BULLETIN Bull. Malaysian Math. Soc. (Second Series) 21 (1998) 21-26 of the
MALAYSIAN MATHEMATICAL SOCIETY
Semi-Invariant Submanifolds of a Lorentzian Para-Sasakian Manifold
BHAGWAT PRASAD
Department of Mathematics, M. M. T. D. College, Ballia, India
Abstract. Recently Matsumoto [1] introduced the idea of Lorentzian para contact structure and studied its several properties. In the present paper we studied the integrability condition of the distribution on semi-invariant submanifolds of LP-Sasakian manifold.
1. Introduction
Let M be an n-dimensional real differentiable manifold of differentiability class C∞ endowed with a C∞-vector valued linear function φ, a C∞ vector field ξ, 1-form η and Lorentzian metric g of type (0,2) such that for each p∈M, the tensor
R M T M T
gp: p × p → is a non-degenerate inner product of signature
(
−, +, +, ", +)
, where TpM denotes the tangent vector space of M at p and R isthe real number space, which satisfies
( ) ( )
,2 η ξ
φ X = X + X (1.1)
( )
ξ = −1,η
(
X, Y)
g(
X,Y)
( ) ( )
X Y ,g φ φ = +η η (1.2)
(
X,) ( )
X ,g ξ = η
for all vector fields X,Y tangent to M. Such structure
(
φ, ξ, η, g)
is termed as Lorentzian para contact [1].In a Lorentzian para-contact structure the following holds
( )
0,,
0 =
= ηφX φξ
rank
( )
φ = n−1.A Lorentzian para contact manifold M is called Lorentzian para-Sasakian (LP-Sasakian) manifold if [1]
(
∇Xφ) ( )Y = g(
X,Y)
ξ + η( )
Y X + 2η( )
X η(Y)ξ, (1.3)
and from (1.3), we find
Xξ = φX
∇ (1.4)
Y X,
∀ tangent to M, where ∇ is the Riemannian connection with respect to g.
Let us put
(
X,Y)
g(
φX,Y)
Φ =
then the tensor field Φ is symmetric (0,2)- tensor field. Thus we have,
(
X,Y)
Φ(
Y,X)
,Φ =
and Φ
(
X,Y)
=(
∇Xη) ( )Y .
Definition 1.1. The submanifold M of the LP-Sasakian manifold M is said to be semi- invariant if it is endowed with the pair of orthogonal distribution (D,D⊥) satisfying the conditions
(i) TM= D⊕D⊥⊕
{ }
ξ ,(ii) the distribution D is invariant under φ, that is
x,
x D
D =
φ for each x∈M,
(iii) the distribution D⊥ is anti-invariant under φ, that is
⊥,
⊥⊂ T M Dx x
φ for each x∈M.
The distribution D (respectively D⊥) is called the horizontal (respectively vertical) distribution. A semi-invariant submanifold M is said to be invariant (respectively anti- invariant) submanifold if we have Dx⊥=
{ }
0 respectively (Dx =0) for each x∈M. We say that M is a proper semi-invariant submanifold if it is a semi-invariant submanifold, which is neither an invariant nor an anti-invariant submanifold.We denote by same symbol g both metrices on M and M. The projection morphisms of TM to D and D⊥ are denoted by P and Q respectively. For any
(
TM)
X∈Γ and N∈Γ
(
TM⊥)
, we have( )
ξ η X QX PXX = + + (1.5)
CN BN N = +
φ (1.6)
where BN (respectively CN) denotes the tangential (respectively normal) component of N.
φ
The equations of Gauss and Weingarten for the immersion of M in Mare given by
(
X,Y)
,h Y
Y X
X = ∇ +
∇ (1.7)
N, X
A
N N X
X = − + ∇⊥
∇ (1.8)
for any X,Y∈Γ
(
TM)
and N∈TM⊥, where ∇ is the Levi-Civita connection on M,∇⊥ is the linear connection induced by ∇ on the normal bundle TM⊥, h is the second fundamental form of M and AN is the fundamental tensor of Weingarten with respect to the normal section N. Also we have
( )
(
h X Y N)
g(
A X Y)
g , , = N , (1.9)
for any X,Y∈Γ
(
TM)
, N∈Γ(
TM⊥)
.2. Basic Lemmas
For X,Y∈Γ
(
TM)
, we put(
X,Y)
PY A X.u = ∇Xφ − φQY (2.1)
We begin with the following lemma.
Lemma 2.1. Let M be a semi-invariant submanifold of LP-Sasakian manifold M. Then we have
(
u X Y)
φP Y ηY PX η X ηY Pξ g X Y PξP ( , ) = ∇X + ( ) + 2 ( ) ( ) + ( , ) (2.2)
(
u X Y)
Bh X Y ηY QX η X η Y Qξ g X Y QξQ ( , ) = ( , ) + ( ) + 2 ( ) ( )( ) + ( , ) (2.3) )
, ( )
,
(X PY QY Q Y Ch X Y
h φ + ∇⊥Xφ = φ ∇X + (2.4)
(
u(X,Y))
g(φX,φY),η = − (2.5)
for all X,Y∈TM.
Proof. By using the decompositions (1.5), (1.6), (1.7), (1.8) in (1.3), we obtain (2.2), (2.3), (2.4) and (2.5) respectively.
Lemma 2.2. Let M be a semi-invariant submanifold of LP-Sasakian manifold M, then we have
(
,)
0,, =
=
∇Xξ φX h X ξ for any X∈Γ
( )
D ; (2.6)(
,)
,,
0 hY Y
Yξ = ξ = φ
∇ for any Y∈Γ
( )
D⊥ ; (2.7)(
,)
0. ,0 =
=
∇ξξ hξ ξ (2.8)
Proof. In consequence of (1.4) and (1.5), we obtain
(
ξ)
ξ
ξ X h X,
X = ∇ +
∇
or ∇Xξ + h
(
X,ξ)
= φPX + φQX. (2.9)Thus,
( ) ( )
2.6 − 2.8 follows from (2.9).Lemma 2.3. Let M be a semi-invariant submanifold of LP-Sasakian manifold M. Then we have
, X A Y
AφX = − φY (2.10)
for all X,Y∈Γ
( )
D⊥ .Proof. By using (1.2), (1.7), (1.9), we get
(
A Y Z)
g(
h(
Y Z)
X)
g(
Y X)
g φX , = , , φ = ∇Z ,φ
(
Y X)
g(
Y X)
gφ∇Z , = ∇Zφ ,
= −g
(
φY,∇ZX)
= −g(
φY,h(X,Z))
=
, ) ,
(A X Z
g φY
−
=
for all X,Y∈Γ
( )
D⊥ and Z∈Γ(
TM)
, which proves (2.10).Lemma 2.4. Let M be a semi-invariant submanifold of LP-Sasakian manifold M. Then we find
( )
D , U Γξ ∈
∇ for any U∈Γ
( )
D ,( )
⊥ ,∈
∇ξV Γ D for any V∈Γ
( )
D⊥ . The proof is obvious.Lemma 2.5. Let M be a semi-invariant submanifold of LP-Sasakian manifold M. Then we obtain
[
X,ξ ∈]
Γ( )
D , for any X∈Γ( )
D , (2.11)[
Y,ξ]
∈Γ( )
D⊥ , for any Y∈Γ( )
D⊥ . (2.12) The proof follows from Lemma 2.4.3. Integrability of distribution on a semi-invariant submanifold in a LP-Sasakian manifold
Theorem 3.1. Let M be a semi-invariant submanifold in LP-Sasakian manifold M. Then the distribution D is integrable if and only if
h
(
X,φY)
= h(
Y,φX)
.Proof. We have from (2.6)
[ ]
(
X,Y ,ξ)
g(
Y X,ξ)
g = ∇X −∇Y
(
Y, ξ)
g(
X,ξ)
g ∇X − ∇Y
=
(
Y X)
g(
X Y)
g ,φ + ,φ
−
= ,
= 0 for all X,Y∈Γ
( )
D .In consequence of (2.4), we find
(
X, Y)
h(
Y, X)
Q[
X,Y]
,h φ − φ = φ
which proves the theorem.
Corollary 3.1. The distribution D⊕
{ }
ξ is integrable if and only if(
X, Y)
h(Y, X)h φ = φ is satisfied.
Proof follows from Theorem 3.1 and (2.11).
Theorem 3.2. Let M be semi-invariant submanifold in LP-Sasakian manifold M. Then the distribution D⊥ is never integrable.
Proof. From (2.1), we have for X,Y∈Γ(D⊥)
(
X,Y)
A X. u = − φYOperating φ in (2.2) and using (1.1), we get
, )
(A X
P Y
P∇X = φ φY (3.1)
for any X,Y∈Γ(D⊥). By virtue of Lemma 2.3, (3.1) reduces to
[ ]
(
X,Y)
2 P(
A X)
,P = φ φY
which proves the statement.
Corollary 3.2. The distribution on D⊥⊕
{ }
ξ is never integrable.Proof follows from Theorem 3.2 and (2.12).
References
1. K. Matsumoto, On Lorentzian paracontact manifolds, Bull. Yamogata Univ. Nat. Sci.
12 (1989).
2. A. Bejancu and N. Papaghiue, Almost semi-invariant submanifolds of a Sasakian manifold, Bull. Math. dela Soc. Sci., Math. dela R.S. Roumanie 28 (1984), 1.
3. N. Papaghiue, Some results on almost semi-invariant submanifolds in Sasakian manifolds, Bull. Math. dela Soc. Sci., Math. dela R.S. Roumanie 28 (1984), 3.
4. M. Kobayashi, Semi-invariant submanifold of a certain class of almost contact manifold, Tensor, N.S. 43 (1986).
5. S. Prasad and R.H. Ojha, Lorentzian para contact submanifolds, Publ. Math. Debrecen 44 (1994), 215-223.
