WARPED PRODUCT BI-SLANT IMMERSIONS IN

Mediterranean Journal of Mathematics

WARPED PRODUCT BI-SLANT IMMERSIONS IN KAEHLER MANIFOLDS SIRAJ UDDIN, BANG-YEN CHEN, AND FALLEH R. AL-SOLAMY

f, g, J) is Abstract. A submanifold M of an almost Hermitian manifold (M called slant, if for each point p ∈ M and 0 6= X ∈ Tp M , the angle between JX and Tp M is constant (see [6]). Later, A. Carriazo [5] defined the notion of bi-slant immersions as an extension of slant immersions. In this paper we study warped product bi-slant submanifolds in Kaehler manifolds and provide some examples of warped product bi-slant submanifolds in some complex Euclidean spaces. Our main theorem states that every warped product bi-slant submanifold in a Kaehler manifold is either a Riemannian product of two slant submanifolds or a warped product hemi-slant submanifold.

1. Introduction R. L. Bishop and B. O’Neill defined warped product manifolds in [3] as follows: Let (M1 , g1 ) and (M2 , g2 ) be Riemannian manifolds, f a smooth positive function on M1 , and πi : M1 × M2 → Mi , i = 1, 2, the canonical projection maps on M1 × M2 . Then the warped product M = M1 ×f M2 is the product manifold M1 × M2 equipped with the Riemannian structure: (1.1)

g(X, Y ) = g1 (π1 ∗X, π1 ∗Y ) + (f ◦ π1 )2 g2 (π2 ∗X, π2 ∗Y )

for any X, Y ∈ T M , where ∗ is the symbol for the tangent maps. The function f is called the warping function. A warped product manifold is called trivial if its warping function is constant. In this special case, the warped product manifold is a Riemannian product. Let X(Mi ) denote the set of tangent vector fields on Mi for i = 1, 2. We denote by L(Mi ) the set of lifts of tangent vector fields on Mi to M1 × M2 . It was well-known in [3] that, for any X ∈ L(M1 ) and Z ∈ L(M2 ) on M1 ×f M2 , we have (1.2)

∇X Z = ∇Z X = (X ln f )Z

where ∇ denotes the Levi-Civita connection on M . It is also well-known that M1 is a totally geodesic submanifold and M2 is a totally umbilical submanifold of M = M1 × f M2 (cf. [3, 12]). Around the beginning of this century, B.-Y. Chen initiated the study of warped product CR-submanifolds of Kaehler manifolds in [8, 9]. He proved that there do 2000 Mathematics Subject Classification. 53C40, 53C42, 53C15. Key words and phrases. Warped product; slant submanifolds; bi-slant submanifolds; warped product bi-slant submanifolds; Kaehler manifolds. 1

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S. UDDIN, B.-Y. CHEN, AND F. R. AL-SOLAMY

not exist warped product CR-submanifolds of the form M⊥ ×f MT in a Kaehler ˜ , where MT is a complex submanifold and M⊥ is a totally real submanmanifold M ifold. On the other hand, he also proved that there do exist many warped products of the form MT ×f M⊥ , which he called CR-warped products. Furthermore, he established several fundamental results for CR-warped products in Kaehler manifolds, including optimal inequalities and characterizations for CR-warped products in Kaehler manifolds (cf. [8, 9, 10, 11]). Later, B. Sahin [16] proved that there do not exist warped product submanifolds ˜ such that MT of the form MT ×f Mθ (or Mθ ×f MT ) in a Kaehler manifold M is a complex submanifold and Mθ is a proper slant submanifold. Moreover, Sahin considered in [17] warped product hemi-slant submanifolds of a Kaehler manifold ˜ . He proved that there are no warped products of the form M⊥ ×f Mθ such M ˜; that M⊥ is a totally real submanifold and Mθ is a proper slant submanifold of M while warped product submanifolds of the form Mθ ×f M⊥ do exist. Furthermore, he proved several interesting results on Mθ ×f M⊥ . For recent detailed survey on warped product submanifolds, see [12, 13]. In this paper, we study warped product submanifolds of the form M1 ×f M2 in a Kaehler manifold, where M1 and M2 are slant submanifolds. We call such submanifolds warped product bi-slant submanifolds. The paper is organized as follows. In section 2, we recall some basic formulas and definitions. The explicit definition and examples of warped product bi-slant submanifolds are given section 3. A useful lemma is proved in section 4. The main result of this paper is proved in section 5 which states that every warped product bi-slant submanifold in a Kaehler manifold is either a Riemannian product of two slant submanifolds or a warped product hemi-slant submanifold. In the last section we provide two additional results for bi-slant submanifolds. 2. Preliminaries ˜ , J, g) be an almost Hermitian manifold with almost complex structure J Let (M and a Riemannian metric g such that (2.1)

J 2 = −I,

(2.2)

g(JX, JY ) = g(X, Y )

˜ ), where I is the identity map. for all X, Y ∈ X(M ˜ denotes the Levi-Civita connection on M ˜ . If the almost complex structure Let ∇ J satisfies (2.3)

˜ X J)Y = 0 (∇

˜ ), then M ˜ is called a Kaehler manifold. for X, Y ∈ X(M ˜ , J, g) is A (differentiable) distribution D defined on a submanifold M of (M called θ-slant if, for each unit vector X ∈ D, the angle θ(X) between JX and D is a constant (cf. [6, 7]). A θ-slant distribution is called holomorphic or complex if θ = 0; and it is called totally real if θ = π2 . A θ-slant distribution is called proper slant whenever θ 6= 0, π2 .

WARPED PRODUCT BI-SLANT IMMERSIONS IN KAEHLER MANIFOLDS

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˜ . Then M is called Let M be a Riemannian manifold isometrically immersed in M a complex submanifold if J(Tx M ) ⊆ Tx M for any x ∈ M , where Tx M is the tangent space of M at x. The submanifold M is called totally real if J(Tx M ) ⊆ Tx⊥ M for any x ∈ M , where Tx⊥ M denotes the normal space of M at x. Besides complex and totally real submanifolds, there are some other important classes of submanifolds determined by the behavior of tangent bundle of the sub˜. manifold under the action of the almost complex structure J of M ˜ (1) A submanifold M of M is called a CR-submanifold if there is a holomorphic distribution D on M whose orthogonal complementary distribution D⊥ is ˜ (cf. [2]). totally real in M ˜ is called a slant submanifold if, for each unit vector (2) A submanifold M of M X ∈ T 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 (cf. [6, 7, 12]). (3) A submanifold M is called semi-slant if there exists a pair of orthogonal distributions D and Dθ such that D is complex and Dθ is proper slant (cf. [15]). ˜ , the formulas of Gauss and For a submanifold M of a Riemannian manifold M Weingarten are given respectively by (2.4)

˜ X Y = ∇X Y + h(X, Y ), ∇

(2.5)

˜ X N = −AN X + ∇⊥ N ∇ X

for tangent vector fields X, Y and normal vector field N of M , where ∇ is the induced Levi-Civita connection on M , h is the second fundamental form, ∇⊥ is the normal connection on T ⊥ M , and AN is the shape operator. It is well-known that the shape operator and the second fundamental form are related by (cf. e.g. [12]): (2.6)

g(AN X, Y ) = g(h(X, Y ), N )

˜. where g denotes the induced metric on M as well as the Riemannian metric on M For a tangent vector field X and a normal vector field N of M , we put (2.7)

JX = P X + F X,

(2.8)

JN = BN + CN,

where P X and F X (respectively, BN and CN ) are the tangential and the normal components of JX (respectively, of JN ). ˜ , we have On a slant submanifold M of an almost Hermitian manifold M (2.9)

P 2 (X) = −(cos2 θ)X,

˜ (see [6, 7]). where θ is the slant angle of M in M As easy consequences of the relation (2.9), we have (2.10)

g(P X, P Y ) = (cos2 θ)g(X, Y ),

(2.11)

g(F X, F Y ) = (sin2 θ)g(X, Y ).

Also, for a slant submanifold, (2.7) and (2.9) yield (2.12)

BF X = −(sin2 θ)X and CF X = −F P X.

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S. UDDIN, B.-Y. CHEN, AND F. R. AL-SOLAMY

3. Bi-slant submanifolds The notion of bi-slant immersions was introduced by A. Carriazo as an extension of slant immersions in [5]. Bi-slant submanifolds in an almost contact metric manifold were studied in [4] by J. L. Cabrerizo et al. ˜ , J, g) is Definition 3.1. A submanifold M of an almost Hermitian manifold (M called bi-slant if there exists a pair of orthogonal distributions D1 and D2 of M such that (a) T M = D1 ⊕ D2 ; (b) JD1 ⊥ D2 and JD2 ⊥ D1 ; (c) The distributions D1 , D2 are slant with slant angle θ1 , θ2 , respectively. The pair {θ1 , θ2 } of slant angles is called the bi-slant angle. A bi-slant submanifold M is called proper if its bi-slant angle satisfies θ1 , θ2 6= 0, π2 . Notice that condition (b) in Definition 3.1 implies that P (Di ) ⊂ Di , i = 1, 2.

(3.1)

Remark 3.1. If we assume θ1 = 0 and θ2 = π2 in Definition 3.1, then M is a CRsubmanifold; and if θ1 = 0 and θ2 6= 0, π2 , then M is called semi-slant. Also, if we assume θ1 = π2 and θ2 6= 0, π2 , then M is called hemi-slant (cf. [17, 18]). Now, we provide an example of bi-slant submanifold in C4 . Example 3.1. Let E2n be the Euclidean 2n-space with the standard metric and let Cn denotes the complex Euclidean n-space (E2n , J) equipped with the canonical complex structure J defined by J(x1 , y1 , . . . , xn , yn ) = (y1 , −x1 , . . . , yn , −xn ). Thus we have  J

∂ ∂xi



∂ = , J ∂yi



∂ ∂yi

 =−

∂ , i = 1, . . . , n. ∂xi

For any pair of real numbers {θ1 , θ2 } satisfying 0 < θ1 , θ2 < the submanifold Mθ1 ,θ2 of C4 defined by

π 2,

let us consider

φθ1 ,θ2 (u, v, w, s) = (u, v cos θ1 , 0, v sin θ1 , w, s cos θ2 , 0, s sin θ2 ). If we put ∂ ∂ ∂ , e2 = cos θ1 + sin θ1 , ∂x1 ∂y1 ∂y2 ∂ ∂ ∂ e3 = , e4 = cos θ2 + sin θ2 , ∂x3 ∂y3 ∂y4 e1 =

then the restriction of {e1 , e2 , e3 , e4 } to M forms an orthonormal frame of the tangent bundle T M . Clearly, we have ∂ ∂ ∂ Je1 = , Je2 = − cos θ1 − sin θ1 , ∂y1 ∂x1 ∂x2 ∂ ∂ ∂ Je3 = , Je4 = − cos θ2 − sin θ2 . ∂y3 ∂x3 ∂x4

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Let us put D1 = Span{e1 , e2 } and D2 = Span{e3 , e4 }. Then obviously D1 and D2 satisfy conditions (a), (b) and (c) of Definition 3.1. Consequently, the submanifold Mθ1 ,θ2 defined by φθ1 ,θ2 is a proper bi-slant submanifold in C4 with {θ1 , θ2 } as its bi-slant angle. Analogous to CR-warped products introduced in [8], we define the notion of warped product bi-slant submanifolds as follows. Definition 3.2. A warped product M1 ×f M2 of two slant submanifolds M1 and ˜ , J, g) is called a warped product bi-slant M2 of an almost Hermitian manifold (M submanifold. A warped product bi-slant submanifold M1 ×f M2 is called proper if both M1 ˜ . Otherwise, M1 ×f M2 is called non-proper. and M2 are proper slant in M Next, we provide an example of warped product bi-slant submanifold of the form Mθ ×f M⊥ whose bi-slant angle satisfy θ1 6= 0, π2 and θ2 = π2 . Such warped product bi-slant submanifolds are called warped product hemi-slant submanifolds in [17]. Example 3.2. Consider a submanifold M of C3 defined by (3.2)

φ(u, v, w) = (v cos u, w cos u, v sin u, w sin u, −v + w, v + w).

It is easy to see that its tangent bundle T M of M is spanned by the following vectors ∂ ∂ ∂ ∂ v1 = −v sin u + v cos u − w sin u + w cos u , ∂x1 ∂x2 ∂y1 ∂y2 ∂ ∂ ∂ ∂ (3.3) v2 = cos u + sin u − + , ∂x1 ∂x2 ∂x3 ∂y3 ∂ ∂ ∂ ∂ v3 = + cos u + sin u + . ∂x3 ∂y1 ∂y2 ∂y3 Then we have ∂ ∂ ∂ ∂ + v cos u + w sin u − w cos u , Jv1 = −v sin u ∂y1 ∂y2 ∂x1 ∂x2 ∂ ∂ ∂ ∂ Jv2 = cos u + sin u − − , ∂y1 ∂y2 ∂y3 ∂x3 ∂ ∂ ∂ ∂ Jv3 = − cos u − sin u − . ∂y3 ∂x1 ∂x2 ∂x3 It is clear that Jv1 is orthogonal to T M . Thus D1 = Span{v1 } is a totally real distribution. Also, it is easy to
verify that D2 = Span{v2 , v3 } is a slant distribution with slant angle θ = cos−1 31 . Hence the submanifold M defined by φ given in (3.2) is a bi-slant submanifold whose bi-slant angle satisfies θ1 = π2 and θ2 6= 0, π2 . It is easy to verify that both distributions D1 and D2 are completely integrable. Let M⊥ and Mθ be integral manifolds of D1 and D2 , respectively. Then it follows from (3.2) that the metric tensor of M is given by (3.4)

g = gMθ + (v 2 + w2 )gM⊥ ,

where (3.5)

gMθ = 3(dv 2 + dw2 ) and gM ⊥ = du2

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S. UDDIN, B.-Y. CHEN, AND F. R. AL-SOLAMY

are the metric tensor of Mθ and M⊥ , respectively. Consequently, M = Mθ√ ×f M⊥ is a warped product bi-slant submanifold of C3 with warping function f = v 2 + w2 and whose bi-slant angle satisfies θ1 = π2 and θ2 6= 0, π2 . Hence M is a warped product hemi-slant submanifold. Remark 3.2. More examples of warped product hemi-slant submanifolds can be found in [17]. 4. A lemma for warped product bi-slant submanifolds To prove our main result, we need the following lemma for warped product bi-slant submanifolds. Lemma 4.1. Let M = M1 ×f M2 be a warped product bi-slant submanifold of a ˜ . Then we have: Kaehler manifold M (i) g(h(X1 , Y1 ), F X2 ) = g(h(X1 , X2 ), F Y1 ), (ii) g(h(X1 , X2 ), F Y2 ) = g(h(X1 , Y2 ), F X2 ) for any X1 , Y1 ∈ L(M1 ) and X2 , Y2 ∈ L(M2 ). Proof. For any X1 , Y1 ∈ L(M1 ) and X2 ∈ L(M2 ), we have ˜ X Y1 , F X2 ) g(h(X1 , Y1 ), F X2 ) = g(∇ 1 ˜ X Y1 , JX2 ) − g(∇ ˜ X Y1 , P X2 ). = g(∇ 1 1 Using (2.2), (2.3) and the orthogonality of vector fields given in condition (b) of Definition 3.1, we find ˜ X JY1 , X2 ) + g(∇X P X2 , Y1 ). g(h(X1 , Y1 ), F X2 ) = −g(∇ 1 1 Thus we derive from (1.2), (2.4), (2.7) and (3.1) that ˜ X P Y1 , X2 ) g(h(X1 , Y1 ), F X2 ) = (X1 ln f )g(P X2 , Y1 ) − g(∇ 1 ˜ X F Y1 , X2 ) − g(∇ 1 ˜ X F Y1 , X2 ). = − g(∇X1 P Y1 , X2 ) − g(∇ 1 Thus, by using (2.5) and (3.1), we arrive at g(h(X1 , Y1 ), F X2 ) = g(∇X1 X2 , P Y1 ) + g(AF Y1 X1 , X2 ) = −g(∇X1 P Y1 , X2 ) + g(h(X1 , X2 ), F Y1 ) = −(X1 ln f )g(P Y1 , X2 ) + g(h(X1 , X2 ), F Y1 ) = g(h(X1 , X2 ), F Y1 ), which proves the first part of the lemma. Similarly, we find from (1.2), (2.1)-(2.3) and (2.5)-(2.7) that ˜ X X2 , JY2 ) − g(∇X X2 , P Y2 ) g(h(X1 , X2 ), F Y2 ) = g(∇ 1 1 ˜ X JX2 , Y2 ) − (X1 ln f )g(X2 , P Y2 ) = −g(∇ 1 = g(AF X2 X1 , Y2 ) − (X1 ln f ){g(P X2 , Y2 ) + g(X2 , P Y2 )} = g(h(X1 , Y2 ), F X2 )

WARPED PRODUCT BI-SLANT IMMERSIONS IN KAEHLER MANIFOLDS

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for X1 ∈ L(M1 ) and X2 , Y2 ∈ L(M2 ). Therefore, we also obtain the second part of the lemma.  5. The main result It was proved in [8] that there do not exist warped product bi-slant submanifolds of the form M⊥ ×f MT in a Kaehler manifold whose bi-slant angle satisfies θ1 = π2 and θ2 = 0. On the other hand, it was also proved in [8] that there do exist many warped product bi-slant submanifolds MT ×f M⊥ with θ1 = 0 and θ2 = π2 . Also, B. Sahin proved in [16] that there do not exist warped product bi-slant submanifolds Mθ1 ×f Mθ2 in a Kaehler manifold with θ1 = 0 and θ2 6= 0, π2 or with θ1 6= 0, π2 and θ2 = 0. The main result of this article is the following. Theorem 5.1. Let M = Mθ1 ×f Mθ2 be a warped product bi-slant submanifold ˜ . Then one of the following with bi-slant angle {θ1 , θ2 } in a Kaehler manifold M two cases must occurs: (1) The warping function f is constant, i.e., M is a Riemannian product of two slant submanifolds; (2) θ2 = π2 , i.e., M is a warped product hemi-slant submanifold such that Mθ2 is a ˜. totally real submanifold M⊥ of M Proof. Let M = Mθ1 ×f Mθ2 be a warped product bi-slant submanifold of a Kaehler ˜ with bi-slant angle {θ1 , θ2 }. Then we have manifold M ˜ X X1 , JY2 ) − g(∇X X1 , P Y2 ), (5.1) g(h(X1 , X2 ), F Y2 ) = g(∇ 2

2

for any X1 ∈ L(M1 ) and X2 , Y2 ∈ L(M2 ). Thus, by applying (1.2) and (2.1)-(2.4), we may derive from (5.1) that ˜ X JX1 , Y2 ) − (X1 ln f )g(X2 , P Y2 ) g(h(X1 , X2 ), F Y2 ) = −g(∇ 2

(5.2)

˜ X P X1 , Y2 ) − g(∇ ˜ X F X1 , Y2 ) = −g(∇ 2 2 − (X1 ln f )g(X2 , P Y2 ).

Thus, it follows from (2.1), (2.5), (2.6), (3.1) and (5.2) that g(h(X1 , X2 ), F Y2 ) = −(P X1 ln f )g(X2 , Y2 ) + g(AF X1 X2 , Y2 ) (5.3)

− (X1 ln f )g(X2 , P Y2 ) = −(P X1 ln f )g(X2 , Y2 ) + g(h(X2 , Y2 ), F X1 ) − (X1 ln f )g(X2 , P Y2 ).

After interchanging X2 by Y2 in (5.3) and using (2.2), we get (5.4)

g(h(X1 , Y2 ), F X2 ) = −(P X1 ln f )g(X2 , Y2 ) + g(h(X2 , Y2 ), F X1 ) + (X1 ln f )g(X2 , P Y2 ).

By subtracting (5.3) from (5.4) and by applying Lemma 4.1(ii), we find (5.5)

(X1 ln f )g(X2 , P Y2 ) = 0.

Now, after interchanging Y2 by P Y2 in (5.5) and using (2.9), we obtain (cos2 θ2 )(X1 ln f )g(X2 , Y2 ) = 0.

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S. UDDIN, B.-Y. CHEN, AND F. R. AL-SOLAMY

Therefore, either f is a constant or cos θ2 = 0 holds. Consequently, either M is a Riemannian product manifold or we have θ2 = π2 . In the later case, M is warped product hemi-slant submanifold.  Remark 5.1. Example 3.1 and Example 3.2 show that both case (1) and case (2) of Theorem 5.1 do occur. Remark 5.2. Hemi-slant warped product of the form Mθ1 ×f M⊥ in a Kaehler manifold have been studied in [17]. 6. Two additional results for bi-slant submanifolds For a proper bi-slant submanifold M of a Kaehler manifold, the normal bundle of M is decomposed as T ⊥ M = F D1 ⊕ F D2 ⊕ ν

(6.1)

where ν is a J-invariant normal subbundle of T ⊥ M . In the following, let Γ(Di ) denotes the space of smooth sections of the distribution Di (i = 1 or 2). For proper bi-slant submanifolds we have the following results. Proposition 6.1. Let M be a proper bi-slant submanifold of a Kaehler manifold ˜ . Then we have: M  g(∇X1 Y1 , P X2 ) = csc2 θ1 g(AF Y1 X2 , X1 ) + g(AF P Y1 P X2 , X1 ) (6.2) + g(∇⊥ X1 F X2 , F P Y1 ) − g(AF X2 X1 , Y1 ) for any X1 , Y1 ∈ Γ(D1 ) and X2 ∈ Γ(D2 ), where θ1 and θ2 are the slant angles of slant distributions D1 and D2 , respectively. Proof. For any X1 , Y1 ∈ Γ(D1 ) and X2 ∈ Γ(D2 ), we have ˜ X Y1 , P X2 ) g(∇X1 Y1 , P X2 ) = g(∇ 1 ˜ X Y1 , JX2 ) − g(∇ ˜ X Y1 , F X2 ). = g(∇ 1 1 Using (2.1), (2.2), (2.4) and (2.7) we get ˜ X Y1 , X2 ) − g(h(X1 , Y1 ), F X2 ). g(∇X Y1 , P X2 ) = −g(J ∇ 1

1

˜ = 0, (2.6) and (2.7), we obtain Then by applying ∇J ˜ X JY1 , X2 ) − g(AF X X1 , Y1 ) g(∇X Y1 , P X2 ) = −g(∇ 1

1

2

˜ X P Y1 , X2 ) − g(∇ ˜ X F Y1 , X2 ) − g(AF X X1 , Y1 ). = −g(∇ 1 1 2 Again using (2.1), (2.2), (2.4) and (2.7), we find ˜ X P Y1 , JX2 ) + g(AF Y X1 , X2 ) − g(AF X X1 , Y1 ). g(∇X1 Y1 , P X2 ) = −g(J ∇ 1 1 2 Since the shape operator A is self-adjoint, it follows from (2.3), (2.6) and (2.7) that ˜ X JP Y1 , P X2 ) − g(∇ ˜ X JP Y1 , F X2 ) g(∇X Y1 , P X2 ) = −g(∇ 1

1

1

+ g(AF Y1 X2 , X1 ) − g(AF X2 X1 , Y1 ) (6.3)

˜ X P 2 Y1 , P X2 ) − g(∇ ˜ X F P Y1 , P X2 ) = −g(∇ 1 1 ˜ X P 2 Y1 , F X2 ) − g(∇ ˜ X F P Y1 , F X2 ) − g(∇ 1 1 + g(AF Y1 X2 , X1 ) − g(AF X2 X1 , Y1 ).

WARPED PRODUCT BI-SLANT IMMERSIONS IN KAEHLER MANIFOLDS

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Since Y1 ∈ Γ(D1 ) and X2 ∈ Γ(D2 ), condition (b) in Definition 3.1 implies that g(P Y1 , X2 ) = 0. Hence we find (6.4)

0 = g(JP Y1 , JX2 ) = g(P 2 Y1 , P X2 ) + g(F P Y1 , F X2 ) = −(cos2 θ1 )g(Y1 , P X2 ) + g(F P Y1 , F X2 ) = g(F P Y1 , F X2 ).

Therefore, we find from (2.5), (2.9) and (6.3) that g(∇X1 Y1 , P X2 ) = (cos2 θ1 )g(∇X1 Y1 , P X2 ) + g(AF P Y1 X1 , P X2 ) (6.5)

˜ X Y1 , F X2 ) + g(F P Y1 , ∇ ˜ X F X2 ) + (cos2 θ1 )g(∇ 1 1 + g(AF Y1 X2 , X1 ) − g(AF X2 X1 , Y1 ).

Now, by applying using (2.3), (2.6), (6.4) and (6.5), we obtain (sin2 θ1 ) g(∇X1 Y1 , P X2 ) = g(AF P Y1 P X2 , X1 ) + (cos2 θ1 ) g(AF X2 X1 , Y1 ) + g(∇⊥ X1 F X2 , F P Y1 ) + g(AF Y1 X2 , X1 ) − g(AF X2 X1 , Y1 ). Now, the desired result follows from the above relation.



Proposition 6.2. Let M be a proper bi-slant submanifold of a Kaehler manifold ˜ whose slant distributions D1 and D2 have slant angles θ1 and θ2 , respectively. M Then we have:  g([X2 , Y2 ], P X1 ) = csc2 θ2 g(AF Y2 X2 , X1 ) − g(AF X2 Y2 , X1 ) (6.6)

+ g(AF P Y2 X2 , P X1 ) − g(AF P X2 Y2 , P X1 ) ⊥ − g(∇⊥ X2 F P Y2 , F X1 ) + g(∇Y2 F P X2 , F X1 )



for any X1 ∈ Γ(D1 ) and X2 , Y2 ∈ Γ(D2 ). Proof. In a similar way as Proposition 6.1, one can derive that  g(∇X2 Y2 , P X1 ) = csc2 θ2 g(AF Y2 X1 , X2 ) + g(AF P Y2 P X1 , X2 ) (6.7) + g(∇⊥ X2 F X1 , F P Y2 ) − g(AF X1 X2 , Y2 ) for X1 ∈ Γ(D1 ) and X2 , Y2 ∈ Γ(D2 ). Interchanging X2 and Y2 in (6.7) yields  g(∇Y2 X2 , P X1 ) = csc2 θ2 g(AF X2 X1 , Y2 ) + g(AF P X2 P X1 , Y2 ) (6.8) + g(∇⊥ Y2 F X1 , F P X2 ) − g(AF X1 Y2 , X2 ). Then after using the symmetry of the shape operator, (6.7) and (6.8), we obtain (6.6). This completes the proof of the lemma.  Acknowledgements The authors thank the reviewers for their valuable suggestions to improve the presentation of this paper.

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S. UDDIN, B.-Y. CHEN, AND F. R. AL-SOLAMY

References [1] Al-Solamy, F.R., Khan, V.A., Uddin, S.: Geometry of warped product semi-slant submanifolds of nearly Kaehler manifolds. Results. Math. (2016). doi:10.1007/s00025-016-0581-4. [2] Bejancu, A.: Geometry of CR-Submanifolds. D. Reidel, Dordrecht, (1986) [3] Bishop, R.L., O’Neill, B.: Manifolds of negative curvature. Trans. Amer. Math. Soc. 145), 1–49 (1969) [4] Cabrerizo, J.L., Carriazo, A., Fernandez, L.M., Fernandez, M.: Semi-slant submanifolds of a Sasakian manifold. Geom. Dedicata 78, no. 2, 183–199 (1999) [5] Carriazo, A.: Bi-slant immersions. in: Proc. ICRAMS 2000, Kharagpur, India, 88–97 (2000) [6] Chen, B.-Y.: Slant immersions. Bull. Austral. Math. Soc. 41, no. 1, 135-147 (1990) [7] Chen, B.-Y.: Geometry of slant submanifolds. Katholieke Universiteit Leuven, Belgium (1990) [8] Chen, B.-Y.: Geometry of warped product CR-submanifolds in Kaehler manifolds. Monatsh. Math. 133, no. 3, 177–195 (2001) [9] Chen, B.-Y.: Geometry of warped product CR-submanifolds in Kaehler manifolds II. Monatsh. Math. 134, no. 2, 103–119 (2001) [10] Chen, B.-Y.: Another general inequality for CR-warped products in complex space forms. Hokkaido Math. J. 32 , no. 2, 415–444 (2003) [11] Chen, B.-Y.: CR-warped products in complex projective spaces with compact holomorphic factor. Monatsh. Math. 141 (2004), no. 3, 177–186. [12] Chen, B.-Y.: Pseudo-Riemannian geometry, δ-invariants and applications. World Scientific, Hackensack, NJ (2011) [13] Chen, B.-Y.: Geometry of warped product submanifolds: A survey. J. Adv. Math. Stud. 6, no. 2, 1–43 (2013) [14] Hiepko, S.: Eine inner kennzeichungder verzerrten produkte. Math. Ann. 241, no. 3, 209–215 (1979) [15] Papaghiuc, N.: Semi-slant submanifolds of a Kaehlerian manifold. An. S ¸ tiint¸. Univ. Al. I. Cuza Ia¸si Sect¸. I a Mat. 40, no. 1, 55–61 (1994) [16] Sahin, B.: Nonexistence of warped product semi-slant submanifolds of Kaehler manifolds. Geom. Dedicata 117, 195–202 (2006) [17] Sahin, B.: Warped product submanifolds of Kaehler manifolds with a slant factor. Ann. Polon. Math. 95, no. 3, 207–226 (2009) [18] Uddin, S., Chi, A.Y.M.: Warped product pseudo-slant submanifolds of nearly Kaehler manifolds. An. S ¸ tiint¸. Univ. “Ovidius” Constant¸a Ser. Mat. 19, no. 3, 195–204 (2011) [19] Uddin, S., Al-Solamy, F.R., Khan, K.A.: Geometry of warped product pseudo-slant submanifolds of nearly Kaehler manifolds. An. S ¸ tiint¸. Univ. Al. I. Cuza Ia¸si Sect¸. I a Mat. (N.S.), 62, no. 3, 223–234 (2016) S. Uddin: Department of Mathematics, Faculty of Science, King Abdulaziz University, 21589 Jeddah, Saudi Arabia E-mail address: [email protected] B.-Y. Chen: Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, Michigan 48824–1027, U.S.A. E-mail address: [email protected] F. R. Al-Solamy: Department of Mathematics, Faculty of Science, King Abdulaziz University, 21589 Jeddah, Saudi Arabia E-mail address: [email protected]