Warped Product Submanifolds of Riemannian Product Manifolds

and Applied Analysis 3 where TX and NX are the tangential and normal components of FX, respectively, and for V ∈ T⊥M,


Introduction
Bishop and O'Neil 1 introduced the notion of warped product manifolds that occur naturally; for example, surface of revolution is a warped product manifold. With regard to physical applications of these manifolds, one may realize that space time around a massive star or a black hole can be modeled on warped product manifolds 2 . CR-warped product was introduced by Chen 3 ; he studied warped product CR-submanifolds in the setting of Kaehler manifolds and showed that there does not exist warped product of the form N ⊥ × f N T ; therefore he considered warped product CR-submanifolds of type N T × N ⊥ and established a relationship between the warping function f and the squared norm of second fundamental form 3 . In 4 Atçeken studied semi-slant warped product of Riemannian product manifolds. In fact they proved that there exists no warped product if spheric submanifold of warped product submanifold is proper slant submanifold. On the other hand they proved the existence of warped product of the type N θ × f N T and N θ × f N ⊥ via some examples. In this continuation we have studied the warped product submanifolds in which proper slant submanifolds are totally geodesic; that is, we study the warped product of the types N θ × N T and N θ × N ⊥ and called them semi-slant warped product and hemi-slant warped product submanifolds, respectively.

Preliminaries
Let M 1 , g 1 and M 2 , g 2 be the Riemannian manifolds with dimensions m 1 and m 2 , respectively, and let M 1 × M 2 be Riemannian product manifold of M 1 and M 2 . We denote projection mapping of T M 1 × M 2 onto TM 1 and TM 2 by σ and π , respectively. Then we have σ π I, σ 2 σ , π 2 π , and σ • π π • σ 0, where denotes the differential. Riemannian metric of the Riemannian product manifold M M 1 × M 2 is defined by for any X, Y ∈ T M. If we set F σ − π , then F 2 I, F / I, and g satisfies the condition for any X, Y ∈ T M; thus F defines an almost Riemannian product structure on M. We denote Levi-Civita connection on M by ∇; then the covariant derivative of F is defined as for any X, Y ∈ T M. We say that F is parallel with respect to the connection ∇ if we have ∇ X F Y 0. Here from 5 , we know that F is parallel; that is, F is Riemannian product structure.
Let M be a Riemannian product manifold with Riemannian product structure F and M an immersed submanifold of M; we also denote by g the induced metric tensor on M as well as on M. If ∇ is the Levi-Civita connection on M, then the Gauss and Weingarten formulas are given, respectively, as for any X, Y ∈ TM and V ∈ T ⊥ M, where ∇ is the connection on M and ∇ ⊥ is the connection in the normal bundle, h is the second fundamental form of M, and A V is the shape operator of M. The second fundamental form h and the shape operator A V are related by For any X ∈ TM, we can write where tV and nV are the tangential and normal components of FV , respectively, and the submanifold M is said to be invariant if N is identically zero. On the other hand M is said to be an anti-invariant submanifold if T is identically zero. The covariant derivatives of T , N, t, and n are defined as 10 Using 2.4 -2.9 we get Let M be an immersed submanifold of a Riemannian product manifold M, for each nonzero vector X tangent to M at a point x, and we denote by θ x the angle between FX and T x M. The angle θ x is called the slant angle of immersion.
Let M be an immersed submanifold of a Riemannian product manifold M. M is said to be slant submanifold of Riemannian product manifold M if the slant angle θ x is constant which is independent of choice of x ∈ M and X ∈ TM.
Invariant and anti-invariant submanifolds are particular cases of slant submanifolds with angles θ 0 and θ π/2, respectively. A slant submanifold which is neither invariant nor anti-invariant is called proper slant submanifold. The following characterization of slant submanifolds of Riemannian product manifolds is proved by Atçeken 6 . Moreover, if θ is the slant angle of M, then it satisfies λ cos 2 θ. Hence, for a slant submanifold we have the following relations which are consequences of the above theorem: for any X, Y ∈ TM.
Papaghuic 7 introduced a class of submanifolds in almost Hermitian manifolds called semi-slant submanifolds; this class includes the class of proper CR-submanifolds and slant submanifolds. Cabrerizo et al. 8 initiated the study of contact version of semi-slant submanifolds and also gave the notion of Bi-slant submanifolds. A step forward Carriazo 9 defined and studied Bi-slant submanifolds and simultaneously gave the notion of anti-slant submanifolds; after that V. A. Khan and M. A. Khan 10 have studied anti-slant submanifolds with the name pseudo-slant submanifolds. Recently, Sahin 11 renamed these submanifolds and studied these submanifolds with the name hemi-slant submanifolds for their warped product.  It is easy to see that semi-invariant submanifolds and slant submanifolds are semislant submanifolds with θ 0 and D ⊥ {0}, respectively. The normal bundle T ⊥ M can be decomposed as follows: As D ⊥ and D θ are orthogonal distributions on M, then it is easy to see that the distributions ND ⊥ and ND θ are mutually perpendicular. In fact, the decomposition 2.18 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 on M.
As a generalization of product manifold and in particular of semi-slant product submanifolds hemi-slant product submanifolds one can consider warped product of manifolds which are defined as.
Definition 2.4. Let B, g B and F, g F be two Riemannian manifolds with Riemannian metric g B and g F , respectively, and f a positive differentiable function on B. The warped product of B and F is the Riemannian manifold B × F, g , where g g B f 2 g F .

2.19
Abstract and Applied Analysis 5 For a warped product manifold N 1 × f N 2 , we denote by D 1 and D 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 of N 2 via vertical lift. In case of semi-slant warped product submanifolds D 1 and D 2 are replaced by D T and D θ , respectively.
The warped product manifold B × F, g is denoted by B× f F. If X is the tangent vector field to M B× f F at p, q , then

2.20
Bishop and O'Neill 1 proved the following.

Theorem 2.5.
Let M B× f F be warped product manifolds. If X, Y ∈ TB and V, W ∈ TF, then ∇f is the gradient of f and is defined as for all X ∈ TM.
Corollary 2.6. On a warped product manifold M N 1 × f N 2 , the following statements hold: i N 1 is totally geodesic in M; ii N 2 is totally umbilical in M.
Throughout, we denote by N T , N ⊥ , and N θ invariant, anti-invariant, and slant submanifolds, respectively, of a Riemannian product manifold M.

Semi-Slant Warped Product Submanifolds
In this section we will consider the warped product of the type N θ × f N T .
For the warped product of the type N θ × f N T by Theorem 2.5 we have for any Z ∈ TN θ and X ∈ TN T . This is part i of the lemma. Now for any X, Y ∈ TN T , from 2.13 and 2.9 , Taking inner product with Z ∈ TN θ , the above equation yields Using 3.1 , the above equation gives In particular TZ ln fg X, X g h X, X , NZ .

3.7
This proves part ii of the lemma. Now we have the following corollary.

Corollary 3.2.
For the warped product of the type N θ × f N T following statements are equivalent: ii θ π/2 or the warping function f is constant; that is, there does not exist warped product.
Proof. Since N T is totally umbilical, then from 3.7 g H, NZ TZ ln f.

3.8
Replacing Z by TZ and using Theorem 2.1, we get g H, NT Z cos 2 θZ ln f.

3.9
The proof follows from 3.9 .
Now we have the following characterization for semi-slant warped product submanifolds.

Theorem 3.3. A semi-slant submanifold M of Riemannian product manifolds M with integrable invariant distribution D T and the slant distribution D θ is locally a semi-slant warped product if and only if ∇ X TZ ∈ D θ and there exist a C ∞ -function α on M with Xα 0 for all X ∈ D T such that
A NZ X TZ ln fX, 3.10 for all X ∈ D T and Z ∈ D θ .
Proof. If M is a semi-slant warped product of the type N θ × f N T , then for any X ∈ TN T and Z ∈ TN θ from 2.9 , 2.13 , and 3.1 , we have TZ ln fX − Z ln fTX A NZ X th X, Z .

3.11
Taking inner product with X, the above equation gives g A NZ X, X TZ ln fg X, X .

3.12
By part i of Lemma 3.1, we also have From 3.12 and 3.13 we have the following equation: A NZ X TZ ln fX.
3.14 Conversely, let M be a semi-slant submanifold of M satisfying the hypothesis of the theorem; then for any Z ∈ TN θ and Y ∈ TN T we have This mean h Z, Y ∈ μ. From 2.14 , we have Comparing components of μ and ND θ , we get It is evident from the above equation that ∇ Z Y ∈ D T ; this means ∇ Z W ∈ D θ for any Z, W ∈ D θ and hence D θ is totally geodesic. Further, let N T be a leaf of D T and h T a second fundamental form of the immersion N T in M; then for any X, Y ∈ D T and Z, W ∈ D θ g h T X, Y , FW g ∇ X Y, FW . Using the hypothesis, we get Finally, the above equation yields That is, N T is totally umbilical and as Xα 0, for all X ∈ D T , this means that mean curvature vector of N T is parallel; that is, the leaves of D T are extrinsic spheres in M. Hence by virtue of result of 12 which says that if the tangent bundle of a Riemannian manifold M splits into an orthogonal sum TM E 0 ⊕ E 1 of nontrivial vector subbundles such that E 1 is spherical and its orthogonal complement E 0 is auto parallel, then the manifold M is locally isometric to a warped product M 0 × f M 1 , we can say M is locally semi-slant warped product submanifold N θ × f N T , where warping function f e α . Let us denote by D T and D θ the tangent bundles on N T and N θ , respectively, and let {X 1 , . . . , X p , X p 1 FX 1 , . . . , X 2p FX p } and {Z 1 , . . . , Z q , Z q 1 TZ 1 , . . . , Z 2q TZ q } be local orthonormal frames of vector fields on N T and N θ , respectively, with 2p and 2q being real dimensions:

3.22
Now, on a semi-slant warped product submanifold of a Riemannian product manifold, we prove the following.

Hemi-Slant Warped Product Submanifolds
In this section we will study the warped product of the type N θ × f N ⊥ . For warped product of type N θ × f N ⊥ from Theorem 2.5 we have for any X ∈ TN θ and Z ∈ TN ⊥ . Now we have the following lemma. Similarly, for any X ∈ TN θ and Z ∈ TN ⊥ by 4.12 , 4.13 , and 4.5 it is easy to see that g h ND θ X r , Z i , h ND θ X r , Z i 0.