Internat. J. Math. & Math. Scl.
Vol. 9 No. 4 (1986) 733-747
733
PSEUDO-SASAKIAN MANIFOLDS ENDOWED
WITH A CONTACT CONFORMAL CONNECTION
VLADISLAV V. GOLDBERG and RADU ROSCA
l)epartment of Matl,emattcs
N.J. Institute of Technology
Newark, N.J. 07|()2
U.S.A.
(Received January 30, 1985)
M(U,E,,,g) endowed wlth a contact conformal
It is proved tlat sucl manifolds are space forms
M(K),K < O, and somo remarkable properttos of the 1,ie algebra of infinitesimal
ABS’I’RACI’.
Pseudo-Sasakian manifolds
connection are defined.
transformatton. of the principal vector feld
of tle leaves of a co-tsotroptc foliation on
TM
bundle manifold
having
lq
U
I’!
M
on
are discussed.
Properties
and properties of the tangent
as a basis nro studied.
Wtt J’’e, CICR sd,m,tiJ’old, relative conac infinitesimal
tvmts fonatio, U-contact cmciculat" pa,{,v., differential form of Godbillon-Vey,
fotn of E. Cat’tan, Finslevian Jb, mech,fca7 sjstem, dynamical system, spray,
KEY WORDS AND PHRASES.
CR product.
1980 SUBJECT CLASSIFICATION CODES.
1.
53C25, 5.’;C40, 53B25
INTRODLICTION.
In tle last years many papers have been concerned with Sasaklan mnifold
M(,,,g)
and related structures.
manifolds M(U,,q,g)
Recently Rosca
and Goldberg and Roca
[2]
[]]
las defined
have studied CICR
8nod
(i.e. co-isotropic CR submanifolds) of H(U,,n,g).
(2m+l)-dime,sional pseudo-Sasaktan manifolds
re+l, m > 4, structured by a eo,,re eonfomna (abr. e.e.) connection. It
is proved that such manifolds are hyperbolic space forms M(K), K < O, and with the
In the present paper we study
of index
e.e. connection (which in fact is a natural generalization of the connection defined
by Rosca
vector
[3])
fZd
osca [3])
is associated (compare witl,
U.
.
a so denominated
I, Section 3 we develop some basic results
The paper is organized as follows.
induced by the c.c. connection and some re,,,.rkable properties of the Lie algebra of
infinitesimal transformations
(resp.
(i)
on
or
(il)
u
U)
defined by
is divergence
J’ee
and all connection forms
(se Li=hnrowicz’
o,
It is sho that
(resp. defines an
M
infinitesil homothety)
are Integral relatlon of invaianoe
[a])"
U-contact cm,imcul pairing (in the sense of
eontaot
Rosca [5])and any contact extension of O Is a relative
and
U
define an
734
V. V. GOLDBERG AND R. ROSCA
infinitesimal transfoation (n
(Ill)
canonlcal
form
U
define botl influ[tesl,,,al automorphlsms of
UU
and
q
Lq
(q<m)
where
(resp. l.)
u
sooOP
c
are
c
CICR
{q},
of
eo-gsot’opie ogatgon
Section 4 is concerned tth
of
submanlfolds of
M
and f
Golon-Vey on blc (see Liclmerowlcz[6])
true integ,al 4nuargant of U
is a
on
M
codlm
,
c
(2+I)-fo
w
UM
Further the
necessa
c
and sufficient conditions for
the isotroplc component
[;]
Yano and Ken
of
U
(2q+l)-forms
(resp. the
?
d/2).
gg
be considered as
of the
Is the dual form of
(l,l)-operator taken with respect to the 2-form
is the exterior dlfferentlal sgste,,, defined by
M
[3])
the ense of Rosca
M
may
The leaves
fo of
which Is a tel.a-
G
feZ{ate
is a CH
Mc
U
and
then the
to be
In thls case
vanishes.
.
Accordlngly, lf
is that
product (see
Rosc
nd
[9] and 1 by
cert properties of he angent bundle
Finally using soe notions Introduced by 7ano and Ishthara
KIetn
[10],
antfold
ue consider in Section 5
’1’
(,,,g)
hatng the manifold
It Is proed that the
c,
as n basis.
ompgt lfJ’t ("
ad
u
of
and
respectively
u
on TM
homogonoo of degree one and tlat tl,..I of E. eat’tan
F{nner{an fo,,. Furthermore, we may associate with
a regulaP
is a
are
2
PRELIMINARIES.
Let
(,)
tgnature
ba a
01,)
(2l)-dimanstonal cmnacted pseudo-Rtemanntan ntfold
> 4.
and suppose that
At each point
p e M
one has the standard decomposition
T()
P
(see Rosca
[1]):
(2. l)
P
P
H,
and T% are the tangent space, a (2m)-dlmensional nut vector
P
t{me-l{ke
llne orthogonal to 1, respectively.
space, and a
P
Let S, S H be two 8eZ-oPthoonZ (abbrevlatlon s.o.) m-distributions
where
T,
P
,
P
P
P
which define an inuolut4ve automorphlsm
operator defined by Libeann
[11])
U
Let
on
which defines a contact structure
of square
,
e r%
P
and
and
+I
(U
e
is the
()
p
be the pairing
be the covartan
tton operator defined by the metric tensor g. The if for
the structure tensors (,,,g) sattsgy
on
any vector fields
’
g(Z,)
%.% %
dn(Z,Z’)
V,
n(Z),
%.%
-2g(UZ
’),
(2.2)
UZ,
,,()
1,
(, ,g) hs been elled pedo-Saoakgan mangod (ee Re [1]).
In rder to study real eo-gott,ope nd gaotopge Jgatgon n
(that
gml’’opev terstons tn ), we consider an adapted gteld of tt
,B,C
2}. The vectors h
0, 1,
and h
(al
re nu
a
a*
and
is the an{8ot’op4e vector field of the W-basls {h
We set
the manifold
;
;aa)
A}.
{hA
ho
Sp
[hal,
P
[ha }
(2.
PSEUDO-SASAKIAN MANIFOLDS
735
and as is known, one ha,
g(ha,hb*)
g( Ii
5al),
0
(2.4)
and
Uh
If
{
d (d
W
meut
{)
ha
a
U
Uh * =-11 *,
a
a
W, we
is tile cobasis associated witll
(2.5)
O.
set
q
and the line ele-
is a canonical vector l-form and is independent on any connection on
is given by
d ’A ’ ’A"
It [ollows fom (2.4) that tle metric tensor
g
2
g
(2.6)
Is:
+
O
(2.7)
a
If
A
A
curvature
C
"Be
h
C())
(BC
are the connection forms and the
B
respectively, then the structure equations
.d
-
()
2-forms on the bundle
.;,.,,
(E. Caftan) may be uitten in the indexlcss form as follows:
vh
d
-t,
d
Referring to (2.4) and (2.8), one
eb
a
,-- O,
+"0
a
o
0
(}
a*
By virtue of (2.8),
0
(2.9),and (2.11)
b*
-
’,a
a*
I,-,,
o,,e
0,
(2.11)
a
O
a
(2.0)
+
and
0
(2.9)
+o
:!
0
(2.8)
(2.12)
(2.13)
a
and
(2.14)
where
and
’
are any vector fields on
In tire following we agree to call tlte
2-form
(2.15)
Since by (2.11) one has
a
a
@a
,= O,
a
(2.16)
we shall call
R
a
(2.17)
a
and
a
a
(2.18)
736
V. V. GOLDBERG AND R. ROSCA
the Rfec
l-oPm
the form
and the
Rcc 2-olvn
defines the
R
i’sA
[12]).
respectively (see Rosca
oy
class
Cheu of
As Is known,
H.
Using (2.10) and referring to (2.12) and (2.15), one quickly obtains
The above equation proves that the
Hence the two
coctdcZ.e8
F
Let now
and
R
c
is a
oonao
f
(1)
c),
is {nvant i.e.
D
H,
o[
T (H
p
p
satisfying:
)
by suppressing
H
f,a,iant i.e.
D
The distribution
distribution.
p c H
UD
c
(resp.
Goldberg and Rosca
H
p--
Da)
[]
that
M
c
D
D
p
D
P
is called the hoPizontal (resp.
submanifolds is called CICR
CR
of
a maximal
(oae denotes the Induced elements on
UI)xC 1x( ).
p- p c
Such type of
H2()
2-cohomoogy 0a88
and denote by
Is there exists a dlfferentlable
tl,.,t
and
p
the complementary orthogonal di.tribution
(ii)
are homologous.
I
It lws been sl,own by Goldberg and Rosca
C
D
D: p
distribution
belong to the
smanJd
CR
and
R
be a colsotroplc foliation on
Integral manifold (leave) of
bl
2-forms
T (H c)
P
is
an-
vertcal)
manod8
(see
[2]).
PSEUDO-SASI bIFOLDS ENDOWED Wl’l’ll A CONTACT CONFOL CONNEXION.
As a natural generalization of the de[[,ttton given by Rosca [3], we assume
tha the structure equations (2.9) are urltte in the
3.
(u+n) A,,
,
(.-,,) a
d
da
where
d/2,
il
and
t,
a,ta,
u e A
e C (H), and
+
+ ?a
(3.)
a
()
Is a closed l-fo.
Note that
are the components of a vector field
U
Z (tl,a+ta, h a ,)
(3.2)
of eonat length.
We shall say (see Rosca [3]) that In tlIs case the pseudo-Saaaklan mnlfold
Is endowed with a eonae$ oonfoa (abr. c.c.) eonneotgon. e aIo agree to
U the princpa vector feld associated with this connection.
Since
(?,71
const, we may write by (3.2) that
[ ta, ta,
c, c
const.
(3.3)
a
Taking exterior diferenttals of (3.1), we get
d?a {+’)?a
%
enote by
and by I
where
2
2a*
(3.4)
g the exterior dtferenttai syte degtned by equatt
(3.1) nd (3.)
the gdea correpondtng to g
The exterior dtferentttt
(3.4)
%a
and m
satisfy (3.1),
Because of this, dl I, that is E
that the system
? d/2, d
s
a
O, leads to the identity.
eZosed system. It follows from this
E defining the pseudo-Sasakian antfold
endowed
enneeton is eompleeZ ;nt’able and ts solution depends on 2 entnts
(the number of equations in (3.4)).
PSEUDO-SASAKIAN MANIFOLDS
737
From (3.4) and (3.3) we also obtain
[
cu
and
()
%
0
a
(t *=-t
a
(3 5)
a
which shows that u is an :,ztegeal reZation
of invoice for
(see Licl,nerowlcz [4]). In the following we agree to call u
the principal
PJhJ’fan associated with the c.c. connectio,t.
Consider no the
U
l-form
a
aktng the exterior dtfgerent(al of v, one finds tth the help og
(3.1) and
(3.4) that c
2. In this case we deduce
d= 2
and this equation asserts that
v
(3.)
is exe’iop PeeuPren (see atta
[13]
2
tth
s the ’eem’enee l- fo.
gy (2.4) and (2.5)
one easily ginds
t
Hence if
Poor’ []4])
T()
T*()
?
a a
,.
(3 8)
Is the musicaZ f:’o,nohgsm with respect to
(see
"’ Since
b(O?), =b (U).
is closed, it follows
ffrom (3.7) that the mntold N under coun lderatton is oltated
by 2-codtmensional
ubmantlds orthgonal t
and
Next if ,:
+
T() T*() is tl,e bundle isomorphism defined by
R
d/2, one readily gtnds
we may write:
.
u
i?,
p(?)
2;
(3.9)
In the following we agree to call the p)’e:)!/m/,let{o form
fundammztat
Let now
(dim
ker()
O)
the
2-form on
f
(
+
C=())
Lle derivative with respect to
Uf.
be a eontt eztengon of
Then by (3.9) one quickly finds
Therefore according to the definition given by Roses
[3],
and
f
f
we may say that
the
O.
f
z’eZaive contact infinitegmaZ transformation of
Denote now by
S (reap. oS) the simple unit form hteh eorropond
). One has
(resp.
a
OS
%
OS*
A...A
(3.10)
1" A...A %m*
and by (3.1) the exterior differentials of (3.10) are
ds ["(+)-] ^ 7s
" " "" OS*
ds* [m (u-n)+vJ
^
,b
(3. II)
Since
o
and Os, are both exterior recurrent, it follows from a well-known
S
property that both co-lsotropic distributions S + {} and
+ {} are
t{ve (orth.
S*). It is worth to emphasize tl,at
orth.
(+{})
;
(*+{.})
this property is true for any pseudo-Sasakian manifold.
*
V. V. GOLDBERG AND R. ROSCA
738
Now wltll the help of (3.1), one flndn tllnt tlle connection forms are given by
a
.,
+
a
/2
+
a’
(,,o ,,,on)
3. 2
,b*
a
’a
By (3.12) and (3.6) one finds
R
and (3.7) shws that
R
is exterior
Comt,g back to reIatto,s (3.12), oe re;dlly [tnds
Therefore we may say that all connection forms of the pseudo-Sasaklan manifold
under consideration are ntt’a ’eZatoa of m,aganoe for the vector field UU.
H.
he volume elemet of
Denote no by
Oue ay take a ieal rtentatton
such tha
o
A 2m+ 1- q*
AqT
,:
and de,ote by
s A oS, A
the aP
n,
opPatoP determined by
e
any vector ield
If, llke
TM, then, a is known, for
means the vector space of sections over
sually,
.
(3.15)
one has
(3.16)
Haktng use of (3.4), (3.11), (3.16), and tho fact that
[ (taha+ta ,ha,.
U
one finds after some calculations:
dlv
llence
U
is
div(U)
d{verenoe fPee and UU
%A’
Now If
(see Poor
0
[14]),
one has
’
2
t3
(3.18)
4
,lff{niesimaZ homohe on
is an
(’)AhA
[ tata,
a
are any vector fields, then, as
Is known
Therefore, by (2.3), (3.4), and (3.12)we get
We also note that since
(()+c) ); ()
is a closed
(U)
According to the defniton
gven
by Rosca
form, we may say (see Poor
[5]
the [ormulae (3.19) show that the vector field
Denote by
D
U
the
3-distrlbution defined by
[14])
and Rosca and Verstraelen
defines a
{,g,}.
[]5],
U-=on =ooz’o=P
By (2.2), (3.5),
and (3.6) one readily finds from (3.19) that
.
[,]=0, [u,]=0.
Hence both vector fields
defines a
see that D
U
U
and
U
3-folat4on
conmute with
on
E
(3.20)
and by (3.19) and (3.20) we
.
PSEUDO-SASAKIAN MANIFOLDS
739
It is worth now to make the following considerations.
Z e
Let
M
be any vector field on
fo,,,uZa (see Poor [14])
:
on
T|len
one has the general
BoohneP
2
"
wlere
do
+
is the
6od
Lapaoe-Bol 1.,n operato (or Laplacian)
and the trace (abr. tr) is calculated
Applying formula (3.21) to the
w|tl,
respect to the metric tensor
g
AT*,
of
and taking into
vector field
l, ri,,ell,.sl
on
account (2.7), one I,as
(3.22)
a
a
q
a
a
ad
11/i2
2
[
a
’
<1, a"h a, J> + <Vr_,U’lU>
(3.23)
Now by (2.14), (3.4), (3.5), (3.16),and (3.19) one finds
*Vhal
V
a
h
tata,U
+
a
Since we have found
(2-a ’a*/2)U[ + (3aa*/2-2) +
(2-tata,/2)UU+(3’a a /2-2) +
[ rata,
2
we dertve from
(3.22)
ta,ha,
(3 24)
(3 23), (3 24), and (3 21)
a
sattsgtes
U
that
L
Let
(3.21) nd this equation
is consistent tth
be the operator of type (1,1) de[tned by the fundental
q
Lq
Denote then by
Since u and
e A2q+l
are both closed,
q
+
td
ence
()
(3.9) and aklng use of the properties f ie Lie
one finds by
dt
that
q
gngnitegma autorphism of all (2q+l)-gor
(
On the other hand, since g(,)
const, e ay ay in tIar
is an
the case
a Sasaktan antfoId that
defines tth
).
an
Like usually denote by
the eueatue
by
and
)
operator. hen, a
t given by
is known, the eetgona
oeat
(,)
degtned
where
R(,U,J ,UU)
’
(R(’,U)U,U).
(3.28)
Making use of (3.5), (3.6), and (3.19), one finds
[, U]
4 (+2)
(3.29)
4 (5+8)
(3.30)
and
R(, U)U’
V. V. GOLDBERG AND R. ROSCA
740
-.
K(II,Ui)
(3.28) one gets
llence by (3.27) and
Now referring to (2.10) and
(3.12) one finds after some calculations
S
a
a
VS, + v s
A
+aa ,
,,
’a
A
a
Vs
A
taU
(.)
(no suaation)
+
where we have set
(3.32)
As is known (see Libermann
b
a
c*
a
bc,
A
a
[]] ]),
R’foef.
the components of the
a*
+
a*
bc
tb
aa
/:.ensor are given by
Because of this, we get from (3.31) that
0).
c
(3.33)
a a
o
I follows from (3.33) tha tle componenC of the Rcci tensor are
Rosca
[16]).
In addition, since the 8eaa cu’vauve
tensor with respect to
of,
the
Rieci
C
4-m
(m > 4).
C
(see
is the trace
g, one finds by (2.7) and (3.3) that
Therefore we conclude that tie pseudo-Sasakian manifold
space fo (4-m) of hpePboe
(U,,,) be a pseudo-Sasaktan manifold
M
under consideration is a
THEOREI 1.
Let
c.c. connection and let
d/2)
(resp. fl
endowed with a
be the principal vector field
associated with this connection (resp. the fundamental 2-fo on
).
One has
the following properties:
(i)
(li)
U
is divergence free, and
U
defines an infinitesimal homothety on
are integral relations of invarlance for
all the connection forms on
UU;
(Ill)
U)
and
define an U-contact couclrcular pairing, and
defines a 3-foliation on
(iv)
f
any contact extension
+
Infinitesimal transformation of
(v)
U
and
is a relative contact
U
q;
define both an 4nflniteslmal automorphism of all
Lq
(vi)
of
where u
q
the Ricci 1-form of
is the dual form of
(2q+l)-forms
U(q<m);
is exterior recurrent, and the Ricci tensor is
M
disjoint;
(vii)
(viii)
M
is a space-fomn of hyperbolic type
any such submanifold
N
_
is defined by a compietely integrable system of
differenttai equations whose solution depends on
co-isooc F0t^o
o
?(v,,n,l.
UD’_ ’)
D=
b)
arbitrary constants.
M the following three distributions:
r (i.e.
of dimension 2(m-)+l
D
An gnarag distribution D
i*=i+m}.
m-;
i=l
by
defined
{hi,hi,, x
of dimension
orth D
(i e
D
An isoropc distribution D
We shall consider on
a)
2m
defined by
A
D
{h.; r=m-E+l
m}.
PSEUDO-SASAKIAN MANIFOLDS
c)
A transversal distribution
by
{h
Dt
r
,; r*
741
Cs,(D’ D)IS *
I)
defined
of dimension
2m).
2m-+1,
a
These three distributions have no conunon direction and tley define on
of
tu’e
rank
(see
2
Sinha
[]7]).
as follows:
Accordingly we shall split the principal vector field
U
where
U
T
e D
"
e D
U
e D
t
-st’uc-
U
q)
t.
’
t
(4.1)
Denote now by
g,2m- +]
the simple unit fo which corresponds to
ell-detned global gor.
Since
x
ctent condition got
eZas of
g
Ltclnerolcz
on [7])
0
.A2m
(4.2)
Because
t.
anntltlates
oo-soeropgo
to be a
[la]
exterior recurrent (see Lichnerowtcz
d
tlence one ust rtte
A.
,
D
ttI(c,R)
t that
be
[7]).
represent the
l(
y defines an eleen og
,R) (see
In the case under dscusston one gnds (co,pare tth gan nd
then the recurrence l-form
[6]).
that the necessary aud sufficient condttten for
Fc D Dx
tropic foliation
is a
the necessary and
ogation
and Yano and Kon
and if
is orientable,
t
case the recurrence
l-form
y
of
t receive a
t
Is that the component
of
vanishes.
In this
is given by
(.3)
(u-n).
Denote by
M
for the
and supress
M
c
According to the considerations of Section
8ubmanoZd.
Fc
(2m-+l)-dlmenslonal leaf of
a
c
induced elements on
By definition we have
I,
it follows that
Mc
is a CICR
Because of this and (3.1), the
O.
du
exterior differentiation of (4.3) gives
dy
Equation (4.4) shows that the restrlction
On the other hand, the form of
the
(2+l)-form
w
G
e
A2+I(M c
w
One knows (see Lichnerowlcz
an element of
H2+I(Mc;R)
[18])
M
Codb[||on-Vey
[ve,
(4.4)
-2.
is an
exact
(see Lichnerowicz
[6])
on
M
c
is
by
yA(^dx).
G
forum.
c
(4.5)
that the class of cohomology of
is an luvarla,t of tle folIaton.
which is
w
G
Using the same
notation as in section 3 and applying (4.4), we may write
w
where we have set
c
_2
G
+1
c(Lu-L,)
c(8-Ln)
(4.6)
Thus it follows from (3.22) that
uwG -CU (L n).
(4.7)
By means of (2.13) and (3.9) one has
d(Ln)
2(A)+I
(4.8)
742
V. V. GOLDBERG AND R. ROSCA
and
dIu(L,)
(4.9)
4u(A)
Therefore we get
LI(LED)
-B
A(A2)
-11
(4.10)
and finally
uWt:-- cB
Since
that
B
(4.11)
closed, the above equation gives
0 and allows us to say
G
’eae
ieg’a
of
int:
U.
G
Further since the submanifold M
is co-isotropic, t follows from ths that
is
duW
is a
w
M
the normal bundle
M
Since
of
M
C
is defined by
C
coincides with
r
m
O, r
D
x.
2m-+l,
and (3.12) that the covariant derivatives
2m, we derive from (2 8)
of the null normal sections
Vl
h
satisfy
Vh
Since
h
h
(4.12)
r
are null vector fields, equation (4.12) shows that
r
dlrFctlons.
has the
v
r
Hence according to the definition of Rosca
,
eoeo opey.
X
Further if
Y
and
are any vector fields of
according to a known definition, the distribution
r
wth the improper inmerslon
x: M
Hoeso
one has
is
(E
is a field of
.
D
one may say hat
VyX
moZe.
D
D
are
emg qmd,= o8
for the meod
-<dp Vh r >
Setting
[]9],
hr
D
Thus
associated
covarlant
setric
M ), we deIve bv a simple argument that all
vanish.
Tlerefoe accodig to a we11-known deflltlon, we agree to say that the improper
tensors of order
2
on
x: M
im,nerslon
is
[2]
It was proved by Coldberg and Rosca
If
involutlve.
oreo.
l,olP oII
are the leaves of
that the distribution
one easily finds that the improper Immersion
es?a.
Slnce
x:
M+
x:
[Z])
Next as it was proved (Coldberg and Rosca
wlich corresponds to
M
D
to be
.
Since obvlously one has
Since by definition in tlis case
(2(m-)+l)-dimensional leaf of
Jl[,
M
is
PoP
o#
M
#o#Z Heo-
ee.
the necessary and sufficient
is that the simple unit fo
D
A
then by (3.1) one finds that the
q equivalent to the condition
[ff
O.
Is i,volutive, let us denote
Is a
subnlfold, M
M, and this implies (see Romca
Because
is as is kown an invariant submanlfold of
tl,at
is always
be exterior recurrent.
property of exterior recurrency for
a
is
is a proper mmrsion, it is
condition for the manifold
D
then in a similar nner as for
I
M
CICR
mmZ.
Coming back to the case under discussion, using (2.8), (3.12) and the fact
that on
that
M
M
Hence
manifolds
U
one has
t
O,
O, we can show by means of a imple calculation
is also totally geodesic.
M
is follated by two familie,q of orthogonal totally geodesic sub-
and
M
r
PSEUDO-SASAKIAN MANIFOLDS
On the other hand, let
[]],
Rosca
one has
Mc
X
PX + FX
UX
the normal) component of
be any vector field on
&
c
=M
H
According to
c
(resp.
M1", one
virtue of tle total geodestcity of
Therefore the tangential component
M
M
is the tangential
VPX
easily finds that
[7],
Yano and Kon
(resp. FX)
I’X
where
gy
UX.
743
PX
it follows from tlts that
of
X
lq
is a
paraZeZ.
is
CR
According to
product
i.e.
1".
Mc
Since
is connected, this property can be checked by
de Rhom decompoo
theo’em.
It is worth
to note tllat this s[tuatlon is quite similar to that of cosubmanlfolds of a para Kaelilerian manifold structured by a geodeio
connection (Rosca [20]).
CR
Isotroplc
TIIEOREM 2.
U
tion and let
M
Let
be a pseudo-Sasaklan manifold structured by a c.c. connec-
be the principal vector field associated with this connection.
Then the necessary and sufficient condition for
c
foliation
to receive a co-lsotroplc
U
is tllat tile transversal component
case tlle leaves
M
F
of
c
c
tile form of Godbillon-Vey on
integral Invariant of
of
CICR
submanlfolds of
Mc
is a
(2+l)-form
[M
U
t
are
WG
In thls
vanishes.
M
and if codlm M
c
which is a relative
c
,
c
In addition, one has the following properties:
(i)
(ll)
x: M
M is improper totally geodesic;
c
is foliated by anti-lnvariant submanifolds M
which are improper
the improper immersion
H
c
totally geodesic and have the geodesic property.
Further the necessary and sufficient condition for
the vertical (or isotropic) component
CR
Mc
is a
5.
TANGENT BUNDLE LMq I FOLD
Let
U
of
U
In this case
TM.
tangent buMZe ,naij’old
be the
to be foliate is that
M c vanishes.
product.
TH
H
laving the pseudo-Sasakian manifold
discussed in Section 3 as a basis.
(A)
the ca,oncaZ Wet,n,
L
Accordingly we may consider the set
Denote by
on
TH.
TM.
cobasis on
the vetieaZ
Following Godbillon
differentiation
taken with respect to
i
v
is a derivative
,[r
8*
we shall
designate by
and the vertical derivation
(d%r
of degree
[2]],
fiel.d (or t/zo vector field of
8’ {A, dA} as an adapted
d,,
opez’aor,
is an antidevuat{on of degree
0
of A
(r,s) on
r
into
the verticaZ and compZete lifts are linear mappings of T
s
Let
.
and
respectively
of
T).
be the set of all tensor fteids of type
LiouvilZe)
AT
and
In general
r rT,
and for
complete lifts one has:
(TtT2)C
With respect to
by
*
T 9
,,(:+.r C e T V
2
2
the complete lift of the fundamental form
%C
n
[ (daAma*+m%a Adva*_
Tile exterior differentiation of (5.1) by means of
(3.1) gives
fl
d/2
is given
(5.1)
V. V. GOLDBERG AND R. ROSCA
744
(5.2)
(5.2),
Using
we find
i (:- ’i (:
[2]]),
As Is known (see Godbillon
of degt’ee
e
(5,3)
e(Imtion (5.3) shows that
.
1.
C
ill no take the complete lift
*.
respect to cobasis
[7])
u
For this purpose e shall
(A=O,I,...,2B) iti
the general theory (Yano and Ishihara
the Pfaffian derivatives of
hB(t A)
aB(t A)
ts
of the principal Pfaffian
associated ith the c.c. connection with sructures
denote by
C
Then according to
t
A
one has
wlere we have set
u
%
,A
C
u
%A
BA
+
UAV
Referrln
UA
(5.4)
B(UA)V
(3.4) and (3.5)
to
(c=2), after some
calculations one finds
u
=[ l(a,,V-t
,
,
+
,,v
I taV-tav
)+
)u
-vv
+v
The exterior differentiation of (5.5) by means of (3.1) gives
dC
I(I (aav+ta v’a) + [ ’vaa*-va*a)
a
.a
dv
Using (5.5) and (5.6), one finds
+t
d a*))
%C
u
%
V
of degree I.
uC
Hence
%a%a*
T:
2
T.
and apply tl,e vertJc.,] differcntlaton of
,
and by
mefllls
dw I
/%t,a
tv
(,,
According to Godbillon
,a %a
+
+v
%0
[2]],
one
(5.8)
v n
of (3. I) o.e gets
"
C TN
respect to
uC is also a homogeneous fo
L
Consider now the following scalar feld on
%
(5.6)
A
8’
-
fia* )-d) +2
is tim operator (,[ Yano and lshihara
one has by
(3.6)
a
a
,v
+t
;I
).
[7],
(5.9)
that is with
(5. o)
One qulckly finds
(5.)
L
a.d since
is closed, it follows from
(5.11) tha
PSEUDO-SASAKIAN MANIFOLDS
I
i.e.
I.
is l,omogeneous of degree
745
blorpover, taking the vertical derivation of
II, one has (see Codblllon
On tle otler land, It is easy to see fr,, (5.9) that
[21])
(;odbillon
II
tl,at
onTH.
FnsZPIn
is a
de
differentiation
Is of maxlmal rank (see
II
Accordingly,as i kuow, equations (5.11) and (5.13) prove
Jtvu
(See EI.I,,
.,,,I Voutier
is an anti-derivatio, ,,f
[2]).
square zoro,
Since tle vertic,l
one easily derives from
(5.8) that
d
’l’l,uq
oJ"
l-fo?rO
cavra,,
V
of
116]).
(see Rose:,
Further one may call
tl,o
B
tle vectorial basis dual to
M.
on
Dv
[9]
(see Yano and Ishiimra
or (:odbilion
[21])
tl,e vertical lift
Tien
() V
is expressed by
()v
(S. IS)
A
Coming back to the case under co,,sd,,ration and using that
(see
of
(U) V
wltlt respect to
II
are both closed, it follows tom this that
and
since
V
@a
v
a
Now, taking the duai V(tl)
(c:2), we quickly find
V
U =-1()
3), we find by (5.15) that
Section
(U)
funot{on (resp.
[19]).
(see Rosca
{h A,
Denote now by
as is know,,
TM
on
"r
on
(rosp. ) tire L?[ouo{ZZe
T
following we shall call
the Lot(t,Ze
2-J>,vm
(5 14)
()
Vl
[21],
;ccording to (;odbillo,
]n tile
o,
r
and referring o (3.5)
(u)V=
0, i.e.
is an infinitesimal aucomorl, ltsm of
and
(’b
are tl,e kineto
u
Since
ezergv and
tire
J’ieZd of Jb’’ns
is closed, one Ires
finds
,
d
-[
T
hand, since
8tl’,tur on
[21])
tle
and
(see Codbillon
and referring to
[21]).
(5.7), one quickly
2,
(5.19)
Lr
Equations (5.19) show that
,r
of
7.
are
lomor,eou8 of ctf?Pee 2. On the other
potent{a sympe{c
is an exact 2-form of maxim,l rank, it defines a
TH.
ilence, according to tie dof[,[tion given by Klein
sysem
Denote no by
is
Z
d
(see Godbtllon
veuZav.
the
dneZ t’m
asctated wltl
s
is knon,Z
d
-
V. V. GOLDBERG AND R. ROSCA
746
is deflned via formula
idl
’[’len:
a)
"’"!
b)
Since
,
and
Since
on
lr
i.e.
[L,d] d"
2-form
is of degree 2
manifold
(U,,n,)
vector field
(i)
Let
TI
L:) + "
(5.20)
are both homoget, ots and of the same degree, Z
d
integral relation of
TIIEOREM 3.
d(q
l-(d-)
the
invaria,,ce
for
+
Z
2
A dt e A (TxR)
(Lichnerowlcz
is a
is an
[5]).
d
be tile tangent bundle manlfold having as a basis tile
(resp.
defined in Section 3 and let
(resp. the fundamental 2-form) on
C
the complete lifts
and
b.
)
be the principal
Then:
--(U)
uC
of
TM
is a Finslerian form;
and
are homogeneous of
degree one;
(il)
(iii)
the 2-form of Cartan
H
system is a spray on
on
a regular mechanical system whose dynamical
one may associate with
M.
RF, FERENCES
On Pseudo-Sasakian Manifolds, Rc,d. Mat. 1984 (to appear).
I’
ROSCA, R.
2.
GOLDBERC, V.V. and ROSCA, R. Contact ’o-lsotroplc CR Submanifolds of a PseudoSasakian Nanifold, lntern. J. Natl. Natl. Sci. !(1984), No. 2, 339-350.
3.
ROSCA, R.
Paris
4.
5.
6.
7.
8.
9.
I0.
II.
Co,te|o, Conforme de Contact, C.R. Acad. Sci.
Vari@ts SasakIenne
Sr. I Math. 294(1982), 43-46.
LICHNEROWICZ, A. Les Relations lntegrales d’Invariance et Leurs’Applications
la Dynamique, Bull. Scl. Mathu_(1946), 82-95.
ROSCA, R. Varits Lorentzienneq h Structure Sasakienne et Admettant up
Cllam I) Vectorie| ]sotrole .l.-quasi Conclrculaire, C.R. Acad. Sci. Paris
Sr. A. 291(1980), 45-47.
LICHNEROW[CZ, A. Vari@tds de Poisso,i et Feuilletages, Ann. Fac. Sci. Toulouse
Math (5) 4(1982), 195-262.
YANO, K. and KON, M. CR Submanifolds of Kaellerian and Sasakian Manifolds,
Birkhuser, Boston-Basel-Stuttgart, 1983.
ROSCA, R. CR-sous-varits Co-isotrol)es d’une Varit Parakhlerienne, C.R.
Acad. Sci Paris Sr. I Math. 298(1984), 149-151.
YANO, K. and ISHIHARA, S. Differential___Ceometry of. Tangent and Cotangent
Bundles, Marcel Dekker Inc., New York, 1973.
KLEIN, I. Espaces Variationels et Mbcanique, Ann. Inst. Fourier .(egpble
12(1962), 1-124.
LIBERblANN, P. Sur le Problme d’quivalence de Certalnes Structures
Infinitsimales, Ann. Mat. Pura AI)I)I. 36(1951), 27-120.
12.
ROSCA, R.
13.
787-796.
DELTA, D.K. Exterior Recurrent Forms
No. I, 115-120.
14.
15.
Codlmension 2 CR Submanfold wth Null Covarlant Decomposable
Vertical Distribution of a Neutral Manifold, Rend. Mat. (4) (1982),
on a
Manifold, Tensor (N.S.) 36(1982),
POOR, W.A. Differential Geometric Structures, McGraw-Hill Book Comp., New
York, 1981.
ROSCA, R. and VERS’rRAELEN, L. On SubmatIfolds Admitting a Normal Section
Which is Quasl-concircular w.r.t.
Correspoudl,g Principal Tangent
20(68)(1976), No. 3-4,
Section, Bull. Math. Soc. Math. R.S. Roumanle
399-402.
PSEUDO-SASAKIAN MANIFOLDS
.
747
16.
ROSCA, R. On Parallel
1-1o.
17.
S1NIIA, ll.B. A Dtfferentiable Hanifold t,’|tit Para f-Structure of Rank r,
Ann. Fac. Sci. Univ. Nat. Zaire (Kisimsa) Sect. Hath.-Piys. 6(1980),
No. I-2, 79-94.
LICHNEROWICZ, A. Feullletages, G@om@trle Riemannle,ne et Cfiomtrie Symplectilue,
C.R. Acad. Sol. Parls Ser.
Matlt. 296(1983), 205-210.
ROSCA, R. Espace-temps Ayant la Propri6t6 :6od6sique, C.R. Acad. Sct. Paris
S6r. A 285(1977), 305-308.
18.
19.
Co, formal Co1,,’t. lott.g, Kodal Matlt.
2(1979), No. I,
20.
ROSCA, R. Sou,-vari6t6s A, ti-invaria,voq d’une Vari6t6 Paraklthlertenne
Structur6e par une Conttexion (;6odesique, C.R. Acad. Sci. Paris S6r. A
287(198), 539-541.
21.
GODBII,I,ON, C. .!6omtrie l)tffaretttieli.,_.o.t_ Mdca,tiq,u_o__Analvtt, Hermann,
Paris, 1969.
22.
KLE[N, [. and VOUTIER, A.
Formes extg, tit,ttres g6u6raLrices de sprays, Arm. lust.
Fourier (Grenoble) 18(1968), 241-28.