An. Ştiinţ. Univ. Al. I. Cuza Iaşi Mat. (N.S.)
Tomul LXII, 2016, f. 2, vol. 3
Geometry of warped product pseudo-slant submanifolds in nearly
Kaehler manifolds
Siraj Uddin · Falleh R. Al-Solamy ·
Khalid Ali Khan
Received: 21.IX.2015 / Revised: 5.XII.2015 / Accepted: 11.I.2016
Abstract Non-existence of warped product pseudo-slant submanifolds of nearly Kaehler manifolds
was proved under some conditions in [16]. In this paper, we continue the study of such warped products
for their existence. We characterise pseudo-slant submanifolds of a nearly Keahler manifold to be
locally warped products. Also, we obtain a geometric inequality for the squared norm of the second
fundamental form of a mixed geodesic warped product submanifold in terms of the warping function
and the slant angle. The equality case is also considered.
Keywords Warped product · Slant submanifolds · Pseudo-slant submanifolds · Warped product
pseudo-slant submanifolds · Nearly Kaehler manifolds
Mathematics Subject Classification (2010) 53C40 · 53C42 · 53C15
1 Introduction
Nearly Kaehler manifolds are exactly the Tachibana manifolds initially studied in
[15]. Nearly Kaehler manifolds form an interesting class of manifolds admitting a metric connection with parallel totally anti-symmetric torsion (see [1]). The best known
example of a nearly Kaehler non-Kaehler manifold is S 6 , a six dimensional sphere.
On the other hand, slant submanifolds of almost Hermitian manifolds were introduced by Chen in [5-6] as a generalization of both holomorphic and totally real submanifolds. In [12], Papaghiuc has introduced another class of submanifolds in almost
Siraj Uddin · Falleh R. Al-Solamy
Department of Mathematics
Faculty of Science
King Abdulaziz University
21589 Jeddah, Saudi Arabia
E-mail: [email protected]; [email protected]
Khalid Ali Khan
International Graduate Centre of Education
Law Education Business and Arts
Casuarina Campus-Orange 2.04.09
Charles Darwin University
Darwin, Nothern Territory 0909, Australia
E-mail: [email protected]
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Siraj Uddin, Falleh R. Al-Solamy, Khalid Ali Khan
Hermitian manifolds, called the semi-slant submanifolds, which are natural generalizations of CR and slant submanifolds. On the similar line of thought, Sahin [14] has
introduced the idea of pseudo-slant submanifolds of Kaehler manifolds to study their
warped products. Initially, these submanifolds were introduced by Carriazo [4] in
almost contact settings.
Recently, warped product pseudo-slant submanifolds of nearly Kaehler manifolds
were studied by the first author in [16]. In this paper, we continue this study for the
existence of warped product pseudo-slant submanifolds. The paper is organized as
follows: in Section 2, we recall some basic formulas and definitions. In Section 3, we
study pseudo-slant submanifolds and their warped products of the forms N⊥ ×f Nθ
f, where N⊥ and Nθ are totally real
and Nθ ×f N⊥ of a nearly Kaehler manifold M
f
and proper slant submanifolds of M , respectively. In the beginning of this section we
obtain some basic results for later use and then obtain a characterization. In Section
4, we derive an inequality for the squared norm of the second fundamental form. The
equality case is also discussed.
2 Preliminaries
f, g ) be an almost Hermitian manifold with almost complex structure J and a
Let (M
Riemannian metric g such that
(a) J 2 = −I,
(b) g (JX, JY ) = g (X, Y ),
(2.1)
f.
for all vector fields X, Y on M
f
f and ∇
e , the covariant
Further, let Γ (T M ) denote the set of all vector fields on M
f with respect to g . Then according to Gray [9], if the almost
differential operator on M
complex structure J satisfies
e X J )X = 0,
(∇
(2.2)
f), then the manifold M
f is called a nearly Kaehler manifold. Equafor any X ∈ Γ (T M
e X J )Y + (∇
e Y J )X = 0, for any X, Y ∈ Γ (T M
f).
tion (2.2) is equivalent to (∇
f
Let M be an almost Hermitian manifold with almost complex structure J , and M a
f. Then M is called holomorphic (or
Riemannian manifold isometrically immersed in M
complex) if J (Tx M ) ⊆ Tx M , for any x ∈ M where Tx M denotes the tangent space of
M at x, and totally real if J (Tx M ) ⊆ Tx⊥ M for any x ∈ M , where Tx⊥ M denotes the
normal space of M at x. There are some other important classes of submanifolds of
an almost Hermitian manifold determined by the behaviour of tangent bundle of the
submanifold under the action of almost complex structure J of the ambient manifold.
Some of them are following:
f is said to be a CR-submanifold
(i) A submanifold M of an almost Hermitian manifold M
f
(see [13]) of M if there exist a differentiable distribution D : x → Dx ⊂ Tx M such
that D is holomorphic distribution and the orthogonal complementary distribution
D⊥ is totally real.
f is said to be slant (see [5]) if
(ii) A submanifold M of an almost Hermitian manifold M
for each non-zero vector X tangent to M the angle θ(X ) between JX and Tx M is
a constant, i.e., it does not depend on the choice of x ∈ M and X ∈ Tx M .
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Geometry of warped product pseudo-slant submanifolds
3
(iii) The submanifold M is called semi-slant (see [12]) if there exist a pair of orthogonal
distributions D and Dθ such that D is holomorphic and Dθ is proper slant, i.e.,
the angle θ(X ) between JX and Dxθ is constant and it is neither 0 nor π/2 for any
X ∈ Dxθ .
f, the Gauss, Weingarten formulas
For a submanifold M of a Riemannian manifold M
are given by
e X Y = ∇X Y + h(X, Y ),
∇
e X N = −AN X + ∇⊥
∇
X N,
(2.3)
for all X, Y ∈ Γ (T M ) and N ∈ Γ (T ⊥ M ), where Γ (T M ) is the Lie algebra of vector
fields in M and Γ (T ⊥ M ) is the set of all vector fields normal to M and ∇ is the
induced Riemannian connection on M , h is the second fundamental form of M , ∇⊥
is the normal connection in the normal bundle T ⊥ M and AN is the shape operator
of the second fundamental form. They are related as g (AN X, Y ) = g (h(X, Y ), N ),
f as well as the metric induced on M .
where g denotes the Riemannian metric on M
Pm
1
The mean curvature vector H of M is given by H = m
i=1 h(ei , ei ), where m is the
dimension of M and {e1 , e2 , . . . , em } is a local orthonormal frame of vector fields on
f is said to be totally umbilical
M . A submanifold M of an almost Hermitian manifold M
if the second fundamental form satisfies h(X, Y ) = g (X, Y )H, for all X, Y ∈ Γ (T M ).
The submanifold M is totally geodesic if h(X, Y ) = 0, for all X, Y ∈ Γ (T M ) and
minimal if H = 0.
Now, let {e1 , . . . , em } be an orthonormal basis of tangent space T M and er belongs
to the orthonormal basis {em+1 , . . . , e2 n} of the normal bundle T ⊥ M , we put
hrij = g (h(ei , ej ), er )
khk2 =
and
m
X
g (h(ei , ej ), h(ei , ej )).
(2.4)
i,j =1
For a differentiable function ϕ on M , the gradient ∇ϕ is defined by
g (∇ϕ, X ) = Xϕ,
(2.5)
for any X ∈ Γ (T M ). As a consequence, we have
k∇ϕk2 =
m
X
(ei (ϕ))2 .
(2.6)
i=1
For any X ∈ Γ (T M ) and N ∈ Γ (T ⊥ M ), the transformations JX and JN are decomposed into tangential, normal part as
(a) JX = T X + F X,
(b) JN = BN + CN.
f, we have
On a slant submanifold M of an almost Hermitian manifold M
T 2 X = − cos2 θX,
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(2.7)
(2.8)
4
Siraj Uddin, Falleh R. Al-Solamy, Khalid Ali Khan
f (see [6]). As a consequence of the relation (2.8),
where θ is the slant angle of M in M
we have
g (T X, T Y ) = cos2 θg (X, Y ),
2
g (F X, F Y ) = sin θg (X, Y ),
(2.9)
(2.10)
for any X, Y ∈ Γ (T M ). Also, for a slant sumanifold from (2.7) (a) and (2.8), we have
BF X = − sin2 θX
and CF X = −F T X,
(2.11)
for any X ∈ Γ (T M ).
e X J )Y , i.e.,
Now, denote by PX Y and QX Y the tangential and normal parts of (∇
e X J )Y = PX Y + QX Y,
(∇
(2.12)
(a) PX Y + PY X = 0, (b) QX Y + QY X = 0,
(2.13)
for all X, Y ∈ Γ (T M ). For the properties of P and Q we refer [11], which we enlist
here for later use.
(p1 ) (i) PX +Y W = PX W + PY W,
(ii) QX +Y W = QX W + QY W,
(p2 ) (i) PX (Y + W ) = PX Y + PX W, (ii) QX (Y + W ) = QX Y + QX W,
(p3 ) (i) g (PX Y, W ) = −g (Y, PX W ), (ii) g (QX Y, N ) = −g (Y, PX N ),
(p 4 )
PX JY + QX JY = −J (PX Y + QX Y ),
for all X, Y, W ∈ Γ (T M ) and N ∈ Γ (T ⊥ M ).
f, by equations
On a Riemannian submanifold M of a nearly Kaehler manifold M
(2.2) and (2.12), we have
for any X, Y ∈ Γ (T M ).
3 Pseudo-slant submanifolds and their warped products
In this section, we study pseudo-slant submanifolds and their warped products in a
f. Pseudo-slant submanifolds were introduced by Carriazo
nearly Kaehler manifold M
under the name anti-slant (see [4]). Later on, these submanifolds were studied by
Sahin under the name hemi-slant submanifolds of Kaehler manifolds for their warped
products (see [14]). We define these submanifolds as follows:
f is said to be
Definition 3.1 A submanifold M of an almost Hermitian manifold M
a pseudo-slant submanifold if there exist a pair of orthogonal distributions D⊥ and Dθ
such that:
(i) T M admits the orthogonal direct decomposition T M = D⊥ ⊕ Dθ ;
(ii) the distribution D⊥ is totally real, i.e., JD⊥ ⊂ T ⊥ M ;
(iii) the distribution Dθ is slant with slant angle θ 6= 0 or π2 .
From the definition, it is clear that if θ = 0, then M is a CR-submanifold. Also, if
we denote the dimensions of D⊥ and Dθ by q and p, respectively. Then, we have:
(i) If p = 0, then M is totally real.
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Geometry of warped product pseudo-slant submanifolds
5
(ii) If q = 0 and θ = 0, then M is holomorphic.
(iii) If q = 0 and θ 6= 0, π/2, then M is a proper slant submanifold with slant angle θ.
(iv) If θ = π/2, then M is again a totally real submanifold.
We say that a pseudo-slant submanifold is proper if neither q = 0 nor θ = 0, π/2.
The normal space of a pseudo-slant submanifold M is decomposed as T ⊥ M = JD⊥ ⊕
F Dθ ⊕ µ, where µ is the invariant normal subbundle of M with respect to J . On a
pseudo-slant submanifold of a nearly Kaehler manifold we have the following results
for later use.
f.
Lemma 3.2 Let M be a pseudo-slant submanifold of a nearly Kaehler manifold M
2
Then g (∇X Y, Z ) = sec θ{g (AJZ X, T Y )−g (AF T Y X, Z )−g (QX Z, F Y )−g (QX Y, JZ )},
for any X, Y ∈ Γ (Dθ ) and Z ∈ Γ (D⊥ ).
e X Y, Z ).
Proof. For any X, Y ∈ Γ (Dθ ) and Z ∈ Γ (D⊥ ), we have g (∇X Y, Z ) = g (∇
Using (2.1) (b), we get
e X Y, JZ )
g (∇X Y, Z ) = g (J ∇
e X JY, JZ ) − g ((∇
e X J )Y, JZ ).
= g (∇
Then from (2.7) (a) and (2.12), we obtain
e X T Y, JZ ) + g (∇
e X F Y, JZ ) − g (QX Y, JZ )
g (∇X Y, Z ) = g (∇
e X JF Y, Z ) + g ((∇
e X J )F Y, Z )
= g (h(X, T Y ), JZ ) − g (∇
− g (QX Y, JZ )
e X BF Y, Z ) − g (∇
e X CF Y, Z )
= g (h(X, T Y ), JZ ) − g (∇
+ g (PX F Y, Z ) − g (QX Y, JZ ).
Using (2.11) and the properties of P -Q, (p3 ) (ii), we arrive at g (∇X Y, Z ) = g (h(X ,
e X F T Y, Z ) − g (QX Z, F Y ) −g (QX Y, JZ ). Thus the
T Y ), JZ ) + sin2 θg (∇X Y, Z ) + g (∇
result follows from the above relation by using (2.3). t
u
The following corollary is a consequence of the above lemma.
f,
Corollary 3.3 On a pseudo-slant submanifold M of a nearly Kaehler manifold M
the slant distribution Dθ defines a totally geodesic foliation if and only if AJZ T X −
AF T X Z ∈ Γ (D⊥ ) and QX U ∈ Γ (µ), for any X ∈ Γ (Dθ ), Z ∈ Γ (D⊥ ) and U ∈
Γ (T M ).
Now, we discuss the warped product submanifolds of a nearly Kaehler manifold.
The warped product manifolds were studied by Bishop and O’Neill [3]. They defined
these manifolds as follows: Let (N1 , g1 ) and (N2 , g2 ) be two Riemannian manifolds and
f , a positive differentiable function on N1 . Then their warped product M = N1 ×f N2
is the product manifold N1 × N2 equipped with the Riemannian structure such that
g = g1 + f 2 g2 . The function f is called the warping function on M . It was proved in
[3] that for any X ∈ Γ (T N1 ) and Z ∈ Γ (T N2 ), the following relation holds
∇X Z = ∇Z X = (X ln f )Z,
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(3.1)
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Siraj Uddin, Falleh R. Al-Solamy, Khalid Ali Khan
where ∇ denotes the Levi-Civita connection on M . A warped product manifold M =
N1 ×f N2 is said to be trivial (or Riemannian product) if the warping function f is
constant. If M = N1 × f N2 be a warped product manifold then N1 and N2 are totally
geodesic and totally umbilical submanifolds of M , respectively [3,7].
In this paper we discuss the warped product pseudo-slant submanifolds of a nearly
f those are either in the form N⊥ × f Nθ or Nθ × f N⊥ , where
Kaehler manifold M
f, respectively. These
N⊥ and Nθ are totally real and proper slant submanifolds of M
two types of warped products are the products between the totally real and proper
f, we call such types of warped products as warped product
slant submanifolds of M
pseudo-slant submanifolds in the same sense of warped product CR-submanifolds (see
[7-8]).
Proposition 3.4 Let M = N⊥ × f Nθ be a warped product submanifold of a nearly
f, where N⊥ and Nθ are totally real and proper slant submanifolds
Kaehler manifold M
f
of M , respectively. Then:
(i) 2g (h(X, Y ), JZ ) = g (h(X, Z ), F Y ) + g (h(Y, Z ), F X ),
(ii) 2g (h(Z, W ), F X ) = g (h(X, Z ), JW ) + g (h(X, W ), JZ ),
for any X, Y ∈ Γ (T Nθ ) and Z, W ∈ Γ (T N⊥ ).
Proof. For any X, Y ∈ Γ (T Nθ ) and Z ∈ Γ (T N⊥ ), we have
e X Z, JY ) − g (∇
e X Z, T Y ).
g (h(X, Z ), F Y ) = g (∇
Using the property of Riemannian metric g , (2.3) and (3.1), we get
e X J )Z, Y ) − g (∇
e X JZ, Y ) − (Z ln f )g (X, T Y ).
g (h(X, Z ), F Y ) = g ((∇
Then from (2.12), we obtain g (h(X, Z ), F Y ) = g (PX Z , Y ) + g (AJZ X, Y ))−
(Z ln f )g (X, T Y ). Then by property (p3 ) of P , we arrive at
g (h(X, Z ), F Y ) = g (h(X, Y ), JZ ) − g (PX Y, Z ) − (Z ln f )g (X, T Y ).
(3.2)
Interchanging X and Y in (3.2), we obtain
g (h(Y, Z ), F X ) = g (h(X, Y ), JZ ) − g (PY X, Z ) + (Z ln f )g (X, T Y ).
(3.3)
Thus, the first part of the lemma follows from (3.2) and (3.3). Now, for any Z, W ∈
e Z W, JX ) − g (∇
e Z W, T X ).
Γ (T N⊥ ) and X ∈ Γ (T Nθ ) we have g (h(Z, W ), F X ) = g (∇
Using the property of Riemannian metric g , (2.3) and (3.1), we get g (h(Z, W ), F X ) =
e Z J )W, X ) − g (∇
e Z JW, X ) + (Z ln f )g (T X, W ). Then from (2.12), we obtain
g ((∇
g (h(Z, W ), F X ) = g (PZ W, X ) + g (AJW Z, X )
= g (PZ W, X ) + g (h(Z, X ), JW ).
(3.4)
Interchanging Z and W in (3.4), we obtain
g (h(Z, W ), F X ) = g (PW Z, X ) + g (h(W, X ), JZ ).
(3.5)
Then from (3.4), (3.5) and (2.13) (a), we derive (ii). This completes the proof. t
u
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Geometry of warped product pseudo-slant submanifolds
7
Now, we discuss the warped product pseudo-slant submanifolds of the form Nθ ×f N⊥
of a nearly Kaehler manifold. First, we give the following two results for later use.
Lemma 3.5 ([2]) Let M = Nθ × f N⊥ be a warped product submanifold of a nearly
f, where N⊥ and Nθ are totally real and proper slant submanifolds
Kaehler manifold M
f
of M , respectively. Then 2g (h(X, Y ), JZ ) = g (h(X, Z ), F Y )+ g (h(Y, Z ), F X ), for any
X, Y ∈ Γ (T Nθ ) and Z ∈ Γ (T N⊥ ).
Lemma 3.6 Let M = Nθ × f N⊥ be a warped product submanifold of a nearly Kaehler
f, where N⊥ and Nθ are totally real and proper slant submanifolds of M
f,
manifold M
respectively. Then
(i) 2g (h(Z, W ), F X ) = g (h(X, Z ), JW ) + g (h(X, W ), JZ )
+2(T X ln f )g (Z, W ),
(ii) 2g (h(Z, W ), F T X ) = g (h(T X, Z ), JW ) + g (h(T X, W ), JZ )
−2 cos2 θ(X ln f )g (Z, W ),
for any X ∈ Γ (T Nθ ) and Z, W ∈ Γ (T N⊥ ).
Proof. For any Z, W ∈ Γ (T N⊥ ) and X ∈ Γ (T Nθ ), we have
e Z W, JX ) − g (∇
e Z W, T X ).
g (h(Z, W ), F X ) = g (∇
Then by the property of Riemannian metric g and covariant derivative of J , we obtain
e Z J )W, X ) − g (∇
e Z JW, X ) + g (∇
e Z T X, W ).
g (h(Z, W ), F X ) = g ((∇
Using (2.3), (2.12) and (3.1), we get
g (h(Z, W ), F X ) = g (PZ W, X ) + g (AJW Z, X ) + (T X ln f )g (Z, W )
= g (PZ W, X ) + g (h(Z, X ), JW ) + (T X ln f )g (Z, W ).
(3.6)
Interchanging Z and W in (3.6), we derive
g (h(Z, W ), F X ) = g (PW Z, X ) + g (h(W, X ), JZ )
+ (T X ln f )g (Z, W ).
(3.7)
First part follows from (3.6) and (3.7). If we interchange X by T X in (i) we get (ii),
which proves the lemma completely. t
u
Now, we prove the following characterization theorem for pseudo-slant submanifolds.
Theorem 3.7 Let M be a proper pseudo-slant submanifold of a nearly Kaehler manf such that PZ W ∈ Γ (D⊥ ), for any Z, W ∈ Γ (D⊥ ) and QU V ∈ Γ (µ), for
ifold M
any U, V ∈ Γ (T M ) where D⊥ and µ are totally real distribution and invariant normal subbundle of M , respectively. Then M is locally a mixed geodesic warped product
submanifold if and only if
AJZ X = 0, AF T X Z = −(Xλ) cos2 θZ,
(3.8)
for any X ∈ Γ (Dθ ), where λ is a differentiable function on M satisfying W 0 λ = 0, for
any W 0 ∈ Γ (D⊥ ).
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Siraj Uddin, Falleh R. Al-Solamy, Khalid Ali Khan
Proof. Let M = Nθ ×f N⊥ be a mixed geodesic warped product submanifold of a
f such that N⊥ and Nθ are totally real and proper slant
nearly Kaehler manifold M
f, respectively. Then by Lemma 3.2 and Lemma 3.3, we get (3.8).
submanifolds of M
Conversely, if M is a proper pseudo-slant submanifold of a nearly Kaehler manifold
f such that PZ W ∈ Γ (D⊥ ), for any Z, W ∈ Γ (D⊥ ) and QU V ∈ Γ (µ), for any
M
U, V ∈ Γ (T M ). Then by Lemma 3.1 and the relation (3.8), we get g (∇X Y, Z ) = 0,
which means that the leaves of Dθ are totally geodesic in M . On the other hand, for
any Z, W ∈ Γ (D⊥ ) and any X ∈ Γ (Dθ ), we have
e Z W, JX ) − g (J ∇
e W Z, JX )
g ([Z, W ], X ) = g (J ∇
e Z JW, JX ) − g ((∇
e Z J )W, JX ) − g (∇
e W JZ, JX )
= g (∇
e W J )Z, JX )
+ g ((∇
e Z JX ) − g (PZ W, T X ) − g (QZ W, F X )
= −g (JW, ∇
e W JX ) + g (PW Z, T X ) + g (QW Z, F X ).
+ g (JZ, ∇
Since PZ W ∈ Γ (D⊥ ), for any Z, W ∈ Γ (D⊥ ) and QU V ∈ Γ (µ), for any U, V ∈
Γ (T M ). Then, the above relation will be
e Z T X ) − g (JW, ∇
e ZF X)
g ([Z, W ], X ) = −g (JW, ∇
e W T X ) + g (JZ, ∇
e W F X)
+ g (JZ, ∇
e Z JF X, W ) − g ((∇
e Z J )F X, W ) − g (h(Z, T X ), JW )
= g (∇
e W JF X, Z ) + g ((∇
e W J )F X, Z ) + g (h(W, T X ), JZ ).
− g (∇
Using (2.7) (a) and (2.12), we get
e Z BF X, W ) + g (∇
e Z CF X, W ) − g (PZ F X, W )
g ([Z, W ], X ) = g (∇
e W BF X, Z ) − g (∇
e W CF X, Z )
− g (AJW T X, Z ) − g (∇
+ g (PW F X, Z ) + g (AJZ T X, W ).
Using (2.11), property (p3 )(ii) and (3.8), we obtain
e Z X, W ) − g (∇
e Z F T X, W ) + g (QZ W, F X )
g ([Z, W ], X ) = − sin2 θg (∇
2
e W X, Z ) + g (∇
e W F T X, Z ) − g (QW Z, F X ).
+ sin θg (∇
Again using the given fact that PZ W ∈ Γ (D⊥ ), for any Z, W ∈ Γ (D⊥ ) and QU V ∈
Γ (µ), for any U, V ∈ Γ (T M ) and then by (2.3), we arrive at
g ([Z, W ], X ) = sin2 θg ([Z, W ], X ) + g (AF T X Z, W ) − g (AF T X W, Z ).
Since AF T X is symmetric and θ 6= π/2, then from above relation we get D⊥ is integrable. If we consider N⊥ be a leaf of D⊥ in M and h⊥ be the second fundamental
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Geometry of warped product pseudo-slant submanifolds
9
e Z W, X ). Using (2.1), we
form of N⊥ in M . Then g (h⊥ (Z, W ), X ) = g (∇Z W, X ) = g (∇
get
e Z W, JX )
g (h⊥ (Z, W ), X ) = g (J ∇
e Z JW, JX ) − g ((∇
e Z J )W, JX )
= g (∇
e Z JX ) − g (PZ W, T X ) − g (QZ W, F X ).
= −g (JW, ∇
Again, from the fact that PZ W ∈ Γ (D⊥ ), for any Z, W ∈ Γ (D⊥ ) and QU V ∈ Γ (µ),
for any U, V ∈ Γ (T M ), we have
e Z T X, JW ) + g (∇
e Z JF X, W ) − g ((∇
e Z J )F X, W )
g (h⊥ (Z, W ), X ) = −g (∇
e Z BF X, W ) + g (∇
e Z CF X, W )
= −g (h(Z, T X ), JW ) + g (∇
− g (PZ F X, W ).
Using property (p3 )(ii), (2.3) and (2.11), we obtain
e Z W, X ) − g (AF T X Z, W )
g (h⊥ (Z, W ), X ) = −g (AJW T X, Z ) + sin2 θg (∇
+ g (QZ W, F X ).
Thus, by the hypothesis of the theorem, we derive
cos2 θg (h⊥ (Z, W ), X ) = (Xλ) cos2 θg (Z, W ),
or equivalently h⊥ (Z, W ) = g (Z, W )∇λ, where ∇λ is gradient of the function λ, which
means that N⊥ is totally umbilical in M with the mean curvature H ⊥ = ∇λ. Also,
since W 0 λ = 0, for all W 0 ∈ Γ (D⊥ ), we can prove that H ⊥ is parallel corresponding
to the normal connection D# of N⊥ in M (see [17]). Thus, N⊥ is an extrinsic sphere
in M . Hence, by a result of Hiepko [10], we conclude that M is a warped product
submanifold. Thus, the proof is complete. t
u
4 Inequality for warped products Nθ ×f N⊥
In this section, we obtain a geometric relationship between the squared norm of the
second fundamental form of a warped product immersion in terms of the warping
function. First, we construct the following orthonormal frame for a warped product
pseudo-slant submanifold M = Nθ ×f N⊥ .
Let M = Nθ × f N⊥ be an m-dimensional warped product pseudo-slant submanifold
f such that Nθ and and N⊥ are 2pof a 2n-dimensional nearly Kaehler manifold M
f, respectively. Let
dimensional slant and q -dimensional totally real submanifolds of M
θ
⊥
us denote by D and D , the tangent bundles of Nθ and N⊥ , respectively and let
{e1 , . . . , ep , ep+1 = sec θT e1 , . . . , e2p = sec θT ep } and {e2p+1 = e∗1 , . . . , em = e2p+q =
e∗q } be the local orthonormal frames of Dθ and D⊥ , respectively.
Then, the orthonormal frames of F Dθ , JD⊥ and µ, respectively are
{em+1 = ẽ1 = csc θF e1 , . . ., em+p = ẽp = csc θF ep , em+p+1 = ẽp+1 = csc θ sec θF T e1 ,
. . ., em+2p = ẽ2p = csc θ sec θF T ep }, {em+2p+1 = Je∗1 , . . ., e2m = Je∗q } and {e2m+1 ,
. . ., e2n }. The dimensions of F Dθ , JD⊥ and µ, respectively are 2p, q and 2n − 2m.
935
10 Siraj Uddin, Falleh R. Al-Solamy, Khalid Ali Khan
Theorem 4.1 Let M=Nθ × f N⊥ be a mixed geodesic warped product pseudo-slant
f such that N⊥ and Nθ are totally real and
submanifold of a nearly Kaehler manifold M
f, respectively. Then
proper slant submanifolds of M
(i) The squared norm of the second fundamental form h of M satisfies
khk2 ≥ q cot2 θk∇ ln f k2 ,
(4.1)
where ∇ ln f is the gradient of ln f and q is the dimension of N⊥ .
(ii) If the equality holds in (4.1), then Nθ and N⊥ are totally geodesic and totally umf, respectively.
bilical submanifolds of M
Proof. By definition, we have
khk2 =
m
X
2n
m
X
X
g (h(ei , ej ), h(ei , ej )) =
g (h(ei , ej ), er )2 .
r=m+1 i,j =1
i,j =1
Now, decompose the above equation according to the frames of D⊥ and Dθ as follows
khk2 =
2p
2n
X
X
g (h(ei , ej ), er )2 + 2
r=m+1 i,j =1
2n
X
+
q
X
2p q
2n
X
X
X
g (h(ei , e∗j ), er )2
r=m+1 i=1 j =1
g (h(e∗i , e∗j ), er )2 .
r=m+1 i,j =1
Since M is mixed geodesic, thus the second term in the right hand side of above
equation is identically zero and then we break the above equation for the frames of
F Dθ , JD⊥ and µ as follows
khk2 =
+
2p 2p
X
X
g (h(ei , ej ), ẽr )2 +
r=1 i,j =1
2n
X
q X
2p
X
r=1 i,j =1
2p
X
g (h(ei , ej ), er )2 +
q X
q
X
q
2p
X
X
g (h(e∗i , e∗j ), ẽr )2
(4.2)
r=1 i,j =1
r=2m+1 i,j =1
+
g (h(ei , ej ), Je∗r )2
g (h(e∗i , e∗j ), Je∗r )2 +
r=1 i,j =1
2n
X
q
X
g (h(e∗i , e∗j ), er )2 .
r=2m+1 i,j =1
Then from Lemma 3.2, the second term of right hand side in (4.2) is zero for a mixed
geodesic warped product submanifold. Since we couldn’t find any relation in terms of
the warping function for the first, third, fifth and sixth terms in the right hand side of
(4.2). Then we shall leave these positive terms and only the fourth term is evaluated,
thus the above equality for the constructed frame change into the following inequality
p X
q
X
2
khk ≥
g (h(e∗i , e∗j ), csc θF er )2
r=1 i,j =1
+
p X
q
X
g (h(e∗i , e∗j ), csc θ sec θF T er )2 .
r=1 i,j =1
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Geometry of warped product pseudo-slant submanifolds
11
Using Lemma 3.3 for mixed geodesic warped products, we derive
2
2
khk ≥ csc θ
p X
q
X
(T er ln f )
2
g (e∗i , e∗j )2
2
+ cot θ
r=1 i,j =1
(er ln f )2 g (e∗i , e∗j )2
r=1 i,j =1
2p
= q csc2 θ
p X
q
X
2p
p
X
X
X
(T er ln f )2 −q csc2 θ
(T er ln f )2 +q cot2 θ
(er ln f )2
r=1
= q csc2 θkT ∇ ln f k2 − q csc2 θ
p
X
2
+ q cot θ
(er ln f )2
r=p+1
p
X
r=1
r=1
g (er+p , T ∇ ln f )2
r=1
2
2
2
2
= q cot θk∇ ln f k − q csc θ sec θ
p
X
+ q cot2 θ
(er ln f )2 .
p
X
r=1
g (T er , T ∇ ln f )2
r=1
Then from (2.9), we get khk2 ≥ q cot2 θk∇ ln f k2 , which is inequality (i). If the equality
holds in (4.1), then from the leaving first and third terms in (4.2), we conclude that
and
g (h(Dθ , Dθ ), F Dθ ) = 0 ⇒ h(Dθ , Dθ ) ⊂ JD⊥ ⊕ µ
(4.3)
g (h(Dθ , Dθ ), µ) = 0 ⇒ h(Dθ , Dθ ) ⊂ JD⊥ ⊕ F Dθ .
(4.4)
h(Dθ , Dθ ) ⊂ JD⊥ .
(4.5)
Then from (4.3) and (4.4), we have
But, from Lemma 3.2 for a mixed geodesic warped product pseudo-slant submanifold,
we have
h(Dθ , Dθ ) ⊥ JD⊥ .
(4.6)
Then, from (4.5) and (4.6), we get
h(Dθ , Dθ ) = 0.
(4.7)
Since Nθ is totally geodesic in M (see [3]), with this fact (4.7) implies that Nθ is
f. Similarly, from the leaving fifth and sixth terms, we conclude
totally geodesic in M
that
g (h(D⊥ , D⊥ ), JD⊥ ) = 0 ⇒ h(D⊥ , D⊥ ) ⊂ F Dθ ⊕ µ
(4.8)
and
g (h(D⊥ , D⊥ ), µ) = 0 ⇒ h(D⊥ , D⊥ ) ⊂ JD⊥ ⊕ F Dθ .
937
(4.9)
12 Siraj Uddin, Falleh R. Al-Solamy, Khalid Ali Khan
Then from (4.8) and (4.9), we derive
h(D⊥ , D⊥ ) ⊂ F Dθ .
(4.10)
Also, by Lemma 3.3, for a mixed geodesic warped product pseudo-slant submanifold,
we have
g (h(Z, W ), F T X ) = − cos2 θ(X ln f )g (Z, W ),
(4.11)
for any X ∈ Γ (Dθ ) and Z, W ∈ Γ (D⊥ ). Thus, N⊥ is totally umbilical submanifold of
f by using the fact that N⊥ is totally umbilical in M (see [3]) with (4.10) and (4.11).
M
This completes the proof of the theorem. t
u
Acknowledgements The authors are thankful to the anonymous referees for their useful suggestions
and constructive comments for improving the presentation of this paper.
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