Internat. J. Math. & Math. Sci.
Vol. 8 No. 2 (1985) 257-266
257
PSEUDO-RIEMANNIAN MANIFOLDS ENDOWED WITH
AN ALMOST PARA f-STRUCTURE
VLADISLAV V. GOLDBERG and RADU ROSCA
Department of Mathematics
New Jersey Institute of Technology
323 Dr. Martin Luther King Jr. Boulevard
Newark, N.J. 07102 U.S.A.
(Received December 27, 1983)
(n+l,n).
(U,,n,,g)
Let
ABSTRACT.
that a manifold
M
be a pseudo-Riemannian manifold of signature
M
One defines on
an almost cosymplectic para
considers on
M
vector field
V.
V
,
P
which are of Otsuki’s type.
2(n-1)-dimensional involuttve distribution
a
of
I/2(n-l)).
Further one
PJ"
It is proved that the maximal integral manifold
as the mean curvature vector (up to
distribution
-Rlcci flat and is follated
endowed with such a structure is
by minimal hypersurfaces normal to
f-structure and proves
and a recurrent
M
of
P
has
If the complimentary orthogonal
is also tnvolutive, then the whole manifold
Different other properties regarding the vector field
M
is foliate.
are discussed.
V
KEY WORDS AND PHRASES. Pseudo-Riemannian manifold, cosymp lectic manifold, para
f-structure, minimal hypersurface.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE. 53C25.
I.
INTRODUCTION.
f-structures or para f-structures
[2]; Yano and Kon [3]; Sinha [4]).
Recently, many papers were devoted to
(Ishichara and Yano
[]];
Kiritchenko
In this paper we consider a
and of inertia index
2n+l
with the
para-complex
suppose that
2
3
.
In (1.1)
M
operator
and such that the
U
component
g
[5])
of square
(,n,)
where
(Libermann
is equipped with a triple
is a canonical
2-form of rank
1-form such that
+I.
of dimension
f
coincides
Furthermore we
exchangeable with the para-Hermitian
2n
of the metric tensor
is a canonical
(M,g)
(l,l)-tensor field
C -pseudo-Riemannian manifold
n+l
g;
(A)nA
# 0;
is the canonical vector field such that
()
d
V
I,
i
0,
(,Z’)
0,
u
0,
(
is the covariant differential operator on
vector fields on
(1.1)
,,Z).
M
and
Z, Z’
are any
M.
If the above conditions are satisfied, we say that
M
is endowed with an
V. V. GOLDBERG AND R. ROSCA
258
almost cosymplectic para
is called an a.c.p,
f-structure
(abr. a.c.p,
f-structure).
In this case
M
f-manifold.
{Z e TM,(Z)
D
The differential distribution
0}
M
on
is involutive
and is called horizontal.
It is proved that an a.c.p,
foliated by minimal hypersurfaces
(Otsuki
M
D
tangent to
[63).
Suppose now that
distributions in
D
D
and
and
Z
are two complementary orthogonal differential
D
u
D
X e D,
+ (R)
(R)
(I.2)
A (M), we say that D is contact covariant
u,u ,v
P be a c.c.d, hyperbolic 2-plane of D
If
and
decomposable (abr. c.c.d.).
+
X
If one has
D.
is a vector field in
VW
for all
-Ricci flat and that it is
which are of Otsuki’s type
f-manifold is always
Let
pairing (Rosca [7]; Morvan and Rosca
[8]), we
form an exterior recurrent
P
the dual forms of two null vector fields which define
M
say that the manifold
admits a
strict c.c.d, hyperbolic 2-plane.
(P,P)
With the paring
are associated a vector field
recurrence vector field) and two vector fields
guished
vector fields
Xn,X2n
E
p
V
(called the
P
E
(called the distin-
).
In the present paper the following properties are proved:
(i)
2(n-l)-distribution
The
P
is always involutive and the mean curvature
vector field of its maximal integral manifold
factor) equal to the induced vector field of
(ii)
p
of
The simple unit form
characteristic vector field of
(iii)
.
Xn
is that
or
be a null vector
X2n
is (up to a constant
IYV is a
is exterior recurrent and
M
The necessary and sufficient condition for
sufficient condition for
M
V.
to be
field and
quasi-minimalChen [9])
the necessary and
to be minimal is that both
Xn
and
X2n
be null vector fields.
(iv)
P
is minimal, then the distribution
If
integral surfaces
of
P
is also involutive and the
are totally geodesic in
M
which in this case
is foliate.
(v)
2.
Both vector fields
Xn
are
and
MOST COSLECTIC P f-IFOLD
Let
(M,g)
inertia index
If
M
be a
U-geodesic
X2M(U,.,,,g).
directions on
C -pseudo-Riemannian nlfold of dimension
2n+l
and of
(I,I)
of
n+l.
is equipped with a non-zero tensor field
f
of type
constant rank and such that
f(f2-1)
(I
is the identity
tensor), then
f
In the following we suppose that
U
(Libermann
(,n,)
o
[5]).
(2.1)
0
is called a
f
para
f-structure
(Sinha
[4]).
coincides with the para-complex operator
In addition, we suppose that
M
is equipped
with the triple
where:
is a canonical 2-form of rank
2n
exchangeable with the para-Hermitian
2
3
..
259
PSEUDO-RIEMANNIAN MANIFOLD WITH PARA f-STRUCTURE
gn
component
of the metric tensor
is a canonical
(Buchner and Rosca
g
(Afl)nA
l-form such that
[]0]).
# 0 everywhere.
is the canonical vector field such that
()
i,
0;
i
(2.2)
i: interior product.
If one has
U
2-I
-n
O,
U
d
(2.3)
O,
(V,’}
(2.4)
g{V,,Z)
{2.5)
Z, Z’ are vector fields
(U,D,n,,g) defines on M an almost cosympletic para f-structure (abr. a.c.p, f-structure) and M(U,,n,,g) is called an a.c.p, f-manlfold.
V
where
in
M
is the covariant differential operator on
and
M, we say that
The differentiable distribution
M
on
{
D
is called
D
O}
(x)
TM,
e
defined by
horizontal.
It is worthwhile to note that equations (2.3), (2.4) and (2.5) show that on
M
(U,,)
the triple
defines a
defines an almost paracontact structure (Slnha
(2n)-foliation, and
E
; a
frames (Vranceanu and Rosca []]]).
One has (Libermann [5]):
Let W
vect
{ha,ha,,h0
Uh a
and at each point
)
e M
S
p
{h ,}
and
h
-ha,, U
Uha,
a
n) S?*
h
{A}
2n; A
a
and
a
g(ha,hA)
g(ha,ha,)
0,1,
(2.7)
B
M.
h
a
,
n-spaces spanned by
{h
a
are normed, one may write
O,
g(ha,,hB)
I,
g(,)
O,
(2.8)
a.
be the dual basis of
the connection forms on
(2.6)
0
P
S are two self-orthogonal vector
p
respectively.
A,B
be a local field of Witt
S
and
Now let
D
one has the splitting
Since the null vector fields
where
a+n}
l,...,n, a
(D
where
[4]),
is a gradient.
W
A
e
and
B
Then the line element
d
A C
YBC m
of
M
(Ce
(dp
C (M))
be
is a canonical
vectorial 1-form) and the connection equations are expressed by
d a@
h
a
+
a
@ ha
,+@
(2.9)
and
%
Vh
where
finds
V
%B
A
e A (R)
h
B
is the covariant differentiation operator on
M.
By (2.8) and (2.10) one
V. V. GOLDBERG AND R. ROSCA
260
O,
a
Oa,
O,
ao
a* +
eb
0
6
e a,
a*
a
e
+
b*
o
+
b
a
a
O,
a
,+
a
b*
a
O,
e oa
0
%a
(2.11)
O,
eb,
and the structure equations (E. Cartan) may be written in the following symbolic
form:
-A
d
(2.12)
and
+
-eAe
de
(2.13)
A are the curvature 2-forms.
where
Further taking into account (2.4), we may set
B
o a
-
a*
o
Ca,
a
(2.14)
Now by means of (2.10), (2.11) and (2.14) one gets
a*
V
h
a
a
h
a
,.
(2.15)
In addition it follows from (2.15) that,
V
is a
which proves that
.
geodes{c direction.
From (2.9) and (2.8) one gets
gn
where
2
g
a a*
of the metric tensor
The 2-form
(2.16)
0
<d,d>
a(R)
2
,
+ n
(2.17)
a
is the
pca-Hert{c (Buchner
and Rosca
[]0])
component
g.
which is exchangeable with
I
g
is then expressed by
(2.18/
A
a
gstng
fo
(2.15)
we can ind the following expression of the quadratic
<V,V>
-g,
(2.19)
Denote by
m
A...A 2n
i
the simple unit form corresponding to
M
(2.20)
One may write the volume element
D
o
of
as
A n.
o
If
Lz
means the Lie derivative in the direction
(2.21)
Z, then by a simple argument
one can find
LE
Using (2.12) and (2.13), one gets
d
d
(div ).
(2.22)
O, and this yields
div
O.
But on a Riemannian or pseudo-Riemannian manifold the following
(2.23)
Yano integral
261
PSEUDO-RIEMANNIAN MANIFOLD WITH PARA f-STRUCTURE
[12]):
formula holds (Yano and Kon
div(VZ)
dlv(div Z)Z
(2.24)
Ric(Z) +
A,B
In (2.24) Z, Ric and
M
{e
g(VeAZ,eB)g(eA,VeB
Z)
M, the Riccl tensor of
are arbitrary vector fields on
A}
(div Z)
and a vectorial basis respectively.
Continuing the consideration, one finds (2.24) and (2.15) by means of (2.5).
2n.
Taking into account (2.8), a short computation gives Ric()
Hence M is Ricci constant in the direction of the structure vector
(or
-Ricci constant).
is coclosed,
On the other hand, by means of (2.19) and (2.4) one sees that
if we
Then
harmonic.
is
O, it follows that
0. Hence since d
i.e.
it follows from the theorem of Tachibana []3] that
the leaf of D
denote by M
is minimal.
M
This property can also be verified by a direct computation.
IM
Since the induced value
almost symplectic, the submanifold
of the almost sympletic form
M
having an almost symplectic structure
.
is an example of a minimal submanifold
(Buchner and Rosca
is endowed with a para co-Kaehlerlan structure
M
If
is also
[]0]),
is a symplectic form.
then
Denote now by III the induced value on
given by (2 19)
Since
is normal to
M,
Mthen,of
the quadratic differential form
as is known, III represents the
third fundamental form of M
Taking into
Thus according to (2 19) III is conformal to the metric of M
and (2.15), it is easy to see that
account of the para-Hermitian form of
M
g
possesses principal curvatures equal to +1 and principal curvatures equal to -1.
Therefore referring to Otsuki [6], we may say that M
is a minimal hypersurface of
Otsuki’s type.
THEOREM i.
a.c.p,
hypersurfaces
field
3.
Let
f-structure.
M(U,,,,g)
be a pseudo-Riemannian manifold endowed with an
Such a manifold is
-Ricci constant and is foliated by minimal
of Otsuki’s type which are orthogonal to the structure vector
M
$.
CONTACT COVARIANT DECOMPOSABLE DISTRIBUTIONS ON
Referring to the definition given by Rosca
DEFINITION.
M
Let
M(U,,g).
[7], we
give now the following
C -Riemannian (resp. C -pseudo-
be an odd-dimensional
Riemannian) manifold equipped with an almost contact (resp. almost para contact)
$. Let
O, a differentiable
m be
and the covariant differentiation operator on M. Let D
structure defined by a structure l-form
D
D
and
V
distribution of
and a structure vector field
be the horizontal distribution defined by
D
the complementary orthogonal distribution of
of
D.
and
in
D
+
v(R)
W
be a vector field
Then if one has
VW
where
D
n
X e D,
e D
and
u(R)
X+ u (R) X-
(3.1)
E
z
u,u ,v e A (M), we say that the distribution
contact covariant decomposable (abr. c.c.d.).
As is known, the null vectorial basis {h
a
ha,}
of
D
N
D
is
admits the orthogonal
V.V. GOLDBERG AND R. ROSCA
262
decomposition
Pa
where
(h a,ha,)
is a
Dn PI ’’’ Pa ’" " Pn
hyperbolic 2-plane.
We say that the a.c.p,
(3.2)
M(U,,n,,g) defined in Section 2, carries
a strict contact covariant decomposable hyperbolic plane P (abr. s.c.c.d, hyperf-manifold
bolic plane) if:
the distribution
P
is contact convariant decomposable;
recurrent pairing (in the sense of
Without loss of
hn,
P
the dual forms of the null vectors which define
2
P
one may suppose that
generalit
exteor
form an
[7]).
Rosca
is defined by
h
n
and
h2n.
In the first place, using (2.10) and (3.1), one finds
Xn Wn
X2n 2n
n
where
, 2n
n
e A
(M);
e C (M)
(3.3)
{i,i*
1,.
Denote by
the complementary orthogonal distribution of P
{h } and we set
Obviously one has P
a
n
px
Xn’X2n
and
e
e
X2n X2nh
Secondly, according to Rosca
corresponding to
P
[7];
(hn,h2n)
i*
in
D
i-I’}.
(3.4)
[8],
the dual forms
n 2n
m ,m
define an exterior recurrent pairing if one has
dn
n 2n n 2n
y ,y ,v ,v
e A
,n;
P
Morvan and Rosca
dn
where
i
n
2n A 2n ]
=y
n
A +
n
=v
An+ 2n
A
n
(3.5)
().
As a consequence of (3.5), using (2.12), (2.11), (2.14), and (3.3), we find:
?n(Z
n
%(It
2n
where
n’2n
e
C’()
M.
vanish nowhere on
dn
d2n
Xnm
2n=( X2na 3
%
(3.6)
Therefore (3.5) become of the form
%n
A
m
2n
=-m
%(]t
+
A
2
A+An
(3 7)
where we have set
n=
Y
2n
2n
n
(3.8)
Denote now by
n A 2n
the simple unit form which corresponds to
P.
d
Since
dim(Ker
)
# 0,
we may also say that
(3.9)
It follows from (3.7) that
0.
is a
(3. I0)
presympZectic form (Souriau
[]4]).
PSEUDO-RIEMANNIAN MANIFOLD WITH PARA f-STRUCTURE
263
Further taking the exterior derivative of equations (3.7) and referring to (2.4), one
gets by an easy argument that
dy
y
dy
(d/)A
’
’’;
It follows from (3.11) that
P.
(3.12)
d/
is a
w
annihilates
as the reorence l-form.
P}
the ideal in
A()
of
belongs to this ideal and by means of (3.10)
Obviously
I(P)
we may say that
A()
{
(P)
Denote now by
the distribution
(3.11)
y
is exterior recurrent and has the exact form
i.e.
C=({).
E:
differentiable
(dI(P)
ideal
P
It follows as is known, that the distribution
l(P)).
is involutive (this can be
also checked by a direct computation with the help of (3.3) and (3.6)).
Let us now denote
+
I
A...A
uo
n-
I*
A
A...A n*-i
the simple unit form corresponding to the distribution
(3.13)
P.
Then by means of (2.12),
(2.11), (2.14), (3.3), (3.4) and (3.6), a straightforward calculation gives
d@
Hence the
2(n-I)
(f n g(Xn,Xn) +
form
(3.14)
is eteior rCaunt and has the form
n
fng(Xn,Xn) + f2ng(X2n,X2n)m
a
as a
f2ng(X2n,X2n)O
(3.15)
form (Datta []]).
In the following we will call the vector field
meu_Penoe
fng(Xn,Xn)h2n + f2ng(X2n,X2n)hn
eor ie on M ((V) g(V,V)) and Xn,X2n
V
the memmmeme
ueoms
(abr. d.v.) of the distribution
(3.16)
the
iiniBhe
P.
By means of (2.6) one has
UV
f2ng(X2n X2n)hn
ng(Xn,Xn)h2n
(3.17)
and according to (2.8) this implies
=(UV)
Since
and the above equations proved that
is an
UV
(3. 19)
0
X e P
corresponding to
(3.12):
P
is any vector field of
is
@nn -W2n)2n’’
(
A
=&nn
.
is a olzraotestio veotor
e,
infinitesimal conformal transformation
Moreover, if
and
(3.18)
UV e P, we have from (3.13), (3.14), and (3.18)
id
X
0.
one gets instantly
of
fieZd of
(X),
t2-form
i.e.
Next the Rioci
and it can be found by means of (2.14), (3.3)
+
+
g(Xn’X2n 2n
A
n.
(3.20)
264
V. V. GOLDBERG AND R. ROSCA
Hence equations (3.12) and (3.10) show that the necessary and sufficient condition
for
to be closed is that the vector fields X
and
are orthogonal
X2n
n
Using now (3.11) and (3.9),one gets
(n 2n (n X2n) <XnAX2n, nA2n
Xn
Therefore, if
and
+
i
(n,X2n) vanishes.
P and let
G defined by
(2n-3)-fo of M
Then
is
n-IAl* A...A n*-I hi,
(-I) i*-ll A...A n-IAl* A...A I* A...A n*-l(R)
hi
(-l)i-l
i
(3.21)
the maximal connected integral manifold of
be the mean curvature
=
.
are orthogonal, then
X2n
M
Denote now by
>
A
A
i
A
A
(R)
(3.22)
,
(the roofs indicate the missing terms and we denote the induced elements on M
supressing
). Since
is the volume element of M
one has (see Chen [9])
dV
H
where
0
is the mean aurvature vector
2(n-l) H
field
(3.23)
IM (Poor
M,
of
+/-, and d V is the
exterior covariant differentiation with respect to V
(2.12) and taking into
account
by
VIM+/-
[18]). Using (2.10),
(2.14), (3.3), (3.6), and (3.16), one finds after
some calculations
2(n-l) V;
H
VIM
V
Hence the mean curvature vector is, up to the factor
(3.24)
.
.
V
value of the recurrence vector field
M.
in
2(n-l)’ equal to the induced
Using the definition given by Rosca
[16], [i?], we obtain the following results:
2
M
The necessary and sufficient condition for
i.e., H
bution
to be
quasi-minimaZ
be a null vector field, is that one of the d.v. of the distri-
P
be a null vector.
M
The necessary and sufficient condition for
both d.v. of
P
to be minimal is that
be null vectors.
We shall now make the following consideration.
(3.13) the volume element of the hypersurface
M
According to (2.21), (3.9) and
defined by
0
may be written
as:
# A
o
(3.25)
and @ are the restrictions of
In (3.25)
and @ on M
It follows from (3.10) that if one has g(Xn,Xn)
g(X2n,X2n) 0, one may
d= + =d is the harmonic operator. Therefore we are
write A
0 where A
in the situation of Tashibana’s theorem (Tashibana
two families of minimal submanifolds, M
Equations (2.6) shows that
d.v.
Xn
and
X2n
variant submanifolds tangent to
foliate.
a
a2n
n’
that II
UP
P
are null vectors, then
P
and
and
and
M
[]3])
and
M, tangent to
UP
P.
M
is covered by
and
P
respectively.
Hence we may say that if both
is foliated by two families of in-
P, and therefore the whole manifold
Moreover, if we consider the immersion of
M
in
M
then the
is
1-forms
given by (3.3) define the normal vector quadratic form II (it is known
is independent of the normal connection).
But by means of (3.6) we can see
PSEUDO-RIEMANNIAN MANIFOLD WITH PARA f-STRUCTURE
that II vanishes
M
is totally geodesic in
M
and therefore
265
We shall give now the following
endowed
be an invariant submanifold of a manifold
of
M. Then
form
quadratic
vector
normal
the
with a para f-structure and II be
an
called
is
0
any tangent vector field X of M such that II(X,fX)
DEFINITION.
M
Let
f-geodesic direction
M.
on
2n
M.
x:
Let us consider now the immersion
-
Denote by
<dp,Vhn >
n
and
<dp’Vh2n > the second quadratic forms associated with x.
By means of (2.9), (2.10), (3.3), and (3.6) one finds after some calculation
n
n
n
2n
f
n
2n
Therefore the normal vector quadratic form II e
II(Xn,UXn)
M
Therefore the d.v. fields on
THEOREM 2.
Let
Xn,X2n
E
P
P
2
(3.26)
(TT*) (R) (TM)
is given by
(3.27)
2n
II(X2n,UX2n)
0
U-geodesic.
are both
be an a.c.p, f-manifold admitting a strict
contact covariant decomposable hyperbolic plane
component of
a
(2.26) and (2.27)
O,
M(U,,,,g)
}
Xn 2)
2n
n
gets by means of
Referring now to (2.4),one
(I
f2n( X2n
2n
f_---2n
n
in the horizontal distribution
P
and
D
P
be the orthogonal
V e P
Further let
and
be the recurrence vector field and the distinguished vector fields
(P,P).
associated with the pairing
Then the following properties hold:
(i)
P
The distribution
is always involutive and the mean curvature vector
field of the maximal integral manifold
(ii)
The simple unit form
of
P
characteristic vector field of
(iii)
(iv)
M
M
to be quasi-minimal is
be null vectors.
If
M
is minimal, then the distribution
P
is also involutive and the
are totally geodesic in
M
is foliate.
(v)
Both d.v. fields on
M
are
is a
be a null vector and the necessary
M
P
UV
to be minimal is that both d.v. fields
of
integral surfaces of
(up to a constant
is exterior recurrent and
M
that one of the d.v. fields of
is
V.
.
The necessary and sufficient condition for
and sufficient condition for
P
of
factor) equal to the induced vector field of
U-geodesic directions on
.
which in this case
M
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f3
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