Bi-warped product submanifolds of nearly Kaehler manifolds
Siraj Uddin, Bang-Yen Chen, Awatif AL-Jedani, Azeb Alghanemi
aa r X i v : . [ m a t h . DG ] F e b BI-WARPED PRODUCT SUBMANIFOLDS OF NEARLYKAEHLER MANIFOLDS
SIRAJ UDDIN, BANG-YEN CHEN, AWATIF AL-JEDANI, AND AZEB ALGHANEMI
Abstract.
We study bi-warped product submanifolds of nearly Kaehler man-ifolds which are the natural extension of warped products. We prove that everybi-warped product submanifold of the form M = M T × f M ⊥ × f M θ in a nearlyKaehler manifold satisfies the following sharp inequality: k h k ≥ p k∇ (ln f ) k + 4 q (cid:18) θ (cid:19) k∇ (ln f ) k , where p = dim M ⊥ , q = dim M θ , and f , f are smooth positive functionson M T . We also investigate the equality case of this inequality. Further, someapplications of this inequality are also given. Introduction
Bi-warped product manifolds are natural extensions of (ordinary) warped prod-uct and Riemannian product manifolds. Let M , M and M be Riemannian man-ifolds and M = M × M × M be the Cartesian product of M , M and M .For each i = 0 , ,
2, we denote by π i : M → M i the canonical projection of M onto M i . For each π i : M → M i , let π i ∗ denote the corresponding tangent map π i ∗ : T M → T M i . Denote by Γ( T M ) the Lie algebra of vector fields of M .If f , f are positive real valued functions on M , then g ( X, Y ) = g ( π ∗ X, π ∗ Y ) + ( f ◦ π ) g ( π ∗ X, π ∗ Y ) + ( f ◦ π ) g ( π ∗ X, π ∗ Y ) ,X, Y ∈ Γ( T M ) , defines a Riemannian metric on M × M × M , called a bi-warped product metric .The product manifold M = M × M × M endowed with this warped productmetric g , denoted by M × f M × f M , is called a bi-warped product manifold .The functions f , f are called the warping functions . Obviously, if f , f areboth constant, M is simply a Riemannian product; and if exactly one of f , f isconstant, then M is an (ordinary) warped product manifold. Further, if none of f , f is constant, then M is called a proper bi-warped product manifold .Let M = M × f M × f M be a bi-warped product submanifold. We put D = T M T , D ⊥ = T M ⊥ , D θ = T M θ , N = f M × f M . Mathematics Subject Classification.
Key words and phrases.
Warped product; bi-warped product; slant submanifold; totally realsubmanifold; nearly Kaehler manifold; semi-slant warped product submanifold.
Then we have (cf. [12] and [22]) ∇ X Z = X i =1 ( X (ln f i )) Z i , (1.1)for X ∈ D and Z ∈ Γ( T N ), where ∇ is the Levi-Civita connection on M and Z i (i=1,2) is the M i -component of Z .Nearly Kaehler manifolds, also known as almost Tachibana manifolds, were firststudied in 1959 by S. Tachibana [19] and then in 1970 by A. Gray [15]. Obviously,Kaehler manifolds are nearly Kaehler, but the converse is not true. Non-Kaehleriannearly Kaehler manifolds are called strict nearly Kaehler manifolds .The best known example of a strict nearly Kaehler manifold is the unit 6-sphere S . More general examples are homogeneous spaces G/K , where G is a compactsemisimple Lie group and K is the fixed point set of an automorphism of G oforder 3 (cf. [24]). In 1985, T. Friedrich and R. Grunewald proved in [14] that aRiemannian 6-manifold is nearly Kaehler if and only if admits a Riemannian Killingspinor. After then, strict nearly Kaehler manifolds obtained a lot of attentions dueto their relation to Killing spinors.The notion of warped products plays very important roles not only in geometrybut also in mathematical physics, especially in general relativity. The term of“warped product” was introduced by R. L. Bishop and B. O’Neill in [2], who usedit to construct a large class of complete manifolds of negative curvature. Inspiredby Bishop and O’Neill’s article, many important works on warped products from intrinsic point of view were done during the last fifty years.On the other hand, the study of warped product submanifolds from extrinsic point of review was initiated around the beginning of this century in [5, 6, 7]. Sincethen warped product submanifolds have became an active research subject (see,e.g., [9, 10, 11, 12, 13, 17, 18, 21]). For instance, B. Sahin studied in [18] warpedproduct pointwise semi-slant submanifolds in Kaehler manifolds. H. M. Tastan [20]extended this study to bi-warped product submanifolds in Kaehler manifolds byconsidering that one of the fiber of warped product is a pointwise slant submanifold.In this article, we study bi-warped product submanifolds in nearly Kaehler man-ifolds. In section 2, we give basic definitions and formulas. In section 3, we provesome useful results for the proof of our main result. In section 4, we prove a sharpinequality for bi-warped product submanifolds in nearly Kaehler manifolds. Wealso discuss the equality case of the inequality. In the last section, we provide someapplications of our main result. 2. Preliminaries
An even-dimensional differentiable manifold N K with Riemannian metric g andalmost complex structure J is called a nearly Kaehler manifold if (cf. [8, 15])(2.1) g ( JX, JY ) = g ( X, Y ) , ( ˜ ∇ X J ) Y + ( ˜ ∇ Y J ) X = 0 , for any vector fields X, Y ∈ Γ( T N K ). I-WARPED PRODUCT SUBMANIFOLDS OF NEARLY KAEHLER MANIFOLDS 3
Let M be a submanifold of a Riemannian manifold ˜ M with induced metric g .Let Γ( T ⊥ M ) denote the set of all vector fields normal to M . Then the Gauss andWeingarten formulas are given respectively by (see, for instance, [5, 10])˜ ∇ X Y = ∇ X Y + h ( X, Y ) , (2.2) ˜ ∇ X ξ = − A ξ X + ∇ ⊥ X ξ, (2.3)for vector fields X, Y ∈ Γ( T M ) and ξ ∈ Γ( T ⊥ M ), where ∇ and ∇ ⊥ denote theinduced connections on the tangent and normal bundles of M , respectively, and h is the second fundamental form, A is the shape operator of the submanifold. Thesecond fundamental form h and the shape operator A are related by g ( h ( X, Y ) , N ) = g ( A N X, Y ) . (2.4)For an n -dimensional submanifold M of an almost Hermitian 2 m -manifold ˜ M ,we choose a local orthonormal frame field { e , · · · , e n , e n +1 , · · · , e m } such that,restricted to M , e , · · · , e n are tangent to M and e n +1 , · · · , e m are normal to M .Let { h rij } , ≤ i, j ≤ n ; n + 1 ≤ r ≤ m, denote the coefficients of the secondfundamental form h with respect to the local frame field. Then, we have(2.5) h rij = g ( h ( e i , e j ) , e r ) = g ( A e r e i , e j ) , k h k = n X i,j =1 g ( h ( e i , e j ) , h ( e i , e j )) . For any X ∈ Γ( T M ), we put JX = T X + F X, (2.6)where
T X and
F X are the tangential and normal components of JX , respectively.A submanifold M of an almost Hermitian manifold ˜ M is said to be holomorphic (resp. totally real ) if J ( T p M ) = T p M (resp. J ( T p M ) ⊆ T p ⊥ M ) ∀ p ∈ M .There are other important classes of submanifolds determined by the behaviourof almost complex structure J acting on the tangent space of M : For a nonzerovector X ∈ T p M , p ∈ M , the angle θ ( X ) between JX and T p M is called theWirtinger angle of X . A submanifold M is said to be slant (cf. [3, 4]) if theWirtinger angle θ ( X ) is constant on M , i.e., it is independent of the choice of X ∈ T p M and p ∈ M . In this case, θ is called the slant angle of M . Holomorphicand totally real submanifolds are slant submanifolds with slant angles 0 and π ,respectively. A slant submanifold is called proper slant if it is neither holomorphicnor totally real. More generally, a distribution D on M is called a slant distribution if the angle θ ( X ) between JX and D p is independent of the choice of p ∈ M andof 0 = X ∈ D p .It is well-known from [3] that a submanifold M of an almost Hermitian manifold˜ M is slant if and only if we have T X = − (cos θ ) X, X ∈ Γ( T M ) . (2.7)From (2.7) we have the following. g ( T X, T Y ) = (cos θ ) g ( X, Y ) , (2.8) SIRAJ UDDIN, BANG-YEN CHEN, AWATIF AL-JEDANI, AND AZEB ALGHANEMI g ( F X, F Y ) = (sin θ ) g ( X, Y ) , (2.9)for any vector fields X, Y tangent to M .3. Bi-warped product submanifolds
Now, we study bi-warped product submanifolds in a nearly Kaehler manifold ˜ M which are of the form M = M T × f M ⊥ × f M θ , where M T , M ⊥ , M θ are holomorphic,totally real and proper slant submanifolds of ˜ M , respectively. If we put D = T M T , D ⊥ = T M ⊥ , D θ = T M θ , then the tangent and normal bundles of M are decomposed as T M = D ⊕ D ⊥ ⊕ D θ , T ⊥ M = J D ⊥ ⊕ F D θ ⊕ µ where µ is an j -invariant normal subbundle of the normal bundle T ⊥ M . Fromnow one, we use the following conventions: X , Y , . . . are vector fields in Γ( D ) and X , Y , . . . are vector fields in Γ( D θ ), while Z, W, . . . are vector fields in Γ( D ⊥ ).We present the following useful results for later use. Lemma 3.1.
Let M = M T × f M ⊥ × f M θ be a bi-warped product submanifold ofa nearly Kaehler manifold ˜ M . Then we have (i) g ( h ( X , Y ) , JZ ) = 0 , (ii) g ( h ( X , Y ) , F X ) = 0 , (iii) g ( h ( X , Z ) , JW ) = − JX (ln f ) g ( Z, W ) , for any X , Y ∈ Γ( D ) , Z, W ∈ Γ( D ⊥ ) and X ∈ Γ( D θ ) .Proof. For any X , Y ∈ Γ( D ) and Z ∈ Γ( D ⊥ ), we have g ( h ( X , Y ) , JZ ) = g ( ˜ ∇ X Y , JZ ) = g (( ˜ ∇ X J ) Y , Z ) − g ( ˜ ∇ X JY , Z ) . Using (1.1), we find g ( h ( X , Y ) , JZ ) = g (( ˜ ∇ X J ) Y , Z ) + X (ln f ) g ( JY , Z ) . By the orthogonality of vector fields, we have g ( h ( X , Y ) , JZ ) = g (( ˜ ∇ X J ) Y , Z ) . (3.1)Interchanging X by Y in (3.1), we get g ( h ( X , Y ) , JZ ) = g (( ˜ ∇ Y J ) X , Z ) . (3.2)Then, first part follows from (3.1) and (3.2) by using (2.1). In a similar fashion, wecan prove (ii). For the third part, we have g ( h ( X , Z ) , JW ) = g ( ˜ ∇ Z X , JW ) = g (( ˜ ∇ Z J ) X , W ) − g ( ˜ ∇ Z JX , W ) , for any X ∈ Γ( D ) and Z, W ∈ Γ( D ⊥ ). Again, using (1.1) and (2.1), we derive g ( h ( X , Z ) , JW ) = − g (( ˜ ∇ X J ) Z, W ) − JX (ln f ) g ( Z, W )= − g ( ˜ ∇ X JZ, W ) + g ( J ˜ ∇ X Z, W ) − JX (ln f ) g ( Z, W ) . I-WARPED PRODUCT SUBMANIFOLDS OF NEARLY KAEHLER MANIFOLDS 5
Using (2.1), (2.2), (2.3) and (2.4), we get2 g ( h ( X , Z ) , JW ) = g ( h ( X , W ) , JZ ) − JX (ln f ) g ( Z, W ) . (3.3)Interchanging Z by W in (3.3), we obtain2 g ( h ( X , W ) , JZ ) = g ( h ( X , Z ) , JW ) − JX (ln f ) g ( Z, W ) . (3.4)Hence, the third part follows from (3.3) and (3.4), which proves the lemma. (cid:3) A bi-warped product submanifold M = M T × f M ⊥ × f M θ in a nearly Kaehlermanifold ˜ M is said to be D ⊕ D ⊥ –mixed totally geodesic (resp., D ⊕ D θ –mixedtotally geodesic ) if its second fundamental h satisfies h ( X , Z ) = 0 ∀ X ∈ Γ( D ) , ∀ Z ∈ Γ( D ⊥ )( resp., h ( X , X ) = 0 ∀ X ∈ Γ( D ) , ∀ X ∈ Γ( D θ )) . Lemma 3.2.
Let M = M T × f M ⊥ × f M θ be a bi-warped product submanifold ofa nearly Kaehler manifold ˜ M . Then we have (i) g ( h ( X , Z ) , F X ) = g ( h ( X , X ) , JZ ) = 0 , (ii) g ( h ( X , X ) , F Y ) = X (ln f ) g ( T X , Y ) − JX (ln f ) g ( X , Y ) , for any X ∈ Γ( D ) , Z ∈ Γ( D ⊥ ) and X , Y ∈ Γ( D θ ) .Proof. For any X ∈ Γ( D ) , Z ∈ Γ( D ⊥ ), and X ∈ Γ( D θ ), we have g ( h ( X , Z ) , F X ) = g ( ˜ ∇ Z X , JX − T X )= g (( ˜ ∇ Z J ) X , X ) − g ( ˜ ∇ Z JX , X ) − g ( ˜ ∇ Z X , T X ) . Using (2.1), (1.1) and the orthogonality of vector fields, we derive g ( h ( X , Z ) , F X ) = − g (( ˜ ∇ X J ) Z, X ) = − g ( ˜ ∇ X JZ, X ) + g ( J ˜ ∇ X Z, X ) . Then, from (2.1)-(2.4), we obtain g ( h ( X , Z ) , F X ) = 12 g ( h ( X , X ) , JZ ) , (3.5)Which is the first equality of (i).On the other hand, we have g ( h ( X , X ) , JZ ) = g ( ˜ ∇ X X , JZ ) = g (( ˜ ∇ X J ) X , Z ) − g ( ˜ ∇ X JX , Z ) . Using (2.1), (1.1) and the orthogonality of vector fields, we find g ( h ( X , X ) , JZ ) = − g (( ˜ ∇ X J ) X , Z ) = − g ( ˜ ∇ X JX , Z ) + g ( J ˜ ∇ X X , Z ) . Then it follows from (2.1) and (2.6) that g ( h ( X , X ) , JZ ) = − g ( ˜ ∇ X T X , Z ) − g ( ˜ ∇ X F X , Z ) − g ( ˜ ∇ X X , F Z ) . Again, using (1.1), (2.2)-(2.4) and the orthogonality of vector fields, we obtain g ( h ( X , X ) , JZ ) = 12 g ( h ( X , Z ) , F X ) . (3.6) SIRAJ UDDIN, BANG-YEN CHEN, AWATIF AL-JEDANI, AND AZEB ALGHANEMI
Hence, the second equality of (i) follows from (3.5) and (3.6). For, the second partof the lemma, we have g ( h ( X , X ) , F Y ) = g ( ˜ ∇ X X , JY − T Y )= g (( ˜ ∇ X J ) X , Y ) − g ( ˜ ∇ X JX , Y ) − g ( ˜ ∇ X X , T Y )= − g (( ˜ ∇ X J ) X , Y ) − JX (ln f ) g ( X , Y ) − X (ln f ) g ( X , T Y )= − g ( ˜ ∇ X T X , Y ) − g ( ˜ ∇ X F X , Y ) − g ( ˜ ∇ X X , T Y ) − g ( ˜ ∇ X X , F Y ) − JX (ln f ) g ( X , Y ) − X (ln f ) g ( X , T Y ) . Using (2.2)-(2.4) and (1.1), we find(3.7) 2 g ( h ( X , X ) , F Y ) = g ( h ( X , Y ) , F X ) − X (ln f ) g ( X , T Y ) − JX (ln f ) g ( X , Y ) . Interchanging X by Y in (3.7), we get(3.8) 2 g ( h ( X , Y ) , F X ) = g ( h ( X , X ) , F Y ) + X (ln f ) g ( X , T Y ) − JX (ln f ) g ( X , Y ) . The second part follows from (3.7) and (3.8). Hence the proof is complete. (cid:3)
The following relations are easily obtained by interchanging X by JX and X and Y by T X and T Y , respectively. g ( h ( X , X ) , F T Y ) = 13 X (ln f ) cos θ g ( X , Y ) − JX (ln f ) g ( X , T Y ) , (3.9) g ( h ( JX , X ) , F T Y ) = 13 JX (ln f ) cos θ g ( X , Y )+ X (ln f ) g ( X , T Y ) , (3.10) g ( h ( X , T X ) , F Y ) = − X (ln f ) cos θ g ( X , Y ) − JX (ln f ) g ( T X , Y ) , (3.11) g ( h ( JX , T X ) , F Y ) = − JX (ln f ) cos θ g ( X , Y )+ X (ln f ) g ( T X , Y ) , (3.12) g ( h ( X , T X ) , F T Y ) = − X (ln f ) cos θ g ( X , T Y ) − JX (ln f ) cos θg ( X , Y ) . (3.13)From Lemma 3.1(iii) we obtain immediately the following. Theorem 3.1.
Let M = M T × f M ⊥ × f M θ be a bi-warped product submanifoldof a nearly Kaehler manifold ˜ M . If M is D ⊕ D ⊥ –mixed totally geodesic, then f is constant, and hence M is an ordianary warped product manifold. Similarly, from Lemma 3.2 (ii), we may obtain the following.
I-WARPED PRODUCT SUBMANIFOLDS OF NEARLY KAEHLER MANIFOLDS 7
Theorem 3.2.
Let M = M T × f M ⊥ × f M θ be a proper bi-warped product sub-manifold of a nearly Kaehler manifold ˜ M . If M is D ⊕ D θ –mixed totally geodesic,then f is constant on M .Proof. From Lemma 3.2 (ii) and (3.10), we have(3.14) (cid:0) cos θ − (cid:1) JX (ln f ) g ( X , Y )= 9 g ( h ( X , X ) , F Y ) + 3 g ( h ( JX , X ) , F T Y ) . If M is D ⊕ D θ –mixed totally geodesic, then we find from (3.14) that(cos θ − JX (ln f ) = 0 , which implies that either cos θ = ±
3, which is not possible or JX (ln f ) = 0, i.e., f is constant. This completes the proof. (cid:3) Remark . Theorems 3.1 and 3.2 imply that a proper bi-warped product subman-ifold M = M T × f M ⊥ × f M θ in a nearly Kaehler manifold is neither D ⊕ D ⊥ –mixedtotally geodesic nor D ⊕ D θ –mixed totally geodesic.4. Inequality for the second fundamental form
Let M = M T × f M ⊥ × f M θ be an n -dimensional proper bi-warped productsubmanifold of a nearly Kaehler manifold ˜ M m . We consider a local orthonormalframe field { e , . . . , e n } of T M such that D = Span { e , · · · , e t , e t +1 = Je , · · · , e t = Je t } , D ⊥ = Span { e t +1 = ˆ e , · · · , e t + p = ˆ e p } , D θ = Span { e t + p +1 = e ∗ , · · · , e t + p + q = e ∗ t ,e t + p + q +1 = sec θe ∗ , · · · , e n = sec θe ∗ q } . Then dim M T = 2 t, dim M ⊥ = p and dim M θ = 2 q . Moreover, the orthonormalframe fields E , . . . , E m − n − p − q of the normal subbundle T ⊥ M are given by J D ⊥ = Span { E = J ˆ e , · · · , E p = J ˆ e p } ,F D θ = Span { E p +1 = csc θF e ∗ , · · · , E p + q = csc θF e ∗ p ,E p + q +1 = csc θ sec θF T e ∗ , · · · , E p +2 q = csc θ sec θF T e ∗ q } ,µ = Span { E p +2 q +1 , · · · , E m − n − p − q } . The main result of this article is the following sharp inequality for bi-warpedproduct submanifolds in a nearly Kaehler manifold.
Theorem 4.1.
Let M = M T × f M ⊥ × f M θ be a bi-warped product submanifoldof a nearly Kaehler manifold ˜ M , where M T , M ⊥ and M θ are holomorphic, totallyreal and proper slant submanifolds of ˜ M , respectively. Then we have: (i) The second fundamental form h and the warping functions f , f satisfy k h k ≥ p k∇ (ln f ) k + 4 q (cid:18) θ (cid:19) k∇ (ln f ) k (4.1) where p = dim M ⊥ , q = dim M θ and ∇ (ln f i ) is the gradient of ln f i . SIRAJ UDDIN, BANG-YEN CHEN, AWATIF AL-JEDANI, AND AZEB ALGHANEMI (ii)
If the equality sign in (4.1) holds identically, then M T is totally geodesicin ˜ M , and M ⊥ , M θ are totally umbilical in ˜ M . Moreover, M is neither D ⊕ D ⊥ –mixed totally geodesic nor D ⊕ D θ –mixed totally geodesic in ˜ M .Proof. From the definition of h , we have k h k = n X i,j =1 g ( h ( e i , e j ) , h ( e i , e j )) = m − n − p − q X r =1 n X i,j =1 g ( h ( e i , e j ) , E r ) . Then we decompose the above relation for the normal subbundles as follows(4.2) k h k = p X r =1 n X i,j =1 g ( h ( e i , e j ) , J ˆ e r ) + p +2 q X r = p +1 n X i,j =1 g ( h ( e i , e j ) , E r )+ m − n − p − q X r = p +2 q +1 n X i,j =1 g ( h ( e i , e j ) , E r ) . Leaving the last µ -components term in (4.2) and using the frame fields of tangentand normal subbundles of M , we derive k h k ≥ p X r =1 2 t X i,j =1 g ( h ( e i , e j ) , J ˆ e r ) + 2 p X r =1 2 t X i =1 p X j =1 g ( h ( e i , ˆ e j ) , J ˆ e r )+ p X r =1 p X i,j =1 g ( h (ˆ e i , ˆ e j ) , J ˆ e r ) + 2 p X r =1 2 t X i =1 2 q X j =1 g ( h ( e i , e ∗ j ) , J ˆ e r )+ p X r =1 2 q X i,j =1 g ( h ( e ∗ i , e ∗ j ) , J ˆ e r ) + 2 p X r =1 2 q X i =1 p X j =1 g ( h ( e ∗ i , ˆ e j ) , J ˆ e r )+ csc θ q X r =1 2 t X i,j =1 (cid:2) g ( h ( e i , e j ) , F e ∗ r ) + sec θ g ( h ( e i , e j ) , F T e ∗ r ) (cid:3) + 2 csc θ q X r =1 2 t X i =1 p X j =1 (cid:2) g ( h ( e i , ˆ e j ) , F e ∗ r ) + sec θg ( h ( e i , ˆ e j ) , F T e ∗ r ) (cid:3) (4.3) + csc θ q X r =1 p X i,j =1 (cid:2) g ( h (ˆ e i , ˆ e j ) , F e ∗ r ) + sec θ g ( h (ˆ e i , ˆ e j ) , F T e ∗ r ) (cid:3) + 2 csc θ q X r =1 p X i =1 2 q X j =1 (cid:2) g ( h (ˆ e i , e ∗ j ) , F e ∗ r ) + sec θ g ( h (ˆ e i , e ∗ j ) , F T e ∗ r ) (cid:3) + csc θ q X r =1 2 q X i,j =1 (cid:2) g ( h ( e ∗ i , e ∗ j ) , F e ∗ r ) + sec θ g ( h ( e ∗ i , e ∗ j ) , F T e ∗ r ) (cid:3) + 2 csc θ q X r =1 2 t X i =1 2 q X j =1 (cid:2) g ( h ( e i , e ∗ j ) , F e ∗ r ) + sec θ g ( h ( e i , e ∗ j ) , F T e ∗ r ) (cid:3) . We have no relation for warped products for the third, fifth, sixth, ninth, tenthand eleventh terms in (4.3), therefore, we leave these positive terms. Moreover, by
I-WARPED PRODUCT SUBMANIFOLDS OF NEARLY KAEHLER MANIFOLDS 9 using Lemma 3.1 and Lemma 3.2 with the relations (3.9)-(3.13), we find that k h k ≥ p t X i =1 h ( − Je i (ln f )) + ( e i (ln f )) i + 4 q csc θ t X i =1 h ( − Je i (ln f )) + ( e i (ln f )) i + 4 q θ t X i =1 h ( − Je i (ln f )) + ( e i (ln f )) i = 2 p t X i =1 ( e i (ln f )) + 4 q (cid:18) csc θ + 19 cot θ (cid:19) t X i =1 ( e i (ln f )) . Then we find the required inequality from the definition of gradient.For the equality case, we have from the leaving third term in (4.2) that h ( T M, T M ) ⊥ µ (4.4)From the vanishing first term and leaving seventh term in (4.3), we find h ( D , D ) ⊥ J D ⊥ and h ( D , D ) ⊥ F D θ . (4.5)Then we find from (4.4) and (4.5) that h ( D , D ) = 0 . (4.6)On the other hand, from the leaving third and ninth terms in (4.3), we get h ( D ⊥ , D ⊥ ) ⊥ J D ⊥ and h ( D ⊥ , D ⊥ ) ⊥ F D θ . (4.7)Again, we conclude from (4.4) and (4.7) that h ( D ⊥ , D ⊥ ) = 0 . (4.8)Also, from the leaving fifth and eleventh terms in the right hand side of (4.3), wehave h ( D θ , D θ ) ⊥ J D ⊥ and h ( D θ , D θ ) ⊥ F D θ . (4.9)Then we obtain from (4.4) and (4.9) that h ( D θ , D θ ) = 0 . (4.10)Moreover, from the leaving sixth and tenth terms in (4.3), we get h ( D ⊥ , D θ ) ⊥ J D ⊥ and h ( D ⊥ , D θ ) ⊥ F D θ . (4.11)Therefore, from (4.4) and (4.11) we obtain h ( D ⊥ , D θ ) = 0 . (4.12)On the other hand, from the vanishing eighth term in (4.3) with(4.4), we have h ( D , D ⊥ ) ⊂ J D ⊥ . (4.13)Similarly, from the vanishing forth term in (4.3) with (4.4), we get h ( D , D θ ) ⊂ F D θ . (4.14) Since M T is totally geodesic in ˜ M (see, e.g., [ ? , 5]), using this fact together with(4.6), (4.8) and (4.12), we know M T is totally geodesic in ˜ M . Also, since M ⊥ and M θ are totally umbilical in M , using this fact together with (4.8), (4.10), (4.13) and(4.14), we conclude that M ⊥ and M θ are both totally umbilical in ˜ M . Furhter, itfollows from Remark 3.1, (4.13) and (4.14) that M is neither D ⊕ D ⊥ –mixed totallygeodesic nor D ⊕ D θ –mixed totally geodesic in ˜ M . Consequently, the theorem isproved completely. (cid:3) Some applications
Theorem 4.1 implies the following.
Theorem 5.1. [5]
Let M = M T × f M ⊥ be a CR -warped product in a Kaeahlermanifold ˜ M . Then the second fundamental form h of M satisfies || h || ≥ p ||∇ (ln f ) || , (5.1) where p = dim M . Moreover, if the equality sign of (5.1) holds identically, then M T is totally geodesic and M ⊥ is totally umbilical in ˜ M . A warped submanifold of the form M = M T × f M θ in a a nearly Kaehler manifold˜ M is called semi-slant if M T is a holomorphic submanifold and M θ is a proper slantsubmanifold in ˜ M .The next result was proved in [16]. Theorem 5.2.
Let M T × f M θ be a semi-slant warped product of a nearly K¨ahlermanifold ˜ M . Then the second fundamental form h of M satisfies || h || ≥ q csc θ (cid:26) θ (cid:27) |∇ (ln f ) | . (5.2)On the other hand, Theorem 4.1 implies the following. Theorem 5.3. [1]
Let M = M T × f M θ be a semi-slant warped product submanifoldof a nearly Kaehler manifold ˜ M . Then second fundamental form h and the warpingfunction f satisfy k h k ≥ q (cid:26) θ (cid:27) k∇ (ln f ) k . (5.3) Moreover, if the equality sign in (4.1) holds identically, then M T is totally geodesicand M θ are totally umbilical in ˜ M .Remark . Theorem 5.3 improves Theorem 5.2 since9 + 10 cot θ > csc θ (9 + cos θ )holds for every θ ∈ (0 , π ). Furthermore, Theorem 5.3 shows that inequality (5.2)in Theorem 5.2 is not sharp. Acknowledgements.
This project was funded by the Deanship of Scientific Research(DSR), King Abdulaziz University, Jeddah, Saudi Arabia under grant no. (KEP-PhD-33-130-38). Therefore, the authors acknowledge their thanks to the DSRtechnical and financial support.
I-WARPED PRODUCT SUBMANIFOLDS OF NEARLY KAEHLER MANIFOLDS 11
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E-mail address : [email protected] B.-Y. Chen: Department of Mathematics, Michigan State University, 619 Red CedarRoad, East Lansing, Michigan 48824–1027, U.S.A.
E-mail address : [email protected] A. AL-Jedani: Department of Mathematics, Faculty of Science, King Abdulaziz Uni-versity, 21589 Jeddah, Saudi Arabia
E-mail address : [email protected] A. Alghanemi: Department of Mathematics, Faculty of Science, King Abdulaziz Uni-versity, 21589 Jeddah, Saudi Arabia
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