finally, i Mis an an i-in a ian submani old, we ob ain:
Theo em 3.6. Le Mbe an (n+s)-dimensional (n≥2) an i-in a ian sub-
mani old o an S-space- o m
M(c) angen o he s uc u e ec o ields. Then:
(i) Fo each uni ec o ield U∈ L, we ha e:
4Ric(U) ≤(n+s)2kHk2+ (n−1)(c+ 3s).(3.11)
(ii) The equali y case o (3.11) holds o all uni ec o ield in Li and only i
ei he Mis a o ally -geodesic submani old o n= 2 and Mis a o ally
–umbilical submani old.
These esul s imp o e hose ones p o ed in [13] o slan submani olds o a
Sasakian space- o m (case s= 1).
9