Bulletin of the Malaysian Mathematical Sciences Society Warped product submanifolds of nearly cosymplectic manifolds with slant immersion

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1 Bulletin of the Malaysian Mathematical Sciences Society Warped product submanifolds of nearly cosymplectic manifolds with slant immersion --Manuscript Draft-- Manuscript Number: Full Title: Article Type: Abstract: Keywords: Corresponding Author: Warped product submanifolds of nearly cosymplectic manifolds with slant immersion Original Article In this paper, we study non trivial warped product pseudo-slant submanifolds of nearly cosymplectic manifolds such that the spheric submanifold is slant. In the beginning, we prove some integrability and totally geodesic foliations results for the distributions of pseudo-slant submanifold and then we give an existence result for warped products isometrically immersed in a nearly cosymplectic manifolds. We develope a general sharp inequality, namely h β cos θ ln f for mixed geodesic warped products pseudo-slant submanifolds. The equality cases are also considered mean curvature; pseudo-slant; warped product; totally geodesic; nearly cosymplectic; mixed geodesic manifold. cenap ozel TURKEY Corresponding Author Secondary Information: Corresponding Author's Institution: Corresponding Author's Secondary Institution: First Author: cenap ozel First Author Secondary Information: Order of Authors: cenap ozel akram ali, Phd Student Wan Ainun Mior Othman, Profesor Order of Authors Secondary Information: Author Comments: I would like to submit our article to your qualified journal. best cenap ozel corresponding author Suggested Reviewers: Siraj Uddin siraj.ch@gmail.com expert Mustafa Kalafat kalafg@gmail.com Expert Muazzez Simsir fsimsir@gmail.com Expert Powered by Editorial Manager suleyman and dirik ProduXion Manager from Aries Systems Corporation slymndirik@gmail.com expert

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3 Manuscript Click here to download Manuscript: pseudo-slant.pdf Click here to view linked References Warped product submanifolds of nearly cosymplectic manifolds with slant immersion Akram Ali, Wan Ainun Mior Othman, Cenap Ozel Abstract In this paper, we study non trivial warped product pseudo-slant submanifolds of nearly cosymplectic manifolds such that the spheric submanifold is slant. In the beginning, we prove some integrability and totally geodesic foliations results for the distributions of pseudo-slant submanifold and then we give an existence result for warped products isometrically immersed in a nearly cosymplectic manifolds. We develope a general sharp inequality, namely h β cos θ ln f for mixed geodesic warped products pseudo-slant submanifolds. The equality cases are also considered. Introduction An almost contact manifold is an odd-dimensional manifold M which carries a field ϕ of endomorpisms of tangent space, vector field ξ, is called characteristic or Reeb vector field and -form η satisfying ϕ = I + η ξ, η(ξ) =, where I : T M T M is the identity mapping. Now, from definition it follows that ϕ ξ = 0 and η ϕ = 0, then the (, ) tensor field ϕ has constant rank n[]. An almost contact manifold ( M, ϕ, η, ξ) is said to be normal when the tensor field N ϕ = [ϕ, ϕ] + dη ξ vanishes identically, where [ϕ, ϕ] is the Nijenhuis tensor of ϕ. An almost contact metric structure (ϕ, ξ, η) is said to be a normal in the form of almost complex structure if almost complex structure J on a product manifold M R given by J(U, f d dt ) = (ϕu fξ, η(u) d dt ), where f is a smooth function on M R has no torsion, i.e., J is integrable. There always exits a Riemannian metric g on an almost contact manifold ( M, ϕ, η, ξ) which is satisfies the following compatibility condition: g(ϕu, ϕv ) = g(u, V ) η(u)η(v ), η(u) = g(u, ξ), (.) for all U, V Γ(T M). Then this metric g is called compatible metric and the manifold M endow together with structure (ϕ, η, ξ, g) is called an almost contact metric manifold. On the other hand, the geometry of warped product submanifolds have been studied actively since B. Y. Chen[]introduced the notion of CR-warped product submanifold in a Kaehler manifold. Basically, he studied different geometric aspect of warping functions in the form of some partial differential equations. In fact, different type warped product submanifolds of different kinds of structures are studied for last fourteen years. Recently, B. Sahin[] established general inequality for warped product pseudo-slant(as the name hemi-slant) This work is supported by the research grant RG-AFR and RG0-AFR 00 Mathematics Subject Classification. C0, C, B. Key words and Phrases. mean curvature, pseudo-slant, warped product, totally geodesic, nearly cosymplectic, mixed geodesic manifold.

4 isometrically immersed in Kaehler manifold for mixed geodesic. Recently, S. Uddin[] has obtained some existence results for warped product pseudo-slant submanifolds in terms of endomorphisms. In the present paper, we study characterization of non trivial warped product pseudo-slant submanifolds of the form M f M θ which are the natural extension of CR-warped product submanifolds. Every CR-warped product submanifold is a non trivial warped product pseudoslant submanifold of the forms M f M θ and M θ f M with slant angle θ = 0. First, we will consider non trivial warped product pseudo-slant submanifold M = M f M θ such that M θ and M are proper slant and anti-invariant submanifolds and then establish an inequality involving the second fundamental form, slant angle and warping functions. The paper is organized as follows: In section, we review some preliminary formulas and definitions. Section, is devoted to the study of pseudoslant submanifolds of nearly cosymplectic manifolds and some theorems on total manifolds are given. In section, we study warped product pseudo-slant submanifolds of a nearly cosymplectic manifold and obtain some results on its characterization. In section, we define orthonormal frames for warped product pseudo-slant subamnifolds and obtain general inequality for the second fundamental form in term of warping functions and slant immersion. Preliminaries As an immediate consequence of almost contact metric structure, i.e, g(ϕu, V ) = g(u, ϕv ) and the fundamental -form Φ is defined by Φ(U, V ) = g(u, ϕv ). An almost contact metric manifold such that both η and Φ are closed are called almost cosymplectic manifold and those for which dη = Φ are called contact metric manifolds. Finally, a normal almost cosymplectic manifold is called cosymplectic manifold and a normal contact manifold is called Sasakian manifold. The cosymplectic and Sasakian manifolds in term of covariant derivative of ϕ can be expressed respectively as: ( U ϕ)v = 0, and ( U ϕ)v = g(u, V )ξ η(v )U, for any U, V Γ(T M). It should be noted that both in cosymplectic and Sasakian manifolds ξ is killing vector field. On the other hand, the Sasakian and the cosymplectic manifolds represent the two external cases of the larger class of quasi-sasakian manifolds. An almost contact metric structure (ϕ, η, ξ) is said to be nearly cosymplectic if ϕ is killing, i.e., if ( U ϕ)u = 0 or equivalently ( U ϕ)v + ( V ϕ)u = 0, (.) for any U, V tangent to M, where is the Riemannian connection on the metric g on M. A manifold M endow with this almost contact metric structure(ϕ, η, ξ) is called nearly cosymplectic manifold. The structure is said to be a closely cosymplectic, if ϕ is killing and η closed. The covariant derivative of tensor field ϕ is defined as: ( U ϕv ) = U ϕv ϕ U V, for any U, V Γ(T M) and is the Riemannian connection on M.

5 Let M be a Riemannian manifold isometrically immersed in almost contact metric manifold M and denote by same symbol g for the Riemannian metric induced on M. Let Γ(T M) and Γ(T M) be the Lie algebra of vector fields tangent to M and normal to M, respectively and the induced connection on T M. Denote by F(M) the algebra of smooth functions on M and by Γ(T M) the F(M)-module of smooth sections of T M over M. Denote by the Levi-Civita connection of M then the Gauss and Weingarten formulas are given by U V = U V + h(u, V ) (.) U N = A N U + U N, (.) for each U, V Γ(T M) and N Γ(T M), where h and A N are the second fundamental form and the shape operator (corresponding to the normal vector field N) respectively for the immersion of M into M. They are related as Now, for any U Γ(T M), we write g(h(u, V ), N) = g(a N U, V ). (.) ϕu = P U + F U, (.) where P U and F U are tangential and normal components of ϕu, respectively. Similarly for any N Γ(T M), we have ϕn = tn + fn, (.) where tn (resp. f N) are tangential (resp. normal) components of ϕn. A submanifold M is said to be totally geodesic and totally umbilical, if h(u, V ) = 0 and h(u, V ) = g(u, V )H, respectively. Let M be a submanifold tangent to the structure vector field ξ isometrically immersed into an almost contact metric manifold M. Then M is said to be contact CR-submanifold if there exists a pair of orthogonal distribution D : p D p and D : p D, for all p M such that (i) T M = D D < ξ >, where < ξ > is the -dimensional distribution spanned by the structure vector field ξ. (ii) D is invariant, i.e., ϕd = D. (iii) D is anti-invariant, i.e., ϕd T M. Invariant and anti-invariant submanifolds are special cases of contact CR-submanifold. If we denote the dimensions of the distributions D and D by d and d, respectively, then M is invariant(resp. anti-invariant) if d = 0(resp.d = 0). There is another class of submanifolds which is called the slant submanifold. For each non zero vector U tangent to M at p, such that U is not proportional to ξ p, we denote by 0 θ(u) π/, the angle between ϕu and T p M is called the Wirtinger angle. If the angle θ(u) is constant for all U T P M < ξ > and p M, then M is said to be a slant submanifold[] and the angle θ is called slant angle of M. Obviously if θ = 0, M is invariant and

6 if θ = π/, M is anti-invariant submanifold. A slant submanifold is said to be proper slant if it is neither invariant nor anti-invariant. In an almost contact metric manifold. In fact, J. L Cabrerizo []obtained the following theorem. Theorem.. [] Let M be a submanifold of an almost contact metric manifold M such that ξ T M. Then M is slant if and only if there exists a constant λ [0, ] such that P = λ( I + η ξ). (.) Furthermore, in such a case, if θ is slant angle, then it satisfies that λ = cos θ. Hence, for a slant submanifold M of an almost contact metric manifold M, the following relations are consequences of the above theorem. g(p U, P V ) = cos θ{g(u, V ) η(u)η(v )}. (.) g(f U, F V ) = sin θ{g(u, V ) η(u)η(v )}. (.) for any U, V Γ(T M). Now we give the another characterization from the consequence of the theorem., Theorem.. Let M be a slant submanifold of an almost contact metric manifold M such that ξ T M. Then (a) tf X = sin θ(x η(x)ξ) and (b) ff X = F P X, (.0) for any X Γ(T M). Proof. From relations (.)-(.) and Theorem., we can derive the proof the theorem. Now, let {e, e...e n } be an orthonormal basis of tangent space T M and e r belong to the orthonormal basis {e n+, e n+,...e m } of the normal bundle T M, then we define h r ij = g(h(e i, e j ), e r ) and h = n g(h(e i, e j ), h(e i, e j )). (.) i,j= As a consequence for a differentiable function λ on M, we have λ = n (e i (λ)), (.) where gradient gradλ is defined by g( λ, X) = Xλ, for any X Γ(T M). i= Pseudo-slant submanifolds In this section, we define pseudo-slant submanifolds of almost contact manifolds by using the slant distribution given by []. However, pseudo-slant submanifolds

7 were defined by Carriazo[] under the name of anti-slant submanifolds as a particular class of bi-slant submanifolds. We investigate the geometry of leaves of distributions involving in the definition and obtain a decomposition theorem for the total manifold and also obtain a necessary and sufficient condition for such submersions to be totally geodesic foliations for later use in characaterization theorem. The definition of pseudo-slant is as follows: Definition.. A submanifold M of an almost contact manifold M is said to be pseudo-slant submanifold, if there exist two orthogonal distributions D and D θ such that (i) T M = D θ D ξ, where < ξ > is -dimensional distribution spanned by ξ. (ii) D is an anti invariant distribution under ϕ i.e., ϕd T M. (iii) D θ is slant distribution with slant angle θ 0, π. Let m and m be dimensions of distributions D and D θ, respectively. If m =0, then M is anti invariant submanifold. If m =0 and θ = 0, then M is invariant submanifold. If m =0 and θ 0, π, then M is proper-slant submanifold, or if θ = π, then M is anti invariant submanifold and if θ = 0, then M is semi-invariant submanifold. If µ is an invariant subspace of normal bundle T M, then in case of pseudo-slant submanifold, the normal bundle T M can be decomposed as follows: T M = F D F D θ µ, (.) where µ is even dimensional invariant sub bundle of T M. A pseudo-slant submanifold is said to be mixed totally geodesic if h(x, Z) = 0, for all X Γ(D ξ) and Z Γ(D θ ). Theorem.. Let M be a pseudo-slant submanifold of a nearly cosymplectic manifold M. Then the distribution D ξ defines as totally geodesic foliation in M if and only if g(h(z, W ), F P Z) = {g(a ϕzw, P X) + g(a ϕw Z, P X)} for all Z, W Γ(D ξ) and X Γ(D θ ). Proof. From the definition of Riemannian metric for almost contact metric manifold and fact that ξ is orthogonal to D θ such that η(x) = 0, then we have g( Z W, X) = g(ϕ Z W, ϕx), for any Z, W Γ(D ξ) and X Γ(D θ ). Using (.), we obtain g( Z W, X) = g(ϕ Z W, P X) + g(ϕ Z W, F X). Then by definition of covariant derivative of structure tensor field ϕ and property of Riemannian metric, we derive g( Z W, X) = g( Z ϕw, P X) g(( Z ϕ)w, P X) g( Z W, ϕf X).

8 Thus by structure equation of nearly cosymplectic manifold and the relations (.), (.), we arrive at g( Z W, X) = g(( W ϕ)z, P X) g( Z W, tf X) g( Z W, ff X) g(a ϕw Z, P X). Again from covariant derivative of structure tensor field and Theorem., we obtain g( Z W, X) = g( W ϕz, P X) g( W Z, ϕp X) + sin θg( Z W, X) +g( Z W, F P X) g(a ϕw Z, P X). Now, from relations (.) and (.), we get cos θg( Z W, X) = g(h(z, W ), F P X) g(a ϕw Z, P X) g(a ϕz W, P X) +g( W Z, P X). Thus by (.) the above equation can be written as cos θg( Z W, X) = g(h(z, W ), F P Z) g(a ϕz W, P X) g(a ϕw Z, P X) cos θg( W Z, X). Assertion follows from last relations. Theorem.. Let M be a pseudo-slant submanifold of a nearly cosymplectic manifold M. Then the distribution D θ is integrable if and only if g( X Y, Z) = sec θ{g(h(x, P Y ) + h(y, P X), ϕz) for any Z Γ(D ξ) and X, Y Γ(D θ ). g(h(x, Z), F P Y ) g(h(y, Z), F P X)} Proof. By using the properties of symmetric torsion and Riemannian metric g, we have g([x, Y ], Z) = g(ϕ X Y, ϕz) + η(z)g( X Y, ξ) g( Y X, Z). From the covariant derivative of endomorphism varphi in the first term and using. [see 0] in the second term, we get g([x, Y ], Z) = g( X φy, ϕz) g(( X ϕ)y, ϕz) g( Y X, Z). Again from covariant derivative and nearly cosymplectic structure (.), we obtain g([x, Y ], Z) = g( X P Y, ϕz) + g( X F Y, ϕz) + g(( Y ϕ)x, ϕz) g( Y X, Z). Since, F Y and ϕz are perpendicular then using the property of Riemannian connection and the relation (.), we arrive at g([x, Y ], Z) = g(h(x, P Y ), ϕz) g(f Y, X ϕz)+g( Y φx, ϕz) g( Y X, Z).

9 Then from (.) and property of Riemannian metric, we can modified as g([x, Y ], Z) = g(h(x, P Y ), ϕz) g(f Y, ( X ϕ)z) + g(ϕf Y, X Z) + g( Y P X, ϕz) + g( Y F X, ϕz) g( Y X, Z). In the second term of above equation can be decomposed by the property of Q [see ] and using the relations (.), (.), we derive g([x, Y ], Z) = g(h(x, P Y ), ϕz)+g(h(p X, Y ), ϕz)+g(f Y, Q X Z)+g(tF Y, X Z) +g(ff Y, X Z) g(f X, Y φz) g( Y X, Z). From (.) and Theorem., then we can write the above equation g([x, Y ], Z) = g(h(x, P Y ), ϕz)+g(h(p X, Y ), ϕz)+g(ϕy, Q X Z) sin θg(y, X Z) g(h(x, Z), F P Y ) g(f X, ( Y ϕ)z)+g(ϕf X, Y Z) g( Y X, Z). Moreover, the fact that Q is a normal part of structure equation then g(ϕy, Q X Z) = g(y, ϕq X Z) = 0, and (.), we derive g([x, Y ], Z) = g(h(x, P Y ), ϕz) + g(h(p X, Y ), ϕz) sin θg(y, X Z) g(h(x, Z), F P Y ) + g(tf X, Y Z) + g(ff X, Y Z) g( Y X, Z). Thus by Theorem. and the fact that X, Y are orthogonal vector fields to Z, then finally, we arrive at g([x, Y ], Z) = g(h(x, P Y )+h(p X, Y ), ϕz)+sin θg(z, X Y ) g(h(x, Z), F P Y ) + sin θg(z, Y X) g(f P X, h(y, Z)) g( Y X, Z). Since, D θ is integrable, we can modified as cos θg(z, X Y ) = g(h(x, P Y ) + h(p X, Y ), ϕz) g(h(x, Z), F P Y ) Which gives our assertion. g(f T X, h(y, Z)). Warped product submanifolds R. L. Bishop and B. O Neill[]initiated the notion of warped product manifolds to construct examples of Riemannian manifolds with negative curvature. These manifolds are natural generalizations of Riemannian products manifolds. They defined these manifolds as: Let (M, g ) and (M, g ) be two Riemannian manifolds and f : M (0, ), a positive differentiable function on M. Consider the product manifold M M with it s canonical projections π : M M M, π : M M M and the projection maps given by π (p, q) = p. and π (p, q) = q for every t = (p, q) M M. The warped

10 product M = M f M is the product manifold M M equipped with the Riemannian structure such that U = π (U) + f (π (p)) π (U) for any tangent vector U Γ(T t M), where is the symbol of tangent maps. Thus we have g = g + f g. The function f is called the warping function on M. It was proved in [] by the following Theorem Lemma..[] Let M = M f M be a warped product manifold. if for any X, Y Γ(T M ) and Z, W Γ(T M ), then (i) X Y Γ(T M ) (ii) Z X = X Z = (X ln f)z (iii) Z W = ZW g(z, W ) ln f where and denote the Levi-Civita connection on M and M respectively. On the other hand, ln f is the gradient of ln f is defined as g( ln f, U) = U ln f. A warped product manifold M = M f M is said to be trivial if the warping function f is constant. If M = M f M warped product manifold then M is totally geodesic and M is totally umbilical submanifolds of M respectively []. We give some preparatory lemmas. Lemma.. Let M = M f M θ be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold M such that structure vector field ξ is tangent to M. Then g(h(x, Z), F P X) = (Z ln f)cos θ X +g(h(x, P X), ϕz)+g(h(z, P X), F X), for any X Γ(T M θ ) and Z Γ(T M ). Proof. Let M = M f M θ be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold M, then by (.), we have g(h(x, Z), F P X) = g( Z X, F P X). From (.), (.), and the fact that ξ is tangent to M, we obtain g(h(x, Z), F P X) = g(ϕ Z X, P X) + cos θg( Z X, X). Then by the definition of covariant derivative of Riemannian connection and lemma.(ii), we can derive g(h(x, Z), F P X) = g(( Z ϕ)x, P X) g( Z φx, P X) + cos θ(z ln f) X. Thus by using the structure equation of nearly cosymplectic and the relation (.), then above equation can written as g(h(x, Z), F P X) = g(( X ϕ)z, P X) g( Z P X, P X) g( Z F X, P X) +cos θ(x ln f) Z.

11 Using the lemma.(ii) and relations (.), (.), we arrive at g(h(x, Z), F P X) = g( X ϕz, P X) g( X Z, ϕp X) cos θ(z ln f) X + g(h(z, P X), F X) + cos θ(z ln f) X. From (.), (.) and the relation (.) for slant submanifold, we obtain that g(h(x, Z), F P X) = g(a ϕz X, P X) + cos θg( X Z, X) g( X Z, F P X) + g(h(z, P X), F X). Finally, by lemma.(ii) and the relation (.), we get g(h(x, Z), F P X) = (Z ln f)cos θ X +g(h(x, P X), ϕz)+g(h(z, P X), F X), which gives our assertion. Lemma.. Let M = M f M θ be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold M, where M θ and M are proper slant and invariant submanifolds, respectively. Then g(h(x, Z), F P X) = g(h(x, P X), ϕz) g(h(z, P X), F X), for any X Γ(T M θ ) and Z Γ(T M ). Proof. From the relations (.), (.) and the fact that ξ is tangent to M, we can write g(h(z, P X), F X) = g(ϕ Z P X, X) g( Z P X, P X), for all X Γ(T M θ ) and Z Γ(T M ). lemma.(ii), (.), we can modified as Then covariant derivative of ϕ and g(h(z, P X), F X) = g(( Z ϕ)p X, X) g( Z ϕp X, X) cos θ(z ln f) X Thus by the relations (.), (.), (.) and the structure equation of nearly cosymplectic manifold (.), we arrive at g(h(z, P X), F X) = g(h(x, Z), F P X) cos θ(z ln f) X g(( P X ϕ)z, X) + cos θg( Z X, X). Using the Lemma.(ii) and covariant derivative property, we obtain g(h(z, P X), F X) = g(h(x, Z), F P X) g( P X ϕz, X) g( P X Z, ϕx). Then from (.), and (.), we derive g(h(z, P X), F X) = g(h(x, Z), F P X) + g(h(x, P X), ϕz) g( P X Z, P X). Again from lemma.(ii) and relation (.), then last equation can be written as g(h(z, P X), F X) = g(h(x, Z), F P X) (Z ln f)cos θ X

12 g(h(x, P X), ϕz). (.) Now, interchanging X by P X and using the relation (.) for slant submanifold, we obtain cos θg(h(x, Z), F P X) = (Z ln f)cos X cos θg(h(z, P X), F X) Which is implies that cos θg(h(p X, X), ϕz). g(h(x, Z), F P X) = (Z ln f)cos X + g(h(z, P X), F X) Then from (.) and (.), we can modified as + g(h(p X, X), ϕz). (.) g(h(z, P X), F X) = g(h(x, P X), ϕz) g(h(x, Z), F P X). Which proves our assertion. Lemma.. Let M = M f M θ be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold M. Then g(h(x, Z), F P X) g(h(p X, X), ϕz) = cos θ(zλ) X, for any X Γ(T M θ ) and Z Γ(T M ). Proof. From lemma. and lemma., we can derive proof the lemma. Theorem.. Let M be a pseudo-slant subamnifold of a nearly cosymplectic manifold M such that the slant distribution is integrable. Then M is locally a warped product of proper slant and anti-invariant submanifolds if and only if A F P X Z A ϕz P X = cos θ(zλ)x, (.) for any Z Γ(D ξ) and any X Γ(D θ ) and for a differentiable function λ on M such that Y λ = 0, for any Y Γ(D θ ). Proof. Let M = M f M θ be a non-trivial warped product proper pseudoslant submanifold of nearly cosymplectic manifold M such that M θ and M are proper slant and anti-invariant submanifolds, then direct part follows from the lemma.. Conversely, suppose that M be a proper pseudo-slant submanifold of a nearly cosymplectic manifold M with (.) holds. Let us take inner product in (.) with W Γ(D ξ) and the fact that X and W are orthogonal then we get g(h(z, W ), F P X) = g(h(p X, W ), ϕz). (.) Thus by polarization identity, we arrive at g(h(z, W ), F P X) = g(h(p X, Z), ϕw ). (.) 0

13 From (.) and (.), we obtain g(h(z, W ), F P X) = g(h(p X, W ), ϕz) + g(h(p X, Z), ϕw ). (.) Then from (.) and Theorem., we conclude that (D ξ) is totally geodesic foliation in M, i.e., it s leaves are totally geodesic in M of the immersion M. So far as the slant distribution D θ is concerned and it is integrable by hypothesis, then by Theorem., the distribution D θ is integrable if and only if g( X Y, Z) = sec θ{g(h(x, P Y ) + h(y, P X), ϕz) g(h(x, Z), F P Y ) g(h(y, Z), F P X)}, for any X, Y Γ(D θ ) and for any Z Γ(D ξ). simplified as Above equation can be g( X Y, Z) = sec θ{g(a F P Y Z A ϕz P Y, X) + g(a F P X Z A ϕz P X, Y )}. Moreover, the distribution D θ is integrable then we can consider M θ is a leaf of D θ or M θ be a integral manifold of D θ and h θ be a second fundamental form of M θ in M of the immersion M. Thus by the hypothesis of Theorem, we arrive at g(h θ (X, Y ), Z) = sec θ{ cos θ(zλ)g(y, X) + cos θ(zλ)g(x, Y )}. Which implies that g(h θ (X, Y ), Z) = (Zλ)g(X, Y ). Thus from the definition of gradient, we derive From the last relation, we obtain that g(h θ (X, Y ), Z) = g(x, Y )g( λ, Z). h θ (X, Y ) = g(x, Y ) λ. (.) The equation (.) shows that the leaves of D θ are totally umbilical in M with mean curvature vector H θ = λ. Moreover, the condition Xλ = 0, for any X Γ(D ) implies that the leaves of D θ are extrinsic spheres in M, that is the integral manifold M θ of D θ is umbilical and it s mean curvature vector field is non zero and parallel along M θ. Hence by the result of Hiepko[0] then M is warped product submanifold of integral manifolds M θ and M of D θ and D, respectively. This completes proof of the Theorem. Note:- We have proved a characterization by theorem.. Now, we have seen that from lemma., if we consider M be a mixed totally geodesic warped product submanifold, then we get a non-trivial warped product submanifold. Next section, we obtain an inequality in this case.

14 Inequality for warped products pseudo-slant submanifolds of the form M f M θ In this section, we obtain a geometric inequality for warped product pseudoslant submanifold in term of second fundamental form and warping function with in fact of mixed geodesic submanifolds. We derive some lemmas and define the orthonormal frame for later use Lemma.. Let M = M f M θ be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold M, where M θ and M are proper slant and anti- invariant submanifolds, respectively. Then (i) g(h(z, Z), F X) = g(h(z, X), ϕz), (ii) g(h(z, Z), F P X) = g(h(z, P X), ϕz), for any X Γ(T M θ ) and Z Γ(T M ). Proof. From relations (.) and (.), we obtain g(h(z, Z), F X) = g( Z Z, ϕx) g( Z Z, P X), for any X Γ(T M θ ) and Z Γ(T M ). Thus by the fact that Z and P X are orthogonal then from property of Riemannian connection for orthogonal vector fields, we arrive at g(h(z, Z), F X) = g(ϕ Z Z, X) + g( Z P X, Z). Using covariant derivative of ϕ and relation (.), we get g(h(z, Z), F X) = g(( Z ϕ)z, X) g( Z ϕz, X) + g( Z P X, Z). From structure equation of nearly cosymplectic manifold and lemma.(ii), we derive g(h(z, Z), F X) = g(a ϕz Z, X) + (Z ln f)g(p X, Z). Which is implies that g(h(z, Z), F X) = g(a ϕz Z, X). This is the first part of the lemma and last part can be obtain by interchanging X by P X. This completes the proof. Lemma.. Let M = M f M θ be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold M. Then (iii) g(h(x, X), ϕz) = g(h(z, X), F X), (iv) g(h(p X, P X), ϕz) = g(h(z, P X), F P X), for any X Γ(T M θ ) and Z Γ(T M ).

15 Proof. For any X Γ(T M θ ) and Z Γ(T M ), we have g(h(x, X), ϕz) = g( X X, ϕz) = g(ϕ X X, Z). From the definition of covariant derivative of tensor field ϕ, we get g(h(x, X), ϕz) = g( X ϕ)x, Z) g( X ϕx, Z). Then by nearly cosymplectic manifold structure and (.), we obtain g(h(x, X), ϕz) = g( X Z, P X) g( X F X, Z). Thus from lemma.(ii) and relation (.), then above equation can be written as g(h(x, X), ϕz) = (Z ln f)g(x, P X) + g(a F X X, Z). Hence, X and P X are orthogonal vector fields and by relation (.), we arrive at g(h(x, X), ϕz) = g(h(x, Z), F X). (.) Which is the first result of the lemma. Interchanging X by T X in (.) then we get the last result of the lemma, which complete proof of the lemma. Let M = M f M θ be an m + -dimensional warped product pseudoslant submanifold of n + -dimensional nearly cosymplectic manifold M with M θ of dimension d = β and M of dimension α +, where M θ and M are the integral submanifolds of D θ and D respectively. We consider the {e, e...e α, e α+ = ξ, } and {e α+ = e,...e α+β+ = e β, e α+β+ = e β+ = secθp e,...e α++β = e β = secθp e β } are orthonormal frames of D and D θ respectively. Thus the orthonormal frames of the normal sub bundles φd, F D θ and µ, respectively are {e m+ = ē = φe,...e m+α = ē α = φe α }, {e m+α+ = ē α+ = ẽ = cscθf e,...e m+α+β = ē α+β = ẽ β = cscθf e β, e m+α+β+ = ē α+β+ = ẽ β+ = cscθsecθf T e,...e m+α+β = ē α+β = ẽ β = cscθsecθf T e β } and {e m = ē n,...e n+ = ē (n m+) }. Theorem.. Let M = M f M θ be a m + -dimensional mixed totally geodesic warped product pseudo-slant submanifold of n + -dimensional nearly cosymplectic manifold M, where M is anti-invariant submanifold of dimension α + and M θ is a proper slant submanifold of dimension d of M. Then (i) The squared norm of the second fundamental form of M is given by h β cos θ ln f. (.) (ii) The equality holds in (.), if h(d θ, D θ ) ϕd and h(d, D ) F D θ. Proof. By the definition of second fundamental form h = h(d θ, D θ ) + h(d, D ) + h(d θ, D ). But, M is mixed totally geodesic then, we have h = h(d, D ) + h(d θ, D θ )

16 Thus by the relations(.), we obtain h = n+ α+ l=m+ r,k= g(h(e r, e k ), e l ) + n+ β l=m+ i,j= g(h(e i, e j ), e l ) The above equation can be expressed as in the components of ϕd, F D θ and ν, then we derive h = α α+ l= r,k= (n m+) + + l=m β+α l=α+ i,j= g(h(e r, e k ), ē l ) + β α+ r,k= β+α α+ l=α+ r,k= g(h(e r, e k ), ē l ) + α g(h(e r, e k ), ē l ) (.) β l= i,j= (n m+) g(h(e i, e j ), ē l ) + l=m β i,j= Leaving all the terms except second and fourth, then we get h β α+ l= r,k= g(h(e r, e k ), ẽ l ) + Then using adapted frame for F D θ, we derive h csc θ + α β l= i,j= β α+ g(h(e r, e r ), F e j ) + csc θsec θ j= r= α β l= i,j= g(h(e i, e j ), ē l ). g(h(e i, e j ), ē l ) g(h(e i, e j ), ē l ). g(h(e i, e j ), ē l ). β α+ g(h(e r, e r ), F T e j ) j= r= After using the lemma. for mixed totally geodesic, then first and second term should be zero, we have h α β l= r,k= g(h(e r, e k), ē l ). Using another adapted frame for D θ, then we derive h α β i= r,k= + sec α g(h(e r, e k), ē i ) + sec θ β i= r,k= α β i= r,k= g(h(e r, P e k), ē i ) + sec θ g(h(p e r, e k), ē i ). p β i= r,k= g(h(p e r, P e k), ē i ). Then for a totally mixed geodesic first and last terms of right hand side in above equation vanishes identically by using the lemma., we obtain h sec θ α i= r= β g(h(p e r, e r), ē i ).

17 By the lemma., for mixed totally geodesic, we arrive at h cos θ α j= r= β (ē j ln f) g(e r, e r). (.) Now, we add and subtract the same term ξ ln f in (.), we get h α+ cos θ j= r= β (ē j ln f) g(e r, e r) cos θ β (ξ ln f) g(e r, e r). Since warped product submanifold of a nearly cosymplectic manifold, we know that ξ ln f = 0, Thus above equation gives h β cos θ ln f. If the equality holds, from the leaving terms in (.), we obtain the following condition from the second and third terms, which means that r= g(h(d, D ), ϕd ) = 0, g(h(d, D ), ν) = 0, h(d, D ) φd, h(d, D ) ν = h(d, D ) F D θ. Similarly, from fifth and sixth terms in (.), we derive which implies that g(h(d θ, D θ ), F D θ ) = 0, g(h(d θ, D θ ), ν) = 0, h(d θ, D θ ) F D, h(d θ, D θ ) ν = h(d θ, D θ ) φd. So the equality case hold. Which completes proof of the theorem. Acknowledgments:- The authors would like express their appreciation to the referees for their comments and valauble suggestions. References [] R. L. Bishop and B. O Neil, Manifolds of negative curvature, Trans. Amer. Math. Soc. (), -. [] D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Vol. 0. Springer-Verlag, New York,. [] D. E. Blair, Riemannian geometry of contact and symplectic manifold, Progress in Mathematics vol. 0, Birkhauser. Boston Inc., Boston, MA 00. [] D. E. Blair and D. K. Showers, Almost contact manifolds with Killing structures tensors II, J. Differ. Geom. (), -. MR 0.

18 [] A. Carriazo, Bi-slant immersions, in : Proc. ICARAMS 000, Kharagpur, India, 000, -. [] A. Carriazo, New developments in slant submanifolds, Narosa Publishing House, New Delhi, 00. [] J. L. Cabrerizo, A. Carriazo, L.M. Fernandez and M. Fernandez, Slant submanifolds in Sasakian manifolds, Glasgow Math. J. (000), -. [] B. Y. Chen, Geometry of warped product CR-submanifold in Kaehler manifolds, Monatsh. Math. (00), no.0, -. [] B.Y. Chen, Slant immersions, Bull. Austral. Math. Soc. (0), -. [0] S. Dirik and M. Atceken, Pseudo-slant submanifolds of a nearly cosymplectic manifold, Turkish Journal of Mathematics and Computer Science. [] S. Hiepko, Eine inner kennzeichungder verzerrten produckt, Math, Ann. (), 0-. [] A. Lotta, Slant submanifolds in contact geometry, Bull. Math. Soc. Roumanie (), -. [] B. Sahin, Warped product submanifolds of a Kaehler manifolds with slant factor, Ann. Pol. Math. (00), 0-. [] S. Uddin, B. R. Wong and A. Mustafa, Warped product pseudo-slant submanifolds of a nearly cosymplectic Manifold, Abstract and Applied Analysis, Volume 0 (0), Article ID 00, pages [] S. Uddin, A Mustafa, B. R. Wong, C. Ozel, A geometric inequality for warped product semi-slant submanifolds of nearly cosymplectic manifolds, Revista Dela Union Matematica Argentina,Vol., No., 0, Pages. Author s address: Akram Ali Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 00 Kuala Lumpur, MALAYSIA akramali@gmail.com Wan Ainun Mior Othman Institute of Mathematical Sciences, Faculty of Science,University of Malaya, 00 Kuala Lumpur, MALAYSIA wanainun@um.edu.my. Cenap Ozel Department of Mathematics, Dokuz Eylul University, Tinaztepe Campus Buca, Izmir,Turkey 0 cenap.ozel@gmail.com