Abstract: The object of the present paper is to find a necessary and sufficient condition for an invariant submanifold of an LP -Sasakian manifold to be totally geodesic. An illustrative example is given to support the obtained result

Avijit Sarkar

Matilal Sen

CiteSeerX

E extracta mathematicae Vol. 27, Núm. 1, 145 – 154 (2012)
On Invariant Submanifolds of LP -Sasakian Manifolds
Avijit Sarkar, Matilal Sen
Department of Mathematics, University of Burdwan, Golapbag, Burdwan 713104,
West Bengal, India, avjaj@yahoo.co.in
Presented by Oscar Garćıa Prada Received March 2, 2011
Abstract : The object of the present paper is to find a necessary and sufficient condition for
an invariant submanifold of an LP -Sasakian manifold to be totally geodesic. An illustrative
example is given to support the obtained result.
Key words: LP -Sasakian manifold, invariant submanifolds, totally geodesic.
AMS Subject Class. (2010): 53C15, 53C40.
1. Introduction
The notion of Lorentzian almost para-contact manifolds was introduced
by K. Matsumoto [6]. In subsequent times, a large number of geometers
studied Lorentzian almost para-contact manifold and their different classes,
viz., Lorentzian para-Sasakian manifolds and Lorentzian special para-Sasakian
manifolds [7], [8], [9], [12]. A beautiful example of a five-dimensional Lorentzian
para-Sasakian manifold has been given by Matsumoto, Mihai and Rosaca [8].
In brief, Lorentzian para-Sasakian manifolds are called LP -Sasakian mani-
folds. The study of LP -Sasakian manifolds is of prime importance due to its
relevant applications in the theory of relativity.
In modern analysis, the geometry of submanifolds have become a subject
of growing interest for its significant application in applied mathematics and
theoretical physics. For instance, the notion of invariant submanifold is used
to discuss properties of non-linear autonomous system [5]. Also, the notion
of geodesics plays an important role in the theory of relativity [8]. For to-
tally geodesic submanifolds, the geodesics of the ambient manifolds remain
geodesics in the submanifolds. Hence, totally geodesic submanifolds have also
importance in physical sciences. The study of geometry of invariant subman-
ifolds was initiated by A. Bejancu and N. Papaghuic [1]. Again N. Papaghuic
has worked on semi-invariant submanifolds of LP -Sasakian manifolds. On
the other hand, a number of works on the geometry of submanifolds of LP -
Sasakian manifolds have been carried out by U.C. De and collaborators [2], [3],
145
146 a. sarkar, m. sen
[4]. B. Prasad [13] also studied semi-invariant submanifolds of LP -Sasakian
manifolds. Recently, in [14] the first author of the present paper studied
submanifolds of Sasakian manifolds.
In [3], it is proved that a submanifold of an LP -Sasakian manifold is invari-
ant if and only if B(X, ξ) = 0, where B is the second fundamental form of the
submanifold. They also obtained some necessary and sufficient conditions for
such submanifolds to be totally umbilical or minimal. In order to enquire into
under what condition an invariant submanifold will become totally geodesic,
the authors of the paper [10], have obtained some necessary and sufficient
conditions under which an invariant submanifold becomes totally geodesic.
But in the present paper, interestingly, we prove that every three-dimensional
invariant submanifold of an LP -Sasakian manifold is totally geodesic and the
converse is also true.
The present paper is organized as follows: In Section 2, we give some
preliminaries which have been used later. Section 3 is devoted to prove that
a three-dimensional submanifold of an LP -Sasakian manifold is invariant if
and only if it is totally geodesic. Section 4 contains an illustrative example to
support the results obtained in Section 3.
2. Preliminaries
Let M̃ 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, an one form η and a Lorentzian metric g of type (0, 2) such that for each
p ∈ M̃, the tensor gp : TpM̃ × TpM̃ → R is a non-degenerate inner product of
signature (−,+,+,+, . . . ,+) where TpM̃ denotes the tangent vector space of
M̃ at p and R is the field of real numbers, which satisfies
ϕ2X = X + η(X)ξ, η(ξ) = −1, (2.1)
g(ϕX, ϕY ) = g(X,Y ) + η(X)η(Y ), g(X, ξ) = η(X), (2.2)
for all vector fields X,Y tangent to M̃. Such structure (ϕ, ξ, η, g) is termed as
Lorentzian para-contact [6]. In Lorentzian para-contact structure the follow-
ing relations hold:
ϕξ = 0, η(ϕX) = 0,
rankϕ = n− 1.
A Lorentzian para-contact manifold M̃ is called Lorentzian para-Sasakian
on invariant submanifolds of LP -sasakian manifolds 147
manifold or LP -Sasakian manifold if [6]
(∇̃Xϕ)Y = g(X,Y )ξ + η(Y )X + 2η(X)η(Y )ξ, (2.3)
∇̃Xξ = ϕX, (2.4)
for all X,Y tangent to M̃, where ∇̃ denotes the Levi-Civita connection with
respect to g. In an LP -Sasakian manifold, we also have that the curvature
tensor R̃ of the manifold M̃ is given by
R̃(ξ,X)Y = g(X,Y )ξ − η(Y )X. (2.5)
Let M be a submanifold immersed in an n-dimensional Riemannian manifold
M̃, we denote by the same symbol g the induced metric on M. Let TM be
the tangent space of M and T⊥M is the set of all vector fields normal to M.
Then Gauss and Weingarten formulae are given by
∇̃XY = ∇XY +B(X,Y ), (2.6)
∇̃XN = −ANX +∇⊥XN, (2.7)
for any X,Y ∈ TM and N ∈ T⊥M, where ∇⊥ is the connection in T⊥M. The
second fundamental form B and AN are related by
g(ANX,Y ) = g(B(X,Y ), N). (2.8)
It is also noted that B(X,Y ) is bilinear, and since ∇fXY = f∇XY, for a C∞
function f on a manifold we have
B(fX, Y ) = fB(X,Y ). (2.9)
Let us now recall the following:
Definition 2.1. Let M be a submanifold of an n-dimensional LP -Sasaki-
an manifold M̃. The submanifold M of M̃ is said to be invariant if the struc-
ture vector field ξ is tangent to M, at every point of M and ϕX is tangent
to M for any vector field X tangent to M, at every point on M, that is,
ϕTM ⊂ TM at every point on M.
Definition 2.2. A submanifold of an LP -Sasakian manifold is called to-
tally geodesic if B(X,Y ) = 0, for any X,Y ∈ TM.
148 a. sarkar, m. sen
3. Invariant submanifolds of LP -Sasakian manifolds
Proposition 3.1. Let M be an invariant submanifold of an LP -Sasakian
manifold M̃. Then there exist two differentiable orthogonal distributions D
and D⊥ on M such that
TM = D ⊕D⊥⊕ < ξ >,
and
ϕ(D) ⊂ D⊥, ϕ(D⊥) ⊂ D.
Proof. For an invariant submanifold M, ξ is tangent to M. Hence, we
can write TM = D1⊕ < ξ >. Let X1 ∈ D1. Now g(X1, ϕX1) = 0 and
g(ξ, ϕX1) = 0. So, ϕX1 is orthogonal to X1 and ξ. Consequently, it is possible
to write D1 = D ⊕ D⊥, where X1 ∈ D ⊂ D1 and ϕX1 ∈ D⊥ ⊂ D1. For
ϕX1 ∈ D⊥, we note that
ϕ(ϕX1) = ϕ
2X1 = X1 + η(X1)ξ = X1 ∈ D.
Let ϕX1 = X2 ∈ D⊥. Hence, for X1 ∈ D, ϕX1 ∈ D⊥ and for X2 ∈ D⊥,
ϕX2 ∈ D. Hence, the proposition follows.
Proposition 3.2. For an invariant submanifold M of an LP -Sasakian
manifold M̃, we have [3] for the two differentiable tangent vector fields X,Y
of M
B(X, ξ) = 0, (3.1)
B(X,ϕY ) = ϕB(X,Y ) = B(ϕX, Y ). (3.2)
Proposition 3.3. ([10]) An invariant submanifold of an LP -Sasakian
manifold is also LP -Sasakian.
Let us consider a three-dimensional invariant submanifold M of an LP -
Sasakian manifold M̃. Then, M is LP -Sasakian. Now it is obvious that
B(X,Y ) satisfies
ϕ2
(
B(X,Y )
)
= B(X,Y ) + η
(
B(X,Y )
)
ξ. (3.3)
Let X1, Y1 ∈ D. Then we have from (3.2)
B(X1, ϕY1) = ϕB(X1, Y1).
on invariant submanifolds of LP -sasakian manifolds 149
Therefore in view of (3.2) and (3.3)
ϕB(X1, ϕY1) = ϕ
2B(X1, Y1) = B(X1, Y1) + η
(
B(X1, Y1)
)
ξ. (3.4)
Since B(X1, Y1) ∈ T⊥M, B(X1, Y1) is orthogonal to ξ ∈ TM. Hence, we
obtain η
(
B(X1, Y1)
)
= 0. Thus, in view of (3.2) and (3.4) it follows that
B(ϕX1, ϕY1) = B(X1, Y1). (3.5)
Let ϕX1 = X2, ϕY1 = Y2. We note that X2 = ϕX1 ∈ D⊥ and Y2 = ϕY1 ∈ D⊥.
Therefore,
B(X2, Y2) = B(X1, Y1), (3.6)
for X1, Y1 ∈ D and X2, Y2 ∈ D⊥. Since B is bilinear, for X1, Y1 ∈ D and
X2, Y2 ∈ D⊥, it follows that
B(X1 +X2 + ξ, Y1) = B(X1, Y1) +B(X2, Y1) +B(ξ, Y1), (3.7)
B(X1 +X2 + ξ,−Y2) = −B(X1, Y2)−B(X2, Y2)−B(ξ, Y2), (3.8)
B(X1 +X2 + ξ, ξ) = B(X1, ξ) +B(X2, ξ) +B(ξ, ξ). (3.9)
Keeping in mind that B(X, ξ) = 0, for X ∈ TM and using (3.7), (3.8), and
(3.9) we get, by virtue of (3.6)
B(X1 +X2 + ξ, Y1 − Y2 + ξ) = B(X2, Y1)−B(X1, Y2).
Now,
TM = D ⊕D⊥⊕ < ξ >,
so, U = X1 + X2 + ξ ∈ TM and V = Y1 − Y2 + ξ ∈ TM. Thus, the above
equation yields
B(U, V ) = B(X2, Y1)−B(X1, Y2). (3.10)
From the above equation it follows that
ϕB(U, V ) = B(X2, ϕY1)−B(ϕX1, Y2) = B(X2, Y2)−B(X2, Y2) = 0.
The above equation gives ϕ2B(U, V ) = 0. Consequently, B(U, V ) = 0. Now,
we are in a position to state the following:
Theorem 3.1. Every three-dimensional invariant submanifold of an LP -
Sasakian manifold is totally geodesic.
150 a. sarkar, m. sen
Next suppose that the submanifold is totally geodesic. If the submanifold
is even dimensional, then it is not invariant, because, an invariant submanifold
of an LP -Sasakian manifold is also LP -Sasakian and an even dimensional sub-
manifold can not admit LP -Sasakian structure. Suppose that the submanifold
is odd dimensional. Then for X,Y ∈ TM, B(X,Y ) = 0. Now we shall show
that ϕX ̸∈ T⊥M. If possible, let ϕX has a component, say FX, along T⊥M.
For X,Y ∈ TM, we note that AFXY ∈ TM. Let us denote AFXY = Z ̸= 0.
Here Z is also not a null vector, in general. Now, by virtue of (2.8) we get
g(Z,Z) = g(Z,Z) = g(AFXY,Z) = g
(
B(Y, Z), FX
)
= g(0, FX) = 0.
Since Z is non-null and non-zero vector, the above equation yields a contradic-
tion. This shows that ϕX has no component along T⊥M. Hence, ϕX ∈ TM.
Therefore, the sub manifold is invariant. The above discussion helps us to
state the following:
Theorem 3.2. Every odd dimensional totally geodesic submanifold of an
LP -Sasakian manifold is invariant.
As a direct consequence of Theorem 3.2, we get the following:
Corollary 3.1. Every three-dimensional totally geodesic submanifold of
an LP -Sasakian manifold is invariant.
Combining Theorem 3.1 and Corollary 3.1, we obtain
Theorem 3.3. A three-dimensional submanifold of an LP -Sasakian man-
ifold is totally geodesic if and only if it is invariant.
4. Example
In this section we like to construct an example of a five-dimensional LP -
Sasakian manifold and there on an example of three-dimensional invariant
submanifold of the manifold.
Let us consider the 5-dimensional manifold M̃ =
{
(x, y, z, u, v) ∈ R5:
(x, y, z, u, v) ̸= (0, 0, 0, 0, 0)
}
, where (x, y, z, u, v) are the standard coordinates
in R5. The vector fields
e1 = −2
∂
∂x
+ 2y
∂
∂z
, e2 =
∂
∂y
, e3 =
∂
∂z
, e4 = −2
∂
∂u
+ 2v
∂
∂z
, e5 =
∂
∂v
on invariant submanifolds of LP -sasakian manifolds 151
are linearly independent at each point of M. Let g be the metric defined by
g(ei, ej) = 1, for i = j ̸= 3,
g(ei, ej) = 0, for i ̸= j,
g(e3, e3) = −1.
Here i and j runs from 1 to 5. Let η be the 1-form defined by η(Z) = g(Z, e3),
for any vector field Z tangent to M̃. Let ϕ be the (1, 1) tensor field defined by
ϕe1 = e2, ϕe2 = e1, ϕe3 = 0, ϕ(e4) = e5, ϕ(e5) = e4.
Then, using the linearity of ϕ and g we have
η(e3) = −1, ϕ2Z = Z + η(Z)e3,
for any vector fields Z,W tangent to M̃. Thus for e3 = ξ, M̃(ϕ, ξ, η, g) defines
an almost para-contact metric manifold. Let ∇̃ be the Levi-Civita connection
on M̃ with respect to the metric g. Then we have
[e1, e2] = −2e3, [e1, e3] = 0, [e1, e4] = 0,
[e1, e5] = 0, [e2, e3] = 0, [e2, e4] = 0,
[e2, e5] = 0, [e3, e4] = 0, [e4, e5] = −2e3.
Taking e3 = ξ and using Koszul’s formula for g, it can be easily calculated
that
∇̃e1e5 = 0, ∇̃e1e4 = 0, ∇̃e1e3 = e2, ∇̃e1e2 = −e3, ∇̃e1e1 = 0,
∇̃e2e5 = 0, ∇̃e2e4 = 0, ∇̃e2e3 = e1, ∇̃e2e2 = 0, ∇̃e2e1 = e3,
∇̃e3e5 = e4, ∇̃e3e4 = e5, ∇̃e3e3 = 0, ∇̃e3e2 = e1, ∇̃e3e1 = e2,
∇̃e4e5 = −e3, ∇̃e4e4 = 0, ∇̃e4e3 = e5, ∇̃e4e2 = 0, ∇̃e4e1 = 0,
∇̃e5e5 = 0, ∇̃e5e4 = e3, ∇̃e5e3 = e4, ∇̃e5e2 = 0, ∇̃e5e1 = 0.
From the above calculations, we see that the manifold under consideration
satisfies η(ξ) = −1 and ∇Xξ = ϕX. Hence, it is an LP -Sasakian manifold.
Let f be an isometric immersion from M to M̃ defined by f(x, y, z) =
(x, y, z, 0, 0).
Let M = {(x, y, z) ∈ R3, (x, y, z) ̸= (0, 0, 0)}, where (x, y, z) are the stan-
dard coordinates in R3. The vector fields
152 a. sarkar, m. sen
e1 = −2
∂
∂x
+ 2y
∂
∂z
, e2 =
∂
∂y
, e3 =
∂
∂z
are linearly independent at each point of M. Let g be the metric defined by
g(e1, e3) = g(e2, e3) = g(e1, e2) = 0,
g(e1, e1) = g(e2, e2) = 1, g(e3, e3) = −1.
Let η be the 1-form defined by η(Z) = g(Z, e3), for any vector field Z tangent
to M. Let ϕ be the (1, 1) tensor field defined by
ϕe1 = e2, ϕe2 = e1, ϕe3 = 0.
Then, using the linearity of ϕ and g we have
η(e3) = 1, ϕ
2Z = Z + η(Z)e3,
for any vector fields Z,W tangent to M. Thus for e3 = ξ, M(ϕ, ξ, η, g) defines
an almost para-contact metric manifold.
Let ∇ be the Levi-Civita connection on M with respect to the metric g.
Then we have
[e1, e2] = −2e3, [e1, e3] = 0, [e2, e3] = 0.
Taking e3 = ξ and using Koszul’s formula for the metric g, it can be easily
calculated that
∇e1e3 = e2, ∇e1e2 = −e3, ∇e1e1 = 0,
∇e2e3 = e1, ∇e2e2 = 0, ∇e2e1 = e3,
∇e3e3 = 0, ∇e3e2 = e1, ∇e3e1 = e2.
We see that the (ϕ, ξ, η, g) structure satisfies the formula ∇Xξ = ϕX, η(ξ) =
−1. Hence M(ϕ, ξ, η, g) is a three-dimensional LP -Sasakian manifold. It is
obvious that the manifold M under consideration is a submanifold of the
manifold M̃.
Let us take D =< e1 >, D
⊥ =< e2 >. Then clearly we see that TM =
D ⊕D⊥⊕ < ξ > . For any X ∈ D and Y ∈ D⊥, we can write X = λe1 and
Y = µe2, where λ, µ are two scalars. Now, ϕ(λe1) = λϕ(e1) = λe2 ∈ D⊥ ⊂
TM. Similarly, ϕ(µe2) ∈ D ⊂ TM. Hence the submanifold is invariant.
on invariant submanifolds of LP -sasakian manifolds 153
Let U = λ1e1 + λ2e2 + λ3e3 ∈ TM and V = µ1e1 + µ2e2 + µ3e3 ∈ TM,
where λi and µi are scalars, i = 1, 2, 3.
Then
B(U, V ) = λ1µ1B(e1, e1) + λ1µ2B(e1, e2) + λ1µ3B(e1, e3)
+ λ2µ1B(e2, e1) + λ2µ2B(e2, e2) + λ2µ3B(e2, e3)
+ λ3µ1B(e3, e1) + λ3µ2B(e3, e2) + λ3µ3B(e3, e3).
From the values of ∇̃eiej and ∇e1ej calculated before and from the relation
B(ei, ej) = ∇̃eiej −∇eiej , we see that B(U, V ) = 0, for all U, V ∈ TM. Hence,
the sub manifold is totally geodesic.
The above arguments tell us that the submanifold M under consideration
agrees with Theorem 3.3 which is the main result of the present paper.
Acknowledgements
The authors are thankful to the referee for his valuable suggestions
for the improvement of the paper.
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On Invariant Submanifolds of LP -Sasakian Manifolds

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On Invariant Submanifolds of LP -Sasakian Manifolds

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