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
¯
.
Let
M
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 field
ξ
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 submanifold
M
, 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 operators
F
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.
Let
M
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 Parallel <inlineformula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M84">
<mml:mrow>
<mml:mi>F</mml:mi></mml:mrow>
</mml:math></inlineformula>
In this section we will study a submanifold
M
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.
Let
M
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.
For
X
∈
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.
For
N
∈
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.
Let
M
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.
Let
C
(
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.
Let
M
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.
Suppose
F
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 operator
C
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.
Let
M
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.
Suppose
tr
·
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.
Let
M
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.
If
F
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 operator
ψ
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.
Let
M
be a submanifold of a Sasakian manifold
(
M
¯
,
φ
,
ξ
,
η
,
g
)
. If
ψ
is parallel, then
M
is a contact CRsubmanifold of
M
¯
.