Hemi-Slant Warped Product Submanifolds of Nearly Kaehler Manifolds

and Applied Analysis 3 A submanifold is called proper slant if it is neither holomorphic nor totally real. More generally, a distribution D θ onM is called a slant distribution if the angle θ(X) between JX and D θ has the same value θ for each x ∈ M and X ∈ D θ and X ̸ = 0. If M is a slant submanifold of an almost Hermitian manifoldM, then we have (cf. [11]) P 2 X = −cos2θX, (21) where θ is the wirtinger angle ofM inM. Hence we have g (PX, PY) = cos2θg (X, Y) , (22) g (FX, FY) = sin2θg (X, Y) , (23) for anyX, Y ∈ TM. Papaghiuc [12] introduced a class of submanifolds in almost Hermitian manifolds called the semislant submanifolds; this class includes the class of proper CR-submanifolds and slant submanifolds. Cabrerizo et al. [13] initiated the study of contact version of semislant submanifold and also bislant submanifolds. As a step forward, Carriazo [14] defined and studied bislant submanifolds and simultaneously gave the notion of antislant submanifolds in almost Hermitian manifolds; after that V. A. Khan and M. A. Khan [15] studied antislant submanifolds with the name pseudo-slant submanifolds in the setting of Sasakian manifolds. Recently, Sahin [4] renamed pseudo-slant submanifolds as hemi-slant submanifolds and studied hemi-slant submanifolds for their warped product. Definition 1. A submanifold M of an almost Hermitian manifold is called hemi-slant submanifold if it is endowed with two orthogonal complementary distributions D⊥ and D θ such thatD is totally real, that is, JD⊥ ⊆ T⊥M andD θ is slant distribution with slant angle θ. It is straight forward to see that CR-submanifolds and slant submanifolds are hemi-slant submanifolds with θ = 0 andD = {0}, respectively. If μ is the invariant subspace of the normal bundle T⊥M then in the case of hemi-slant submanifold, the normal bundle T⊥M can be decomposed as follows: T ⊥ M = μ ⊕ FD ⊥ ⊕ FD θ . (24) As D⊥ and D θ are orthogonal distributions on M, then it is easy to see that the distributions FD⊥ and FD θ are mutually perpendicular. In fact, the decomposition (24) is an orthogonal direct decomposition. A hemi-slant submanifold M is called a hemi-slant product if the distributions D⊥ and D θ are parallel onM. In this caseM is foliated by the leaves of these distributions. In particular ifM is CR-submanifold with parallel distribution then it is called CR-product. In general, if M 1 and M 2 are Riemannian manifolds with Riemannian metrics g 1 and g 2 , respectively, then the product manifold (M 1 × M 2 , g) is a Riemannian manifold with Riemannian metric g defined as g (X, Y) = g 1 (dπ 1 X, dπ 1 Y) + g 2 (dπ 2 X, dπ 2 Y) , (25) where π i (i = 1, 2) are the projectionmaps ofM ontoM 1 and M 2 , respectively, and dπ i (i = 1, 2) are their differentials. As a generalization of the product manifold and in particular of a hemi-slant product submanifold, one can consider warped product of manifolds which are defined in the following. Definition 2. Let (B, g B ) and (C, g C ) be two Riemannian manifolds with Riemannian metrics g B and g C , respectively, and f a positive differentiable function on B. The warped product of B and C is the Riemannian manifold (B × C, g), where g = g B + f 2 g C . (26) For awarped productmanifoldN 1 × f N 2 , we denote byD 1 andD 2 the distributions defined by the vectors tangent to the leaves and fibers, respectively. In other words,D 1 is obtained by the tangent vectors of N 1 via the horizontal lift and D 2 is obtained by the tangent vectors ofN 2 via vertical lift. In case of semi-invariant warped product submanifolds D 1 and D 2 are replaced byD andD, respectively. The warped product manifold (B × C, g) is denoted by B× f C. IfX is the tangent vector field toM = B× f C at (p, q) then ‖X‖ 2 = 󵄩󵄩󵄩󵄩dπ1X 󵄩󵄩󵄩󵄩 2 + f 2 (p) 󵄩󵄩󵄩󵄩dπ2X 󵄩󵄩󵄩󵄩 2 . (27) Bishop and O’Neill [1] proved the following. Theorem 3. LetM = B× f C be warped product manifolds. If X, Y ∈ TB and V,W ∈ TC then (i) ∇ X Y ∈ TB, (ii) ∇ X V = ∇ V X = (Xf/f)V, (iii) ∇ V W = (−g(V,W)/f)∇f. ∇f is the gradient of f and is defined as g (∇f,X) = Xf, (28) for allX ∈ TM. Corollary 4. On a warped product manifoldM = N 1 × f N 2 , the following statements hold: (i) N 1 is totally geodesic inM (ii) N 2 is totally umbilical inM. In what follows, N ⊥ and N θ will denote a totally real and slant submanifold, respectively, of an almost Hermitian manifoldM. A warped product manifold is said to be trivial if its warping function f is constant. More generally, a trivial warped product manifold M = N 1 × N 2 is a Riemannian product N 1 × N f 2 , where Nf 2 is the manifold with the Riemannian metric f2g 2 which is homothetic to the original metric g 2 ofN 2 . For example, a trivial CR-warped product is CR-product. Sahin [4] extended the study of warped product hemislant submanifolds and hemi-slant warped product of Kaehler manifolds introducing warped product submanifolds asN θ × f2 N ⊥ andN ⊥ × f N θ , where θ is the slant angle. 4 Abstract and Applied Analysis 3. Hemi-Slant Warped Product Submanifolds In [5] Uddin and Chi investigated warped product pseudoslant (hemi-slant) submanifolds of nearly Kaehler manifolds and they only showed that there do not exist warped products of the formN ⊥ × f N θ in nearly Kaehler manifolds, whereN ⊥ is totally real submanifold andN θ is slant submanifold. In this section we study the warped products of the typesN θ × f N ⊥ . LetM = N θ × f N ⊥ be a hemi-slant warped product of a nearly Kaehler manifoldM. Then byTheorem 3, ∇ X Z = ∇ Z X = X lnfZ, (29) for anyX ∈ TN θ , Z ∈ TN ⊥ . Now by formula (12) andTheorem 3, (∇ Z P)W = g (Z,W)P (∇ lnf) − g (Z, PW)∇ lnf, (30) for each Z, W ∈ TN θ . Now we will investigate some interesting results of the second fundamental form. Proposition 5. On a hemi-slant warped product submanifold M = N θ × f N ⊥ of a nearly Kaehler manifoldM, one has (i) 2g(h(X, Y), FZ) = g(h(X, Z), FY) + g(h(Y, Z), FX), (ii) g(h(X, Z), FX) = g(h(X,X), FZ), for anyX, Y ∈ N θ and Z ∈ N ⊥ . Proof. As N θ is totally geodesic inM then (∇ X P)Y ∈ TN θ and therefore by formula (17), g (P X Y,Z) = −g (A FX Y,Z) − g (th (X, Y) , Z) , (31) or g (h (X, Y) , FZ) = g (P X Y,Z) + g (h (Y, Z) , FX) . (32) Similarly, we have g (h (X, Y) , FZ) = g (P Y X,Z) + g (h (X, Z) , FY) . (33) Adding above two equations and using (20)(a), we get part (i). Now by formula (17) and (20)(a), 0 = (∇ X P)Z + (∇ Z P)X − 2th (X, Z) − A FZ X − A FX Z, (34) and by (29) the above equation gives (PX lnf)Z = 2th (X, Z) + A FZ X + A FX Z. (35) Taking inner product of (35) withX ∈ D θ , we get g (h (X, Z) , FX) = g (h (X,X) , FZ) , (36) which proves part (ii). Theorem 6. For a hemi-slant warped product submanifold N θ × f N ⊥ of a nearly Kaehler manifold the warping function satisfies the following relation: cos2θX lnf‖Z‖2 = g (h (PX,Z) , FZ) − g (H, FPX) ‖Z‖2, (37) for anyX ∈ TN θ and Z ∈ TN ⊥ . Proof. If M is a hemi-slant warped product submanifold N θ × f N ⊥ of a nearly Kaehler manifold then (∇ X P)Z = 0 for eachX ∈ TN θ and Z ∈ TN ⊥ , and thus by (17), P X Z = −A FZ X − th (X, Z) . (38) On the other hand P Z X = ∇ Z PX − P∇ Z X − A FX Z − th (X, Z) . (39) Now using (29), the above equation takes the form P Z X = PX lnfZ − A FX Z − th (X, Z) . (40) Adding (38) and (40) and using (20)(a), A FZ X + A FX Z = PX lnfZ − 2th (X, Z) , (41) taking inner product with Z ∈ TN ⊥ , and using the fact that N ⊥ is totally umbilical, one gets the following equation: PX lnf‖Z‖2 = −g (h (X, Z) , FZ) + g (H, FX) ‖Z‖2. (42) By replacingX by PX the required result follows. Remark 7. In [4] Sahin proved that hemi-slant warped products of the type N ⊥ × f N θ do not exist in the setting of Kaehler manifolds. Therefore, in the following Corollary we discuss the warped products of the typeN θ × f N ⊥ . Corollary 8. For a hemi-slant warped product submanifold N θ × f N ⊥ of a Kaehler manifold the warping function satisfies the following relation: cos2θX lnf‖Z‖2 = g (h (PX,Z) , FZ) − g (H, FPX) ‖Z‖2, (43) for any X ∈ TN θ and Z ∈ TN ⊥ . Proof. SinceM is a hemi-slant warped product submanifold of a Kaehler manifold, then by tensorial equation of Kaehler manifold, it is easy to see that P Z X = 0, for any X ∈ TN θ and Z ∈ TN ⊥ , and using this fact in (40) and taking inner product with Z ∈ TN ⊥ , we get the required result. Let us denote by D θ and D⊥ the tangent bundles on N θ and N ⊥ , respectively, and let {X 1 , X 2 , . . . , X p , X p+1 = PX 1 , . . . , X 2p = PX p } and {Z 1 , Z 2 , . . . , Z q } be local orthonormal frames of vector fields on N θ and N ⊥ , respectively, with 2p and q being their real dimensions; then


Introduction
In [1] Bishop and O'Neill introduced the notion of warped product manifolds as a natural generalization of Riemannian product manifolds.For instance, a surface of revolution is a warped product manifold.So far as its applications are concerned, it has been shown that warped product manifolds provide excellent setting to model space time near black holes or bodies with large gravitational forces (see [1,2]).Due to wide applications of warped product submanifolds, this becomes a fascinating and interesting topic for research and many articles are available in literature (see [1,[3][4][5]).Chen [6] initiated the study of warped product submanifolds by showing that there do not exist warped product CR-submanifolds of the type  ⊥ ×    , and he considered warped product CR-submanifolds of the types   ×   ⊥ and established a relationship between the warping function  and the squared norm of the second fundamental form.Extending the study of Chen, Sahin [7] proved that there exist no semislant warped product submanifolds in a Kaehler manifold.In [8], V. A. Khan and K. A. Khan studied generic warped product submanifolds of nearly Kaehler manifolds and obtained an inequality for squared norm of second fundamental form in terms of warping function.Recently, Sahin [4] investigated hemi-slant warped product for Kaehler manifolds and obtained an inequality for squared norm of second fundamental form for mixed totally geodesic submanifolds.In view of the interesting geometric characteristic of nearly Kaehler manifolds and the nonexistence of CRproduct submanifolds in  6 [9], it will be significant to explore hemi-slant warped product submanifolds of a nearly Kaehler manifold.In this continuation we have achieved success in extending the results of Sahin [4] and Chen [6] to the setting of nearly Kaehler manifolds.

Preliminaries
Let (, , ) be a nearly Kaehler manifold with an almost complex structure  and Hermitian metric  and a Levi-Civita connection ∇ such that for all vector fields  and  on .Six-dimensional sphere  6 is a classic example of a nearly Kaehler non-Kaehler manifold.It has an almost complex structure  defined by the vector cross product in the space of purely imaginary Cayley numbers which satisfies the condition (∇  ) = 0.
Let  be the Cayley division algebra generated by { 0 = 1,   , (1 ≤  ≤ 7)} over  and  + the subspace of  consisting of all purely imaginary Cayley numbers.We may identify  + with a 7-dimensional Euclidean space  7 with the canonical inner product  = (, ).The automorphism group of  is the compact simple Lie group  2 and furthermore the inner product  is invariant under the action of  2 and hence, the group  2 may be considered as a subgroup of  (7).A vector cross product for vectors in  7 (=  + ) is defined by Then the multiplication table for   ×   is given by Considering  6 as { ∈  + : (, ) = 1}, the almost complex structure  on  6 is defined by where  ∈  6 and  ∈    6 .The almost complex structure defined by the above equation together with the induced metric on  6 from  on  7 (=  + ) gives rise to a nearly Kaehler structure  6 [10].
Let  be a submanifold of .Then the induced Riemannian metric on  is denoted by the same symbol  and the induced connection on  is denoted by the symbol ∇.If  and  denote the tangent bundle on  and , respectively, and  ⊥ , the normal bundle on , then the Gauss and Weingarten formulae are given by for ,  ∈  and  ∈  ⊥  where ∇ ⊥ denotes the connection on the normal bundle  ⊥ .ℎ and   are the second fundamental form and the shape operator of immersions of  into .Corresponding to the normal vector field  they are related as The mean curvature vector  of  is given by where  is the dimension of  and { 1 ,  2 , . . .,   } is a local orthonormal frame of vector fields on .The squared norm of the second fundamental form is defined as (ℎ (  ,   ) , ℎ (  ,   )) .
For  ∈  and  ∈  ⊥  we write where  and  are tangential components of  and , respectively, and  and  are the normal components of  and .
The covariant differentiation of the tensors , , , and  is defined, respectively, as Furthermore, for any ,  ∈ , the tangential and normal parts of (∇  ) are denoted by P   and Q  ; that is, On using ( 6)-( 13), we may obtain that Similarly, for  ∈  ⊥ , denoting by P   and Q  , respectively, the tangential and normal parts of (∇  ), we find that On a submanifold  of a nearly Kaehler manifold by ( 2) and ( 16) for any ,  ∈ .
Let  be an almost Hermitian manifold with an almost complex structure  and Hermitian metric  and let  be a submanifold of .Submanifold  is said to be CRsubmanifold if there exist two orthogonal complementary distributions  and  ⊥ such that  is holomorphic distribution, that is,  ⊆  and  ⊥ is totally real distribution, that is,  ⊥ ⊆  ⊥ .
A submanifold is called proper slant if it is neither holomorphic nor totally real.More generally, a distribution   on  is called a slant distribution if the angle () between  and   has the same value  for each  ∈  and  ∈   and  ̸ = 0.
If  is a slant submanifold of an almost Hermitian manifold , then we have (cf.[11]) where  is the wirtinger angle of  in .Hence we have for any ,  ∈ .
Papaghiuc [12] introduced a class of submanifolds in almost Hermitian manifolds called the semislant submanifolds; this class includes the class of proper CR-submanifolds and slant submanifolds.Cabrerizo et al. [13] initiated the study of contact version of semislant submanifold and also bislant submanifolds.As a step forward, Carriazo [14] defined and studied bislant submanifolds and simultaneously gave the notion of antislant submanifolds in almost Hermitian manifolds; after that V. A. Khan and M. A. Khan [15] studied antislant submanifolds with the name pseudo-slant submanifolds in the setting of Sasakian manifolds.
Recently, Sahin [4] renamed pseudo-slant submanifolds as hemi-slant submanifolds and studied hemi-slant submanifolds for their warped product.Definition 1.A submanifold  of an almost Hermitian manifold is called hemi-slant submanifold if it is endowed with two orthogonal complementary distributions  ⊥ and   such that  ⊥ is totally real, that is,  ⊥ ⊆  ⊥  and   is slant distribution with slant angle .
It is straight forward to see that CR-submanifolds and slant submanifolds are hemi-slant submanifolds with  = 0 and  ⊥ = {0}, respectively.
If  is the invariant subspace of the normal bundle  ⊥  then in the case of hemi-slant submanifold, the normal bundle  ⊥  can be decomposed as follows: As  ⊥ and   are orthogonal distributions on , then it is easy to see that the distributions  ⊥ and   are mutually perpendicular.In fact, the decomposition (24) is an orthogonal direct decomposition.
A hemi-slant submanifold  is called a hemi-slant product if the distributions  ⊥ and   are parallel on .In this case  is foliated by the leaves of these distributions.In particular if  is CR-submanifold with parallel distribution then it is called CR-product.In general, if  1 and  2 are Riemannian manifolds with Riemannian metrics  1 and  2 , respectively, then the product manifold ( 1 ×  2 , ) is a Riemannian manifold with Riemannian metric  defined as where   ( = 1, 2) are the projection maps of  onto  1 and  2 , respectively, and   ( = 1, 2) are their differentials.As a generalization of the product manifold and in particular of a hemi-slant product submanifold, one can consider warped product of manifolds which are defined in the following.Definition 2. Let (,   ) and (,   ) be two Riemannian manifolds with Riemannian metrics   and   , respectively, and  a positive differentiable function on .The warped product of  and  is the Riemannian manifold ( × , ), where For a warped product manifold  1 ×   2 , we denote by  1 and  2 the distributions defined by the vectors tangent to the leaves and fibers, respectively.In other words,  1 is obtained by the tangent vectors of  1 via the horizontal lift and  2 is obtained by the tangent vectors of  2 via vertical lift.In case of semi-invariant warped product submanifolds  1 and  2 are replaced by  and  ⊥ , respectively.
The warped product manifold ( × , ) is denoted by  ×  .If  is the tangent vector field to  =  ×   at (, ) then Bishop and O'Neill [1] proved the following.∇ is the gradient of  and is defined as for all  ∈ .
Corollary 4. On a warped product manifold  =  1 ×   2 , the following statements hold: In what follows,  ⊥ and   will denote a totally real and slant submanifold, respectively, of an almost Hermitian manifold .
A warped product manifold is said to be trivial if its warping function  is constant.More generally, a trivial warped product manifold  =  1 ×  2 is a Riemannian product  1 ×   2 , where   2 is the manifold with the Riemannian metric  2  2 which is homothetic to the original metric  2 of  2 .For example, a trivial CR-warped product is CR-product.
Sahin [4] extended the study of warped product hemislant submanifolds and hemi-slant warped product of Kaehler manifolds introducing warped product submanifolds as   × 2  ⊥ and  ⊥ ×    , where  is the slant angle.

Hemi-Slant Warped Product Submanifolds
In [5] Uddin and Chi investigated warped product pseudoslant (hemi-slant) submanifolds of nearly Kaehler manifolds and they only showed that there do not exist warped products of the form  ⊥ ×    in nearly Kaehler manifolds, where  ⊥ is totally real submanifold and   is slant submanifold.In this section we study the warped products of the types   ×   ⊥ .
Let  =   ×   ⊥ be a hemi-slant warped product of a nearly Kaehler manifold .Then by Theorem 3, for any  ∈   ,  ∈  ⊥ .Now by formula (12) and Theorem 3, for each ,  ∈   .Now we will investigate some interesting results of the second fundamental form.Adding above two equations and using (20)(a), we get part (i).Now by formula ( 17) and (20)(a), and by (29) the above equation gives Taking inner product of (35) with  ∈   , we get which proves part (ii).
Theorem 6.For a hemi-slant warped product submanifold   ×   ⊥ of a nearly Kaehler manifold the warping function satisfies the following relation: for any  ∈   and  ∈  ⊥ .
Proof.If  is a hemi-slant warped product submanifold   ×   ⊥ of a nearly Kaehler manifold then (∇  ) = 0 for each  ∈   and  ∈  ⊥ , and thus by (17), On the other hand Now using (29), the above equation takes the form Adding ( 38) and (40) and using (20)(a), taking inner product with  ∈  ⊥ , and using the fact that  ⊥ is totally umbilical, one gets the following equation: By replacing  by  the required result follows.
Remark 7. In [4] Sahin proved that hemi-slant warped products of the type  ⊥ ×    do not exist in the setting of Kaehler manifolds.Therefore, in the following Corollary we discuss the warped products of the type   ×   ⊥ .
Corollary 8.For a hemi-slant warped product submanifold   ×   ⊥ of a Kaehler manifold the warping function satisfies the following relation: for any  ∈   and  ∈  ⊥ .
Proof.Since  is a hemi-slant warped product submanifold of a Kaehler manifold, then by tensorial equation of Kaehler manifold, it is easy to see that P   = 0, for any  ∈   and  ∈  ⊥ , and using this fact in (40) and taking inner product with  ∈  ⊥ , we get the required result.
To discuss the equality case we will explore the expression ‖ℎ   (  ,   )‖ 2 as follows.
Remark 10.Since (37) is the key result of the paper which helps to get the inequality in Theorem 9 and moreover (37) is also true for the Kaehler manifolds, hence the results in Theorem 9 are also true for hemi-slant warped product submanifolds of a Kaehler manifold.
we calculate the inequality for the squared norm of second fundamental form in the following theorem.