A Geometric Obstruction for CR-Slant Warped Products in a Nearly Cosymplectic Manifold

: In the early 20th century, B.-Y. Chen introduced the concept of CR-warped products and obtained several fundamental results, such as inequality for the length of second fundamental form. In this paper, we obtain B.-Y. Chen’s inequality for CR-slant warped products in nearly cosymplectic manifolds, which are the more general classes of manifolds. The equality case of this inequality is also investigated. Furthermore, the inequality is discussed for some important subclasses of CR-slant warped products.

and hemi-slant warped products are particular classes of CR-slant warped products. We refer Chen's books [17,18] for up-to-date survey on warped product manifolds and warped product submanifolds.
In this paper, we study CR-slant warped product submanifolds of nearly cosymplectic manifolds which are the more general classes of contact metric manifolds. We prove that every CR-slant warped product M = B × f N θ in a nearly cosymplectic manifold satisfies the following inequality: where B = N T × N ⊥ , a contact CR-product and 2s = dim N θ , while ∇ T (ln f ) and ∇ ⊥ (ln f ) are the gradient components of ln f along N T and N ⊥ , respectively.

Preliminaries and Basic Results
An odd dimensional almost contact metric manifold is a smooth manifoldM of dimension 2n + 1, endowed with a structure (φ, ξ, η, g), given by a (1, 1) tensor field φ, a vector field ξ, a 1-form η and a Riemannian metric g satisfying [19] for all vector fields X, Y onM (see [20] for more details). From the definition it follows that φξ = 0 and η • φ = 0. Furthermore, φ is skew-symmetric with respect to g, so that the bilinear form Φ(X, Y) := g(X, φY) defines a 2-form onM, called fundamental 2-form. An almost contact metric manifold with dη = 2Φ is called a contact metric manifold. In this case, η is a contact form, i.e., η ∧ (dη) n = 0 everywhere onM. An almost contact metric manifold (M, φ, ξ, η, g) is called a cosymplectic manifold if dη = 0, dΦ = 0 and N φ = 0, where N φ is the Nijenhuis tensor of φ. Equivalently, we have∇φ = 0. It is known that a cosymplectic manifold is locally a Riemannian product of the real line R and a Kaehler manifoldÑ, which is an integral submanifold of the distribution D = Ker(η) (see [21] for further details).
A nearly cosymplectic manifold is an almost contact metric manifold (M, φ, ξ, η, g) such that for all vector fields X, Y onM. It is known that in a nearly cosymplectic manifold, the structure vector field ξ is Killing and satisfies∇ ξ ξ = 0,∇ ξ η = 0 and moreover g(∇ X ξ, X) = 0 for any vector field X tangent to a nearly cosymplectic manifoldM. Let M be a m-dimensional manifold isometrically immersed into a Riemannian manifoldM; denote by the same symbol g the induced metric on M. Let Γ(TM) be the Lie algebra of vector fields on M and Γ(T ⊥ M), the set of all vector fields normal to M. If we denote by ∇ and∇, the Levi-Civita connections of M andM, respectively, then the Gauss and Weingarten formulas are respectively given bỹ for any vector field X, Y ∈ Γ(TM) and N ∈ Γ(T ⊥ M), where ∇ ⊥ is the normal connection in the normal bundle, σ is the second fundamental form and A N is the shape operator (corresponding to the normal vector field N) for the immersion of M intoM. They are related by g(σ(X, Y), N) = g(A N X, Y).
For any X tangent to M and N normal to M, we write where TX (respectively, FX) is the tangential (respectively, normal) component of φX and tN (respectively, f N) is the tangential (respectively, normal) component of φN. Then T is an endomorphism on TM and F is a normal valued 1-form. For any p ∈ M and {E 1 , · · · , E m , · · · , E 2n+1 } is an orthonormal frame of T pM such that E 1 , · · · , E m are tangent to M at p and E m+1 , · · · , E 2n+1 normal to M. Then, There are two well-known classes of submanifolds, namely φ-invariant submanifolds and φ-anti-invariant submanifolds due to the behaviour of the tangent bundle of a submanifold under the action of the almost contact structure tensor φ of the ambient manifold. In the case of invariant submanifolds, the tangent space remains invariant under the action of the almost contact structure tensor φ whereas in case of anti-invariant submanifolds it is mapped into the normal space. As a generalized class of invariant and anti-invariant submanifolds, B.-Y. Chen introduced slant submanifolds of almost Hermitian manifolds. Later, A. Lotta [22] and Cabrerizo et al. [23] in separate articles extended this study to almost contact metric manifolds.
A submanifold M tangent to the structure vector field ξ is called slant if for every non-zero tangent vector X which is not proportional to ξ p , the angle 0 ≤ θ(X) ≤ π/2 between φX and T p M is constant (called, slant angle), i.e., θ is independent of the choice of X ∈ Γ(TM) \ {ξ, 0} and p ∈ M. If the slant angle is different from 0 and π/2, then it is called proper slant.
Another, generalized class of CR-submanifolds and slant submanifolds introduced as semi-slant submanifolds by N. Papaghuic [24]. Later, these submanifolds studied by Cabrerizo et al. [25] in almost contact metric manifolds.
A submanifold M of an almost contact metric manifoldM is a semi-slant submanifold if there exist two orthogonal distribution D and D θ on M such that: Hemi-slant submanifolds were defined by Carriazo in [26] under the name of anti-slant submanifolds as a particular class of bi-slant submanifolds. A submanifold M of an almost contact metric manifoldM is said to be a hemi-slant submanifold if there exists a pair of orthogonal distributions D ⊥ and D θ on M such that Now, we recall the following useful characterization theorem proved in [23].
Furthermore, if θ is the slant angle of M, then λ = cos 2 θ.
The following relations are easily obtained from Theorem 1: and for any X, Y ∈ Γ(TM).

Definitions and Lemmas on CR-Slant Warped Products
A warped product B × f F of two Riemannian manifolds (B, g B ) and (F, g F ) is the product manifold B × F equipped with the warped product metric where f : B → (0, ∞) is a positive differentiable function on B and π 1 : M → B, π 2 : M → F are projection maps given by π 1 (p, q) = p and π 2 (p, q) = q for any (p, q) ∈ B × F and * denotes the symbol for the tangent map. If function f (called, warping function) is constant, then M is simply a Riemannian product. We know that, for any vector field X on B and a vector field Z on F, we have where ∇ is the Levi-Civita connection on M. Notice that on a warped product manifold M, B is totally geodesic and F is totally umbilical in M.
In this section, we study CR-slant warped product submanifolds of the form M = B × f N θ of a nearly cosymplectic manifoldM, where B = N T × N ⊥ , a contact CR-product of invariant and anti-invairant submanifolds ofM, and N θ is a slant submanifold. For the simplicity, throughout this paper we denote the corresponding tangent spaces of N T , N ⊥ and N θ by D, D ⊥ and D θ , respectively. For a CR-slant warped product M = B × f N θ of an almost contact metric manifoldM, the tangent space is decomposed as: where D is an invariant distribution, D ⊥ is an anti-invariant distribution and D θ is a proper slant distribution and < ξ > is the 1-dimensional distribution spanned by the structure vector field ξ.
Clearly, we observe that if ξ along D θ then the CR-slant warped product M = B × f N θ is trivial as follows: Since ξ ∈ Γ(D θ ) is killing on a nearly cosymplectic manifold, from (3) and (10), we find Furthermore, the normal bundle T ⊥ M is decomposed as where µ is the invariant normal subbundle of T ⊥ M under φ. From now, we use the following conventions: be a CR-slant warped product submanifold of a nearly cosymplectic manifoldM such that B = N T × N ⊥ and ξ ∈ Γ(TB). Then, we have for any X 1 , Y 1 tangent to N T and X 3 , Y 3 tangent to N θ .
Following relations are easily obtained by interchanging X 1 with φX 1 ; X 3 with TX 3 and Y 3 with TY 3 with the help of (1) and (6) in Lemma 1 (iii). and for any X 1 tangent to N T ; X 2 , Y 2 tangent to N ⊥ and X 3 tangent to N θ .

Proof. From (3) and (4), we have
Interchanging X 2 and Y 2 , we find Then, the first statements follows from (24) and (25) together with (2). For the second part, we have By orthogonality of distributions, we find On the other hand, we also have Again, by orthogonality of the distributions, we get Hence, from (26), (27) and (2), we get the desired result.

Lemma 3.
For a CR-slant warped product M = B × f N θ in a nearly cosymplectic manifoldM, we have for any X 2 tangent to N ⊥ and X 3 , Y 3 tangent to N θ .
By interchanging X 3 with TX 3 and Y 3 with TY 3 , one can get the following relations.

Main Results
In this section, we present our main results of the paper. First, we have the following non-existence theorem of proper CR-slant warped products. Proof. For a D ⊕ D θ -mixed totally geodesic CR-slant warped product, from Lemma 1 (iii) and (21), we derive for any X 1 ∈ Γ(D) and X 3 , Y 3 ∈ Γ(D θ ). Since g is a Riemannian metric, then we find either cos θ = ±3 which is impossible, or φX 1 (ln f ) = 0, i.e., f is constant along N T , which proves the theorem completely.
Now, we establish a sharp estimation for the length of the second fundamental form by using the following frame field for a CR-slant warped product.
Let M = B × f N θ be a m-dimensional CR-slant warped product submanifold of a 2n + 1 dimensional nearly cosymplectic manifoldM such that B is the Riemannian product of an invariant submanifold N T and an anti-invariant submanifold N ⊥ inM. Let the corresponding tangent space of N T , N ⊥ and N θ respectively are D, D ⊥ and D θ . If dim D = 2α + 1, dim D ⊥ = γ and dim D θ = 2β, then the tangent bundle TM is spanned by the following orthonormal frame fields D = Span{E 1 , · · · , E α , E α+1 = φE 1 , · · · , E 2α = φE α , E 2α+1 = ξ}, where β = 1 2 dim N θ and ∇ ⊥ (ln f ) and ∇ T (ln f ) are the gradient components of ln f along N ⊥ and N T , respectively.
Moreover, if the equality sign holds in (37), then N T and N ⊥ are totally geodesic submanifolds ofM and N θ is a totally umbilical submanifold ofM. Furthermore, M is also a D ⊕ D ⊥ -mixed totally geodesic submanifold ofM but never be a D ⊕ D θ -mixed totally geodesic and hence M is not minimal inM.

Proof. From (5), we have
Leaving the third term and decompose first two terms in the right hand side of (39) for the considered orthonormal frame fields, we derive Using Lemma 1, relations (17)-(23), Lemmas 2 and 3 with the relations (33)-(35), after computations, we derive Using this fact in (40), we obtain The required inequality follows from (41) by using the fact ξ(ln f ) = 0. For the equality, from the leaving third term in r.h.s. of (37), we find σ(X, Y) has no components in µ for all X, Y tangent to M. Furthermore, from the leaving first term and vanishing seventh term in r.h.s. of (39) with the above fact that σ has no components in µ, we find Also, from the leaving fourth term in r.h.s. of (39) and the second term in r.h.s. of (40) with the fact that σ has no components in µ, we find From the hypothesis of the theorem And from the leaving second term and vanishing eighth term in r.h.s. of (39) with this fact that σ has no components in µ, we conclude that Furthermore, from the leaving ninth term in r.h.s. of (39) with the fact that σ has no components in µ, we obtain With the help of above facts and the fact that B is totally geodesic and N θ is totally umbilical in M [7,8], we conclude that N T and N ⊥ are totally geodesic submanifolds ofM, while N θ is a totally umbilical submanifold ofM. Furthermore, from (43) M is also D ⊕ D ⊥ -mixed totally geodesic. Moreover, from Theorem 2, M can never be a D ⊕ D θ -mixed totally geodesic. Hence, the theorem is proved completely.
As applications of Theorem 3, we have the following results. If dim N T = 0 in Theorem 3, then we have where 2s = dim N θ and ∇ ⊥ (ln f ) is the gradient of ln f . Moreover, if the equality holds in (47), then N ⊥ is a totally geodesic submanifold ofM and N θ is a totally umbilical submanifold ofM. Furthermore, M is minimal inM.
If N ⊥ = {0} in Theorem 3, then we state the following theorem.
where s = 1 2 dim N θ and ∇ T (ln f ) is the gradient of ln f . Moreover, if the equality holds in (48), then N T is a totally geodesic submanifold ofM and N θ is a totally umbilical submanifold ofM. Furthermore, M is never a mixed totally geodesic submanifold and hence M is not minimal inM.
Notice that Theorem 5 was proved in [27] which is a special case of Theorem 3. Also, in the above statement we improve the equality case of the main theorem of [27].
Theorem 5 implies the following theorem proved in [28]. Theorem 6. Let M = N T × f N ⊥ be a contact CR-warped product submanifold of a nearly cosymplectic manifoldM. Then, the second fundamental form σ of M satisfies where s = dim N ⊥ and ∇ T (ln f ) is the gradient of ln f . Moreover, if the equality holds in (49), then N T is a totally geodesic submanifold ofM and N ⊥ is a totally umbilical submanifold ofM. Furthermore, M is a minimal submanifold ofM.

Conclusions
In [27,28], we studied contact CR-warped product and semi-slant warped product submanifolds of nearly cosymplectic manifolds and obtained B.-Y. Chen's inequalities. As a generalised class of these submanifolds, in this paper, we study CR-slant warped products in nearly cosymplectic manifolds and establish a geometric inequality (Theorem 3) which generalizes Theorem 6 for contact CR-warped products, Theorem 5 for semi-slant warped products and Theorem 4 for hemi-slant warped products.