23
no mal
connec ion
in
S-mani olds
whose
in a ian
-sec ional
L. M.
FERNÁNDEZ
i n a i an
has
de ined
[ 1
])
,
(Blai ,
o
·
an
i n a i an
submani old
o
Blai ,
in
an
S-mani old
o
cons an
INTRODUCTIO
N
.-
D.
-sec ional
cu a u e
o
be
o ally
geodesic.
codimension
2
ob ained
a
condi
ion
o
in a ian
submani o
lds
o
S-mani o
ls.
Specially
,
hey
ha e
Tsuchiya
,
[4])
,
ha e
in es iga ed
some
opics
i n
he
geo
me y
[5]
and
Kon,
[6]).
Kobayashi
and
Tsuchiya,
(Koba
yashi
and
submani olds
o
codimension
g ea e
ha
2
and
wi h
la
The
pu pose
o
he
p esen
no e
is
o
s udy
in
a
ian
cu a u e
is
cons an
and
o
ob ain
a
condi ion
o
hem
o
be
mani olds.
On
he
o he
hand,
many
au ho s
ha e
s udied
in a ian
sub
mani olds
o
Sasak
ian
mani olds,
(see,
e.g.
, Kon ,
o.
S-mani olds
whi ch
educe,
in
a
specia
l
case,
o
Sasakian
I n
[1]
, S-m
an
i o
lds
, w
hic
h e
du
ce i n a s
pecial
case
o
Sasa-
ki
an
m
an
i ol ds , we e de i ne d .
In
his
no
e,
a
co
n
di io
n
o
an
i
n a
ia
n
s ubma
ni ol
d
o
co di
m~n
s
io
n
g ea e
ha
n 2 i n a n
S-mani-
ol
d
o
be
o a
l
ly
geo
desic
i s o
b ai
ne
d .
ON
TOTALLY
GEODESIC
INVAR
IANT
SUB
MA
NIFOLDS
OF ANS-
MANIFOLD
Dp
o.:
Alg
eb a
, Comp
u ació
n,
Ge ome
í
a y To
pología
.
Fa c
ul a
d de Ma emá i c as .
Uni
e s
id
ad de
Se illa
.
Apdo . de Co eos
1.
16
0.
410
80 SEVI LLA (Es
paña)
.
Re . Acad; Ci encias Za agoza , 43 (1988)
24
The
Gauss
-
Weinga en
o mulas
a e
gi en
by
~
is
he
Weinga en
R(X,Y,U,V)
=
RO(X,y,U,V)
-
g([~,~]X,y),
X,YET(M), U,VET(M).l,
We
deno e
by
~
he
co a ian
di e en ia ion
in
Nn
and
~.
(1.
2)
gi en
by
endomo phism
associa ed
wi h
V
and
i
sa is ies:
whe e
[~,~]X =
~X
-
~X.
g(~X,y)
=g«(}'(X,Y)
,V).
We
deno e
by
R, R
and
RO
he
cu a u e
enso s
associa ed
wi h
~,
'V
and
°
espec i ely.
I
RO
anishes
iden ically
he
no mal
connec ion
°
is
said
o
be
la o
The
Ricci
equa ion
is
(
1.1)
~XV
-~X
+0XV,
X,YeT(M),
VET(M).l,
whe e
D
is
he
connec ion
in
he
no mal
bundle,
a
is
he
by
'V
he
co a ian
di e en ia ion
in
Mm
de e mined
by
he
second
undamen al
o m
o
Mm,
l.
PRELIMINARIES.-
Le
Nn
be
a
Riemannian
mani old
o
on
induced
me ic.
Le
T(N)
( esp.
T(M»
be
he
Lie
algeb a
o
ec o
ields
on
Nn
( esp.
on
~)
and
T (M).l
he
se
o
all
ec o
ields
no mal
o
~.
dimension
n
and
Mm
an
m-dimensional
submani old
o
N
n.
Le
g
be
he
me ic
enso
ield
on
Nn
as
well
as
he
induced
me ic
sec ion
2,
de ini ions
and
so ne
p ope ies
o
S~mani olds.
In
su nma y
o
no a ions
and
o mulas
o
submani olds
and,
in
o ally
geodesic.
To
his
end,
in
sec ion
1,
we
gi e
a
b ie
sec ion
3we
ge
he
main
esul o
ze o.
a e
dual
on
g
i
a
Riemannian
me ic
such
ha ,
exis s
-s uc u e
wi h
F
closed
is
called
a
g(X,Y)
=
g( X
, Y)
+
~(X,Y),
X,YeT(N),
ields
A
no mal
(2.2)
F(X,Y)
=
g(X, Y),
X,YeT(N).
~a(~~)
=
°a~i
~a
=Di
~ao
=Oi
2
=
-I
+¿
~a
0
~a'
a,~e{l,
...
,s},
a
N2n+s
is
said
o
ha e
an
-s uc u e
wi h
complemen ed
ames.
is
de ined
by
K-s uc u e
and
N2n+s
is
called
a
K-mani old.
In
such
a
25
[ , ]
+
2¿
~
0
d~
=O,
a a a
whe e
[ ,
]
is
he
Nij
enhuis
o sion
o
.
Mo eo e ,
i
is
Fu he ,
he
-s uc u e
is
said
o
be
no mal
i
1- o ms,
hen
mani old,
he
~
a e
Killing
ec o
ields,
(Blai ,
[1]).
. a
Le
~
deno e
he
dis ibu ion
de e mined
by
_
2
and
~
. he
complemen a y
dis ibu ion.
~
is
de e mined
by
2
+ I
and
whe e
~(X,Y)
=¿
~D(X)~D(Y)
.
The
undamen al
2- o m
F
on
N2n+s
D
sa i ying,
(Yano,
[7]):
spanned
by
~l'
..
.
'~s.
I
Xe~,
hen
~a(X)
=O,
o
any
a
and
i
known
ha
he e
(2.1)
ec o
Finally,
he
submani old
MID
is
said
o
be
o ally
geodesic
in
Nn
i
i s
second
undamen al
o m
is
iden ically
2.
S-MANIFOLDS. -
Le
N2n+s
be
a
(2n+s)
-dimensional
mani old
wi h
an
-s uc u e
o
ank
2n.
I
he e
exis
on
N2n+s
26
in a ian
-sec ional
cu a u e
k,
hen
i s
cu a u e
enso
he
N2n+s
.,
[
1]
) .
Fo
he
X, YeT(N) .
an
S-mani old
on
o
g
- X,
XeT(N),
ae{l,
...
,s},
=
L[g( X, Y)~a
+
~a(Y) 2xJ,
a .
connec ion
~
-
2F(X,Y)F(Z,W)},
X,Y,Z,WeT(N).
R(X,y,Z,W)
=¿
{g( X, W)~a(Y)~~(Z)
-
a,~
-
g( X, Z)~a(Y)~~(W)
+
g( y, Z)~a(X)~~(W)
-
-
g( y, W)~a(X)~~(Z)}
+
+
«1/4)
(k+3s)
{g(X,W)g( Y, Z)
-
g(X,Z)g( Y, W)
+
+
g( Y, W)~(X,Z)
-
g( Y, Z)~(X,W)}
+
+
(1/4)
(k-s)
{F(X,W)F(Y,Z)
-
F(X,Z)F(Y,W)
-
In
he
case
s=
1,
an
S-mani old
is
a
Sasakian
mani old.
(2.5)
wi h
ce ain
condi ions
is
an
S-mani old.
In
his
way,
a
space
.
o
a
p incipal
o oidal
bundle
o e
a
Kaehle
mani old
(Blai ,
[2]),
(Blai ,
Ludden
and
Yano,
[3]).
Thus,
he
bundle
Fo
s~2,
examples
o
S-mani olds
a e
gi
en
in
(Blai ,
[1]
) ,
has
he
o m,
(Kobayashi
and
Tsuchiya,
[4])
an
o hono mal
pai
spanning
he
sec ion.
The
sec ional
Aplane
sec ion
is
called
an
in a ian
-sec ion
i
i
2n+s
is
de e mined
by
a
ec o
Xe~(p),
peN ,
such
ha
{X, X}
is
cu a u e
K(X,
X),
deno ed
by
H(X),
is
called
an
in a ian
-sec ional
cu a u e.
I
N2n+s
is
an
S-mani old
o
cons an
ollowing
we e
also
p o ed:
Riemannian
mani olds
ha e
been
s udied
in
(Blai ,
A
K-s uc u e
such
ha
F=
d~a'
a=
1,
...
,s,
is
called
an
S-s uc u e
and
N2n+ s
is
called
an
S-mani old.
These
(2.3)
(2.4)
XeM,
hen
X =O.
27
an
in a ian
submani old
o
an
S-mani old
is
such
ha
an
X =TX + NX,
(2.6)
(2.7)
and,
so,
m=
2p+s.
Fo
la e
use,
we
p o e
he
ollowing
Lemma
2.1.-
Le
M
2p+s
be
an
in a ian
submani old
o
P oo :
By
using
he
Weinga en
o mula
(1.1),
(2.4)
and
he
ac
ha
M
m+s
is
an
in a ian
submani old,
i
is
easy
o
show
ha
A VX =
~X.
Now,
i
YeT(M), we
ha e
g(A VX,y)
=
g(X,A VY)
=
g(X, ~Y)
=
-g(~ X,y)
l l
VeT(M) ,
o
any
VeT(M) .
Mo eo e ,
i
is
an
S-mani old
oo
.
2n+s
l
S-man~ old
N.
Then,
o
any
XeT(M),
VeT(M)
,
ae{l,
...
,s}:
i
de ines
an
-s uc u e
in
he
angen
bundle.
On
he
o he
hand,
i
one
o
he
~a
is
no mal
o
Mm,
hen
T=O,
because
g(X, Y)
=
F(X,Y)
=
d~a(X,y)
=O,
X,YeT(M),
The
submani old
~
is
said
o
be
in a ian
i
all
o
~
a
(a
1,
...
,s)
a e
always
angen
o
~
and
N
is
Lde
Lca
L
y
ze o,
i.e.,
XeT(M),
o
any
XeT(M).
I
is
easy
o
show
ha
angen
bundle.
I
is
easy
o
show
ha
i
T
does
no
anish,
no mal
componen
o
X.
Then,
T
is
an
endomo phism
o
he
angen
bundle
and
N
is
a
no mal-bundle
alued
1- o m
on
he
whe e
TX
is
he
angen ial
componen
o
X
and
NX
is
he
in oduced
as
a
canonical
example
o
an
S-mani old
playing
he
ole
o
complex
p ojec i e
space
in
Kaehle
geome y
and
he
odd-dimensional
sphe e
in
Sasakian
geome y.
Now,
le
~
be
an
m-dimensional
submani old
imme sed
in
an
S-mani old
N2n+
s.
Fo
any
XeT(M), we
w i e
gene aliza ion
o
he
Hop
ib a ion
':S2n+1~~~n
is

28
Ricci
equa ion
(1.2)
and
Lemma
2.1,
we
ha e
and
(2.7)
holds.
an
Le
M
2p+s
be
such
ha
he
no mal
(s-k)g( X, Y)
=
4g(~X,~y).
Now, we
p o e
Theo em
3.2.
-
Le
M2p+s
be
an
in a ian
submani old
o
an
.
2n+s
2p+s
S-man~ old
N
(k).
I
he
codimension
o
M
is
g ea e
han
2,
hen
he
no mal
con~ec ion
o
M2p+s
is
la
i
and
only
i
k=s
and
M
2p+s
is
o ally
geodesic.
P oo :
F om
he
Ricci
equa ion
(1.2)
and
(2.5),
i
is
clea
ha
i
M
2p+s
is
o ally
geodesic,
hen
i s
no mal
connec ion
is
la o
Now, we
suppose
ha
M
2p+s
is
no
o ally
geodesic.
We
can
choose
a
local
ield
o
o hono mal
ames
o
ec o
ields
in
M
2p+s
in
he
o m
R(X, y,V, V)
=
2g(~X,~y),
o
any
ec o
ield
X,YeT(M)
and any
uni
ec o
ield
V~T(M)i.
Now,
om
(2.5)
we
ob ain
Then,
we
ge
he
ollowing
p oposi ion
3.1.-
Le
M2p+s
be
an
in a ian
submani old
o
an
S-mani old
N2n+
s(k)
wi h
la
no mal
connec ion.
Then,
k~s
and
he
equali y
holds
i
and
only
i
M2p+s
is
o ally
geodesic.
(3.1)
in a ian
submani old
o
N2n+
s(k)
connec ion
o
M
2p+s
is
la ,
i.
e.,
RD=O.
Then,
by
using
he
-sec ional
cu a u e
is
a
cons an
k.
3 . INVARIANT
SUBMANIFOLDS
WITH
FLAT
NORMAL
CONNECTION. -
In
his
sec ion,
le
N2n+s
(k)
be
an
.
S-mani old
whose
in a ian
29
is
o
cons an
in a ian
-sec ional
cu a u e.
again,
ha
~
+
~
=O,
and
V.
Rega ding
o
(3.3),
i
ollows
ha
g(~X,~y)
=
O,
o
any
X,YeT(M).
Consequen ly,
he
ec o
ields
~(E1)'"
.,~(E2p)
,~(E1)'··
·
,~(E2p)
a e
linea ly
independen ,
which
"is
a
con adic ion.
The e o e,
M
2p+s
is
o ally
geodesic
and
k=
s.
in a ian
submani old
o
codimension
2
in
an
S-mani old
N2n+
s(k).
Then,
1 "l+s
is
o ally
geodesic
i
and
only
i
1 "l+s
Finally,
o
codi oension
2,
we
ha e
he
ollowing
Theo e o
3.3.
-
(Kobayashi
and
Tsuchiya,
[4])
Le
1 "l+s
be
an
"
~
o
any
o hono oal
ec o
ields
V,WeT(M) .
Thus,
o o
(1.2)
and
(3.2),
we
ge
R(X,y,V,W)
=
2g(~X,~y),
X,YeT(M).
Using
(2.5),
we
ob ain
{E
1,···,E
p,E
p+1=
E
1,···,E
2p = E
p'€l'···'€S
}·
I
~(Ei)
=O,
o
so oe
uni
ec o
ield
VeT(M)~,
he
~,
o o
(3.1),
we
ge
ha
M
2p+s
is
o ally
geodesic,
by
i ue
o
P oposi ion
3.1.
Thus,
~(Ei)
~
O,
o
any
Ei
and
V,
and
so,
~(E1),
...
,~(E2p)
a e
linea ly
independen .
on
he
o he
hand,
i
is
easy
o
show,
by
using
(3.1)
(3
.3)
(s-k)g(X, Y)g(V, W)
=
4g(~X,~y).
I
he
codi oension
o
M
2p+s
is
g ea e
han
2,
we
can
ake
a
uni
ec o
ield
W
in
T(M)~
which
is
,
o hogonal
o
V
(3.2)
[2]
REFERENCES
[1]
Blai ,
O.E.:
1970,
J.
oi .
Geom.
~,
(155-167).
:
1971,
An.
i.
Uni .
"Al.
l.
Cuza"
la
i,
(Se ie
no a)
XVII,
Fase.
~,
(171-177).
[3]
Blai ,
O.E.,
Ludden,
G.O.
and
Yano,
K.:
1973,
T ans.
'
Am
.
Ma h.
Soco
181,
(175-184).
[4]
Kobayashi,
M.
and
Tsuchiya,
S.:
1972,
Kodai
Ma h.
Sem.
Rep.
24,
(
430
..:
45
~)
.
[5]
Kon,
M.:
1973,
Kodai
Ma h.
Sem.
Rep.
25,
(330-336).
[6]
1974,
Tenso
N.S.
28,
(133-138).
[7]
Yano,
K. :
1963,
Tenso
14,
(99-109).
30