# Submanifolds of Sasakian Manifolds with Certain Parallel Operators

Authors: Amira A. Ishan
Journal: ISRN Geometry
 (2014)

---

## Abstract

We study submanifolds of Sasakian manifolds and obtain a condition under which certain naturally defined symmetric tensor field on the submanifold is to be parallel and use this result to obtain conditions under which a submanifold of the Sasakian manifold is an invariant submanifold.


---

## Body

## 1. Introduction

Differential geometry of submanifolds of Sasakian manifold is an interesting branch and has been subject of investigations of many mathematicians. There are four important types of submanifolds of a Sasakian manifold, namely, invariant submanifolds, anti-invariant submanifolds, CR submanifolds, and slant submanifolds. All these types of submanifolds have been studied quite extensively; for invariant and anti-invariant submanifolds one can refer to [1–5] and for slant submanifolds one can refer to [6–8] and references therein. A submanifold M of a Sasakian manifold (
M
¯
,
φ
,
ξ
,
η
,
g
) is said to be a CR-submanifold if there is a pair of orthogonal complementary distributions 𝒟 and 𝒟
⊥ defined on M of which 𝒟 is invariant under φ and 𝒟
⊥ is totally real distribution so that φ
𝒟
⊥ is subbundle of the normal bundle. CR submanifolds of a Sasakian manifold have been studied in [9–13] getting various geometric properties of the CR-submanifold. A submanifold of a Sasakian manifold naturally carries four operators, ϕ, F, ψ, and G, defined on this submanifold. In [14], it has been shown that a submanifold of a Sasakian manifold with parallel ϕ is essentially a CR-submanifold. In this paper, we are interested in finding conditions under which a submanifold of a Sasakian manifold with parallel operator F is an invariant submanifold. First we show the existence of a symmetric tensor field C on any submanifold of a Sasakian manifold and study its properties and use these properties to obtain conditions under which a submanifold is an invariant submanifold (cf. Theorem 5).


## 2. Preliminaries

Let(
M
¯
,
φ
,
ξ
,
η
,
g
) be a (
2
m
+
1
)-dimensional almost contact metric manifold. Then the structure tensor satisfies (cf. [15])
(1)
φ
2
=
-
I
+
η
⊗
ξ
,
η
(
ξ
)
=
1
,
φ
ξ
=
0
,
η
∘
φ
=
0
,
g
(
φ
X
,
φ
Y
)
=
g
(
X
,
Y
)
-
η
(
X
)
η
(
Y
)
,
g
(
X
,
ξ
)
=
η
(
X
)
,
where X and Y are smooth vectors fields on M
¯. An almost contact metric manifold (
M
¯
,
φ
,
ξ
,
η
,
g
) is said to be a Sasakian manifold if it satisfies
(2)
(
∇
¯
X
φ
)
(
Y
)
=
g
(
X
,
Y
)
ξ
-
η
(
Y
)
X
,
∇
¯
X
ξ
=
-
φ
X
,
for smooth vector fields X and Y, where ∇
¯ is the Riemannian connection on M
¯.LetM be an n-dimensional submanifold of the Sasakian manifold (
M
¯
,
φ
,
ξ
,
η
,
g
). Denote by the same g the Riemannian metric induced on the submanifold M and by Γ
(
T
⊥
M
) the space of smooth sections of the normal bundle T
⊥
M of M. Then we define operators ϕ, F, ψ, and G as follows:
(3)
φ
X
=
ϕ
X
+
F
X
,
φ
N
=
ψ
N
+
G
N
,
where X
∈
χ
(
M
) is the Lie algebra of smooth vector fields on M and N
∈
Γ
(
T
⊥
M
); ϕ
X (resp., F
X) denotes the tangent part (resp., normal part) of φ
X; and ψ
N (resp., G
N) denotes the tangent part (resp., normal part) of φ
N.Also for the structure vector fieldξ of the Sasakian manifold M
¯, we set
(4)
ξ
=
u
+
t
,
where u
∈
χ
(
M
) and t
∈
Γ
(
T
⊥
M
). Define a smooth one form α on the submanifold M and β
:
Γ
(
T
⊥
M
)
→
C
∞
(
M
) by α
(
X
)
=
g
(
X
,
u
), X
∈
χ
(
X
), and β
(
N
)
=
g
(
N
,
t
), N
∈
Γ
(
T
⊥
M
). It can be easily checked that the operators ϕ, F, ψ, and G satisfied the relations
(5)
g
(
ϕ
X
,
Y
)
=
-
g
(
X
,
ϕ
Y
)
,
X
,
Y
∈
χ
(
M
)
,
(6)
g
(
G
N
1
,
N
2
)
=
-
g
(
N
1
,
G
N
2
)
,
N
1
,
N
2
∈
Γ
(
T
⊥
M
)
,
(7)
g
(
F
X
,
N
)
=
-
g
(
X
,
ψ
N
)
,
X
∈
χ
(
M
)
,
N
∈
Γ
(
T
⊥
M
)
,
(8)
ϕ
2
X
+
ψ
(
F
X
)
=
-
X
+
α
(
X
)
u
,
F
(
ϕ
X
)
=
-
G
(
F
X
)
+
α
(
X
)
t
,
X
∈
χ
(
M
)
,
(9)
G
2
N
+
F
(
ψ
N
)
=
-
N
+
β
(
N
)
t
,
ϕ
(
ψ
(
N
)
)
=
-
ψ
(
G
N
)
+
β
(
N
)
u
,
N
∈
Γ
(
T
⊥
M
)
,
(10)
ϕ
u
=
-
ψ
t
,
F
u
=
-
G
t
.Also for the submanifoldM, we have the following Gauss and Weingarten formulas (cf. [16]):
(11)
∇
¯
X
Y
=
∇
X
Y
+
h
(
X
,
Y
)
,
X
,
Y
∈
χ
(
M
)
,
∇
¯
X
N
=
-
A
N
X
+
∇
X
⊥
N
,
N
∈
Γ
(
T
⊥
M
)
,
where ∇
⊥ is the normal connection, h is the second fundamental form and A
N is the Weingarten map with respect to the normal bundle T
⊥
M of M and we have
(12)
g
(
A
N
X
,
Y
)
=
g
(
h
(
X
,
Y
)
,
N
)
,
X
,
Y
∈
χ
(
M
)
,
N
∈
Γ
(
T
⊥
M
)
.For the operatorsF and ψ on the submanifold M of the Sasakian manifold M
¯, we define the following covariant derivatives:
(13)
(
D
X
F
)
Y
=
∇
X
⊥
F
Y
-
F
(
∇
X
Y
)
,
X
,
Y
∈
χ
(
M
)
,
(
D
X
ψ
)
N
=
∇
X
ψ
N
-
ψ
(
∇
X
⊥
N
)
,
X
∈
χ
(
M
)
,
N
∈
Γ
(
T
⊥
M
)
.
Then using (2)–(4) and (11) we immediately get the following.Lemma 1.
LetM be a submanifold of the Sasakian manifold M
¯. Then one has
(14)
∇
X
u
=
-
ϕ
X
+
A
t
X
,
∇
X
⊥
t
=
-
h
(
X
,
u
)
-
F
X
,
X
∈
χ
(
M
)
,
(
∇
X
ϕ
)
Y
=
g
(
X
,
Y
)
u
-
α
(
Y
)
X
+
A
F
Y
X
d
d
+
ψ
(
h
(
X
,
Y
)
)
,
X
,
Y
∈
χ
(
M
)
,
(
D
X
F
)
Y
=
g
(
X
,
Y
)
t
-
h
(
X
,
ϕ
Y
)
d
d
d
d
d
d
d
+
G
(
h
(
X
,
Y
)
)
,
X
,
Y
∈
χ
(
M
)
,
(
D
X
ψ
)
N
=
-
β
(
N
)
X
+
A
G
N
X
-
ϕ
(
A
N
X
)
,
X
∈
χ
(
M
)
,
N
∈
Γ
(
T
⊥
M
)
,
(
∇
X
⊥
G
)
N
=
-
F
(
A
N
X
)
-
h
(
X
,
ψ
N
)
,
X
∈
χ
(
M
)
,
N
∈
Γ
(
T
⊥
M
)
.


## 3. Submanifolds with ParallelF

In this section we will study a submanifoldM of the Sasakian manifold (
M
¯
,
φ
,
ξ
,
η
,
g
) with structure vector field ξ tangent to M and the operator F is parallel. Note that it is customary to require that ξ is tangent to M (cf. [13]), rather than normal as it is too restrictive, as in this case M must be totally real [13, Proposition 1.1, page 43] or leads to too complicated embedding equations.If we define the operators(15)
B
=
ψ
∘
F
:
χ
(
M
)
⟶
χ
(
M
)
,
C
=
F
∘
ψ
:
Γ
(
T
⊥
M
)
⟶
Γ
(
T
⊥
M
)
,
by using (7) it is easy to see that B and C are symmetric tensor fields.In [17], Yano and Kon have studied the submanifolds of a Sasakian manifold with operator F is parallel and have proved that these submanifolds are contact CR submanifolds [17, Proposition 3.3, page 241]. Thus all our results for submanifolds with parallel operator F of a Sasakian manifold are automatically the results of contact CR submanifolds of the Sasakian manifold. First we will prove the following.Theorem 2.
LetM be an n-dimensional submanifold of a (
2
m
+
1
)-dimensional Sasakian manifold (
M
¯
,
φ
,
ξ
,
η
,
g
) with structure vector field ξ tangent to M. If the operator F is parallel, then C is also parallel.Proof.
ForX
∈
χ
(
M
) and N
∈
Γ
(
T
⊥
M
) we have
(16)
∇
X
⊥
(
C
N
)
=
∇
X
⊥
F
(
ψ
(
N
)
)
=
(
D
X
F
)
(
ψ
(
N
)
)
+
F
(
∇
X
ψ
(
N
)
)
=
(
D
X
F
)
(
ψ
(
N
)
)
+
F
(
(
D
X
ψ
)
N
)
+
F
(
ψ
(
∇
X
⊥
N
)
)
=
(
D
X
F
)
(
ψ
(
N
)
)
+
F
(
(
D
X
ψ
)
N
)
+
C
(
∇
X
⊥
N
)
,
and consequently
(17)
(
∇
X
⊥
C
)
N
=
(
D
X
F
)
(
ψ
(
N
)
)
+
F
(
(
D
X
ψ
)
N
)
.
Using Lemma 1 in above equation and noting that β
(
N
)
=
g
(
N
,
t
)
=
0, we get
(18)
(
∇
X
⊥
C
)
N
=
-
h
(
X
,
ϕ
(
ψ
(
N
)
)
)
+
G
(
h
(
X
,
ψ
(
N
)
)
)
+
F
(
A
G
N
X
-
ϕ
(
A
N
X
)
)
.
Now since F is parallel then by Lemma 1 with t
=
0, we get
(19)
h
(
X
,
ϕ
Y
)
=
G
(
h
(
X
,
Y
)
)
.
Also, using (6) we get
(20)
g
(
h
(
X
,
ϕ
Y
)
,
N
)
=
-
g
(
h
(
X
,
Y
)
,
G
N
)
;
by using the above equation and (12) we get
(21)
A
G
N
X
=
ϕ
A
N
X
.
Using (21) and (19) in (18), we conclude that (
∇
X
⊥
C
)
N
=
0 and this proves the theorem.ForN
∈
Γ
(
T
⊥
M
), let C
(
N
)
=
λ
N, λ
∈
C
∞
(
M
); that is, N is the eigenvector of C corresponding to an eigenvalue λ.Lemma 3.
LetM be an n-dimensional submanifold of the Sasakian manifold (
M
¯
,
φ
,
ξ
,
η
,
g
) with structure vector field ξ tangent to M. If the operator F is parallel, then the eigenvalues of C are constant.Proof.
LetC
(
N
)
=
λ
N, λ
∈
C
∞
(
M
), N
∈
Γ
(
T
⊥
M
). Without loss of generality we can assume that N is a unit normal vector field. As F is parallel then, by Theorem 2, we have C being parallel and consequently
(22)
0
=
(
∇
X
⊥
C
)
(
N
)
=
∇
X
⊥
(
C
N
)
-
C
(
∇
X
⊥
N
)
=
X
(
λ
)
N
+
λ
(
∇
X
⊥
N
)
-
C
(
∇
X
⊥
N
)
.
Taking inner product with N, we get
(23)
0
=
X
(
λ
)
,
which proves that λ is a constant.For a local orthonormal frame{
N
1
,
…
,
N
k
} (where 2
m
+
1
=
n
+
k) on M, we define smooth function ∥
G
∥
:
M
→
R by
(24)
∥
G
∥
2
=
∑
i
=
1
k
‍
g
(
G
(
N
i
)
,
G
(
N
i
)
)
;
then using (6), (7), and (9), we obtain
(25)
∥
G
∥
2
=
k
-
∥
ψ
∥
2
,
where ∥
ψ
∥
2
=
∑
i
=
1
k
‍
g
(
ψ
(
N
i
)
,
ψ
(
N
i
)
). Also, we define
(26)
∥
C
∥
2
=
∑
i
=
1
k
‍
g
(
C
(
N
i
)
,
C
(
N
i
)
)
;
since C is symmetric we can choose an orthonormal frame {
N
1
,
…
,
N
k
} that diagonalizes C.Lemma 4.
LetM be n-dimensional submanifold of the Sasakian manifold (
M
¯
,
φ
,
ξ
,
η
,
g
) with structure vector field ξ tangent to M. If the operator F is parallel, then ∥
ψ
∥
2 is a constant and ∥
G
∥
2
+
∥
ψ
∥
2
=
k holds; consequently ∥
G
∥
2 is also a constant.Proof.
SupposeF is parallel; then by (25), we have
(27)
∥
G
∥
2
+
∥
ψ
∥
2
=
k
.
By (7) we have
(28)
X
(
∥
ψ
∥
2
)
=
-
X
(
∑
i
=
1
k
‍
(
g
(
C
(
N
i
)
,
N
i
)
)
)
;
since the operator C is symmetric we get
(29)
X
(
∥
ψ
∥
2
)
=
-
∑
i
=
1
k
‍
{
g
(
(
∇
X
⊥
C
)
N
i
,
N
i
)
+
2
g
(
C
(
N
i
)
,
∇
X
⊥
N
i
)
}
;
since by Theorem 2 since F is parallel implies C also parallel and the local orthonormal frame {
N
1
,
…
,
N
k
} on normal vector fields that diagonalizes C with the equation C
(
N
i
)
=
λ
i
N
i,  thus we get that
(30)
X
(
∥
ψ
∥
2
)
=
-
∑
i
=
1
k
‍
λ
i
X
(
g
(
N
i
,
N
i
)
)
=
0
,
which proves that ∥
ψ
∥
2 is a constant. Finally as ∥
ψ
∥
2 is a constant, we get that ∥
G
∥
2 is also a constant.Note that the operatorC plays an important role in the geometry of submanifolds of a Sasakian manifold; for instance the following theorem shows that tr
·
C
=
0 implies that the submanifolds are invariant submanifolds.Theorem 5.
LetM be an n-dimensional connected submanifold of the Sasakian manifold (
M
¯
,
φ
,
ξ
,
η
,
g
) with structure vector field ξ tangent to M. If  tr
·
C
=
0, then M is an invariant submanifold.Proof.
Supposetr
·
C
=
0; then we have
(31)
0
=
tr
·
C
=
∑
i
=
1
k
‍
g
(
N
i
,
C
N
i
)
=
-
∑
i
=
1
k
‍
g
(
ψ
N
i
,
ψ
N
i
)
=
-
∥
ψ
∥
2
,
and that gives ψ
=
0. This also gives B
=
0 and consequently
(32)
0
=
g
(
B
X
,
X
)
=
-
g
(
F
X
,
F
X
)
=
-
∥
F
X
∥
2
,
X
∈
χ
(
M
)
;
that is, F
X
=
0, X
∈
χ
(
M
). That is, M is an invariant submanifold.Theorem 6.
LetM be an n-dimensional submanifold of the Sasakian manifold (
M
¯
,
φ
,
ξ
,
η
,
g
) with structure vector field ξ tangent to M. The operator F is parallel if and only if ψ is parallel.Proof.
IfF is parallel then by Lemma 1 with t
=
0 we get
(33)
h
(
X
,
ϕ
Y
)
=
G
(
h
(
X
,
Y
)
)
.
Taking inner product with N
∈
Γ
(
T
⊥
M
) and using (6) we get
(34)
g
(
h
(
X
,
ϕ
Y
)
,
N
)
=
-
g
(
h
(
X
,
Y
)
,
G
N
)
,
and using (12) and (5) we get
(35)
g
(
ϕ
(
A
N
X
)
,
Y
)
=
g
(
A
G
N
X
,
Y
)
,
which implies that
(36)
A
G
N
X
=
ϕ
(
A
N
X
)
.
Now using the above equation in Lemma 1 and the fact that t
=
0 we get that β
(
N
)
=
0 and (
D
X
ψ
)
N
=
0 which means that ψ is parallel.
Conversely, if we suppose that the operatorψ is parallel then by Lemma 1 with t
=
0, we get
(37)
A
G
N
X
=
ϕ
(
A
N
X
)
.
Also using (5) to get
(38)
g
(
A
G
N
X
,
Y
)
=
-
g
(
A
N
X
,
ϕ
Y
)
,
which together with (12) and (6) gives
(39)
g
(
G
(
h
(
X
,
Y
)
)
,
N
)
=
g
(
h
(
X
,
ϕ
Y
)
,
N
)
,
we get that
(40)
G
(
h
(
X
,
Y
)
)
=
h
(
X
,
ϕ
Y
)
,
and using the above equation in Lemma 1 with t
=
0 we get (
D
X
F
)
=
0 which proves that F is parallel.As a direct consequence of the above theorem and Proposition 3.3 in [17], we have the following.Corollary 7.
LetM be a submanifold of a Sasakian manifold (
M
¯
,
φ
,
ξ
,
η
,
g
). If ψ is parallel, then M is a contact CR-submanifold of M
¯.



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