An inequality for warped product pseudo-slant submanifolds of nearly cosymplectic manifolds

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1 Al-Solamy Journal of Inequalities and Applications (2015) 2015:306 DOI /s y R E S E A R C H Open Access An inequality for warped product pseudo-slant submanifolds of nearly cosymplectic manifolds Falleh R Al-Solamy * * Correspondence: falleh@hotmail.com Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia Abstract Warped product submanifolds of nearly cosymplectic manifolds were studied in Uddin et al. (Math. Probl. Eng. 2011, doi: /2011/230374), UddinandKhan (J. Inequal. Appl. 2012:304, 2012) and Uddin et al. (Rev. Unión Mat. Argent. 55:55-69, 2014). In this paper, we study warped product submanifolds of nearly cosymplectic manifolds in which the base manifold is slant and thus we derive a sharp relation for the squared norm of the second fundamental form. The equality case is also considered. MSC: 53C40; 53C42; 53C15 Keywords: slant submanifold; pseudo-slant submanifold; nearly cosymplectic manifold; warped products 1 Introduction The almost contact manifolds with Killing structures tensors were defined in [1]as nearly cosymplectic manifolds. Later, these manifolds were studied by Blair and Showers from the topological point of view [2]. A totally geodesic hypersurface S 5 of a 6-dimensional sphere S 6 is a nearly cosymplectic manifold. A normal nearly cosymplectic manifold is cosymplectic (see [3]). On the other hand, pseudo-slant submanifolds of almost contact metric manifolds were studied by Carriazo [4] under the name of anti-slant submanifolds. Later on, Sahin studied these submanifolds for their warped products [5]. Recently, Uddin et al. studied warped product semi-invariant and semi-slant submanifolds of nearly cosymplectic manifolds [6 8]. In this paper, we study the warped product pseudo-slant submanifolds of the type N θ f N of a nearly cosymplectic manifold, where N and N θ are anti-invariant and proper slant submanifolds of a nearly cosymplectic manifold, respectively. We derive an inequality for the second fundamental form of such warped product immersions in terms of the warping function and the slant angle. The equality case is also discussed. 2 Preliminaries Let M be a (2n + 1)-dimensional C manifold with almost contact structure (ϕ, ξ, η) i.e., a (1, 1) tensor field ϕ,avectorfieldξ and a 1-form η on M such that 2015 Al-Solamy. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Al-Solamy Journal of Inequalities and Applications (2015) 2015:306 Page 2 of 9 ϕ 2 = I + η ξ, ϕξ =0, η ϕ =0, η(ξ) = 1. (2.1) There always exists a Riemannian metric g on an almost contact manifold M satisfying the following compatibility condition: η(x)=g(x, ξ), g(ϕx, ϕy )=g(x, Y ) η(x)η(y ), (2.2) where X and Y are vector fields on M [2]. An almost contact structure (ϕ, ξ, η) issaidtobenearly cosymplectic if ϕ is Killing, i.e., if ( X ϕ)y +( Y ϕ)x =0, (2.3) or any X, Y tangent to M, where denotes the Riemannian connection of the metric g. Equation(2.3) isequivalentto( X ϕ)x =0,foreachX tangent to M. Anormal nearly cosymplectic structure is cosymplectic. It is well known that an almost contact metric manifold is cosymplectic if and only if ϕ vanishes identically, i.e.,( X ϕ)y =0and X ξ =0. On a nearly cosymplectic manifold the structure vector field ξ is Killing [9], that is, g( Y ξ, Z)+g( Z ξ, Y )=0 (2.4) for any Y, Z tangent to M. Let M be submanifold of an almost contact metric manifold M with induced metric g and let and be the induced connections on the tangent bundle TM and the normal bundle T M of M, respectively.denotebyf(m) the algebra of smooth functions on M and by Ɣ(TM)theF(M)-module of smooth sections of TM over M. Then the Gauss and Weingarten formulas are given by X Y = X Y + h(x, Y ), (2.5) X N = A N X + X N, (2.6) for each X, Y Ɣ(TM) andn Ɣ(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 g ( h(x, Y ), N ) = g(a N X, Y ), (2.7) where g denotes the Riemannian metric on M as well as the one induced on M.Themean curvature vector H of M is given by H = 1 m m i=1 h(e i, e i ), where n is the dimension of M and {e 1, e 2,...,e m } is a local orthonormal frame of vector fields on M. A submanifold M of an almost contact metric manifold M is said to be totally umbilical if the second fundamental form satisfies h(x, Y )=g(x, Y )H, for all X, Y Ɣ(TM). The submanifold M is totally geodesic if h(x, Y )=0,forallX, Y Ɣ(TM) and minimal if H =0.

3 Al-Solamy Journal of Inequalities and Applications (2015) 2015:306 Page 3 of 9 Now, let {e 1,...,e m } be an orthonormal basis of tangent space TM and e r belong to the orthonormal basis {e m+1,...,e 2n+1 } of the normal bundle T M,weput h r ij = g( h(e i, e j ), e r ) m and h 2 = g ( h(e i, e j ), h(e i, e j ) ). (2.8) i,j=1 For a differentiable function ϕ on M, the gradient ϕ is defined by g( ϕ, X)=Xϕ (2.9) for any X Ɣ(TM). As a consequence, we have ϕ 2 = m ( ei (ϕ) ) 2. (2.10) i=1 For any X Ɣ(TM), we write ϕx = PX + FX, (2.11) where PX is the tangential component and FX is the normal component of ϕx.asubmanifold M of an almost contact metric manifold M is said to be invariant if F is identically zero, that is, ϕx Ɣ(TM)andanti-invariant if P is identically zero, that is, ϕx Ɣ(T M), for any X Ɣ(TM). Let M be a submanifold tangent to the structure vector field ξ isometrically immersed into an almost contact metric manifold M.ThenM is said to be a contact CR-submanifold if there exists a pair of orthogonal distributions D : p D p and D : p Dp, p M such that: (i) TM = D D ξ,where ξ is the 1-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 the special cases of a contact CR-submanifold. If we denote the dimensions of the distributions D and D by d 1 and d 2,respectively. Then M is invariant (resp. anti-invariant)ifd 2 = 0 (resp. d 1 =0). There is another class of submanifolds that is called the slant submanifold. For each non-zero vector X tangent to M at x, suchthatx is not proportional to ξ x,wedenoteby 0 θ(x) π 2,theanglebetweenϕX and T xm is called the Wirtinger angle. Iftheangle θ(x) is constant for all nonzero X T x M ξ x and x M, thenm is said to be a slant submanifold [10] and the angle θ is the slant angle of M. Obviously if θ =0,M is invariant and if θ = π, M is an anti-invariant submanifold. A slant submanifold is said to be proper 2 slant if it is neither invariant nor anti-invariant. We recall the following result for a slant submanifold of an almost contact metric manifold. Theorem 2.1 [10] Let M be a submanifold of an almost contact metric manifold M, such that ξ is tangent to M. Then M is slant if and only if there exists a constant λ [0, 1] such

4 Al-Solamy Journal of Inequalities and Applications (2015) 2015:306 Page 4 of 9 that P 2 = λ( I + η ξ). (2.12) Furthermore, if θ is slant angle of M, then λ = cos 2 θ. The following relations are straightforward consequences of (2.12): g(px, PY )=cos 2 θ ( g(x, Y ) η(y )η(x) ), (2.13) g(fx, FY)=sin 2 θ ( g(x, Y ) η(y )η(x) ), (2.14) for all X, Y Ɣ(TM). Now, we give the brief introduction of pseudo-slant submanifolds introduced by Carriazo in [4] underthenameofanti-slant submanifolds, which are the generalization of contact CR-submanifolds and slant submanifolds [10]. He defined these submanifolds as follows. Definition 2.1 AsubmanifoldM of an almost contact metric manifold M is said to be a pseudo-slant submanifold if there exists a pair of orthogonal distributions D and D θ on M such that: (i) TM admits the orthogonal direct decomposition TM = D D θ ξ. (ii) The distribution D is anti-invariant, i.e., ϕ(d ) T M. (iii) The distribution D θ is slant with angle θ π 2. The normal bundle T M of a pseudo-slant submanifold is decomposed as T M = FD θ ϕd μ, (2.15) where μ is an invariant normal subbundle under ϕ. 3 Warped product pseudo-slant submanifolds In this section, we discuss the warped product submanifolds of a nearly cosymplectic manifold. These manifolds were studied by Bishop and O Neill [11]. They defined these manifolds as follows: Let (N 1, g 1 )and(n 2, g 2 ) be two Riemannian manifolds and f a positive differentiable function on N 1. Then their warped product M = N 1 f N 2 is the product manifold N 1 N 2 equipped with the Riemannian structure such that g = g 1 + f 2 g 2. The function f is called the warping function on M. Itwasprovedin[11] that for any X Ɣ(TN 1 )andz Ɣ(TN 2 ), the following holds: X Z = Z X =(X ln f )Z, (3.1) where denote the Levi-Civita connection M. A warped product manifold M = N 1 f N 2 is said to be trivial if the warping function f is constant. If M = N 1 f N 2 is a warped

5 Al-Solamy Journal of Inequalities and Applications (2015) 2015:306 Page 5 of 9 product manifold then the base manifold N 1 is totally geodesic and the fiber N 2 is a totally umbilical submanifold of M,respectively[11]. Now, we discuss the warped product pseudo-slant submanifolds of the type N θ f N of a nearly cosymplectic manifold M. We consider the structure vector field ξ tangent to the base manifold N θ of the warped products. If ξ is tangential to N then the warped product is trivial [6]. We have the following results for later use. Lemma 3.1 Let M = N θ f N be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold M, then: (i) ξ ln f =0, (ii) 2g(h(X, Y ), ϕz) =g(h(x, Z), FY) +g(h(y, Z), FX). for any X, Y Ɣ(TN θ ) and Z Ɣ(TN ). Proof For any Z, W Ɣ(TN )andξ tangential to N θ,wehave g( Z ξ, W)=g( Z ξ, W). Then from (3.1), we obtain g( Z ξ, W)=ξ ln fg(z, W). (3.2) By the polarization identity, we derive g( W ξ, Z)=ξ ln fg(z, W). (3.3) Thus the first part followsfrom (3.2) and(3.3) by using(2.4). For the second part, consider X, Y Ɣ(TN θ )andz Ɣ(TN ), we have g ( h(x, Y ), ϕz ) = g( X Y, ϕz)= g(ϕ X Y, Z). Then by the covariant derivative property of ϕ,wederive g ( h(x, Y ), ϕz ) = g ( ( X ϕ)y, Z ) g( X ϕy, Z) Using (2.5)and(2.7), weget = g ( ( X ϕ)y, Z ) g( X PY, Z) g( X FY, Z) = g ( ( X ϕ)y, Z ) + g(py, X Z)+g(A FY X, Z). g ( h(x, Y ), ϕz ) = g ( ( X ϕ)y, Z ) + g(py, X Z)+g ( h(x, Z), FY ). From (3.1), we obtain g ( h(x, Y ), ϕz ) = g ( ( X ϕ)y, Z ) +(X ln f )g(py, Z)+g ( h(x, Z), FY ). The second term of right hand side is identically zero by the orthogonality of vector fields, thus we have g ( h(x, Y ), ϕz ) = g ( ( X ϕ)y, Z ) + g ( h(x, Z), FY ). (3.4)

6 Al-Solamy Journal of Inequalities and Applications (2015) 2015:306 Page 6 of 9 Then by the polarization identity, we obtain g ( h(x, Y ), ϕz ) = g ( ( Y ϕ)x, Z ) + g ( h(y, Z), FX ). (3.5) Then from (3.4)and(3.5), we get 2g ( h(x, Y ), ϕz ) = g ( ( X ϕ)y +( Y ϕ)x, Z ) + g ( h(x, Z), FY ) + g ( h(y, Z), FX ). The first term of right hand side is identically zero by (2.3), thus we get (ii), which proves the lemma completely. Lemma 3.2 Let M = N θ f N be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold M, where N and N θ are anti-invariant and proper slant submanifolds of M, respectively. Then: (i) 2g(h(Z, W), FX)=g(h(X, Z), ϕw)+g(h(x, W), ϕz)+2(px ln f )g(z, W), (ii) 2g(h(Z, W), FPX)=g(h(PX, Z), ϕw)+g(h(px, W), ϕz) 2cos 2 θ(x ln f )g(z, W) for any X Ɣ(TN θ ) and Z, W Ɣ(TN ). Proof For any Z, W Ɣ(TN )andx Ɣ(TN θ ), we have g ( h(z, W), FX ) = g( Z W, ϕx) g( Z W, PX) = g ( ( Z ϕ)w, X ) g( Z ϕw, X)+g(W, Z PX). Using (2.5)and(2.6), we obtain g ( h(z, W), FX ) = g ( ( Z ϕ)w, X ) + g(a ϕw Z, X)+g(W, Z PX). Then from (2.7)and(3.1), we get g ( h(z, W), FX ) = g ( ( Z ϕ)w, X ) + g ( h(x, Z), ϕw ) +(PX ln f )g(z, W). (3.6) Then by the polarization identity we derive g ( h(z, W), FX ) = g ( ( W ϕ)z, X ) + g ( h(x, W), ϕz ) +(PX ln f )g(z, W). (3.7) From (3.6)and(3.7), we get 2g ( h(z, W), FX ) = g ( ( W ϕ)z +( Z ϕ)w, X ) + g ( h(x, Z), ϕw ) + g ( h(x, W), ϕz ) +2(PX ln f )g(z, W). Then, from the above relation, (i) holds by using (2.3). If we interchange X by PX in (i) we get (ii) by using Theorem 2.1 and Lemma 3.1(i). Thus, the proof is complete. Now, we construct the following frame for a warped product pseudo-slant submanifold M = N θ f N of a (2n + 1)-dimensional nearly cosymplectic manifold.

7 Al-Solamy Journal of Inequalities and Applications (2015) 2015:306 Page 7 of 9 Let M = N θ f N be a m-dimensional warped product pseudo-slant submanifold of a(2n + 1)-dimensional nearly cosymplectic manifold M such that N is a q-dimensional anti-invariant submanifold and N θ is a (2p + 1)-dimensional slant submanifold tangent to the structure vector field ξ of M, respectively. Then the orthonormal frame fields of the tangent spaces of N and N θ,respectively,are{e 1,...,e q } and {e q+1 = e 1,...,e q+p = e p, e q+p+1 = e p+1 = sec θpe 1,...,e q+2p = e 2p = sec θpe p, e q+ = e m = ξ}. The orthonormal frames of ϕ(tn ), F(TN θ ), and μ, respectively,are{e m+1 = ϕe 1,...,e m+q = ϕe q }, {e m+q+1 = ẽ 1 = csc θfe 1,...,e m+p+q = ẽ p = csc θfe p, e m+p+q+1 = ẽ p+1 = csc θ sec θfpe 1,...,e m+2p+q = ẽ 2p = csc θ sec θfpe p } and {e 2m,...,e 2n+1 }. The dimensions of ϕ(tn ), F(TN θ ), and μ, respectively, are q,2p,and2(n m +1). Theorem 3.1 Let M = N θ f N be a mixed geodesic warped product pseudo-slant submanifold of a nearly cosymplectic manifold MsuchthatN and N θ are anti-invariant and proper slant submanifolds of M, respectively. Then: (i) The squared norm of the second fundamental form h of M satisfies h 2 q cot 2 θ θ ln f 2 where θ ln f is the gradient of ln f over N θ and q is the dimension of N. (ii) If the equality holds in (i), then h(z, W) lies in F(TN θ ) for any Z, W Ɣ(TN ) and h(x, Y ) lies in ϕ(tn ), for any X, Y Ɣ(TN θ ). Proof From (2.8), we have h 2 = m g ( h(e i, e j ), h(e i, e j ) ) = i,j=1 2n+1 m g ( ) 2. h(e i, e j ), e r Then using the frame fields of TN and TN θ,weget h 2 = 2n+1 + 2n+1 g ( 2n+1 ) 2 h(e i, e j ), e r +2 q g ( h ( e i, ) ) 2. e j, er r=m+1 i=1 j=1 q g ( ) 2 h(e i, e j ), e r Since M is mixed geodesic, the second term of right hand side is identically zero and break the above relation for the frames of F(TN θ ), ϕ(tn ), and μ.thenwederive h 2 = m+q + + 2n+1 r=m+q+ i,j=1 2m 1 g ( 2m 1 ) 2 h(e i, e j ), e r + q r=m+q+1 i,j=1 r=m+q+1 i,j=1 g ( m+q ) 2 h(e i, e j ), e r + g ( 2n+1 ) 2 h(e i, e j ), e r + g ( ) 2 h(e i, e j ), e r r=m+q+ i,j=1 q g ( ) 2 h(e i, e j ), e r q g ( ) 2. h(e i, e j ), e r (3.8)

8 Al-Solamy Journal of Inequalities and Applications (2015) 2015:306 Page 8 of 9 The first term of right hand side is identically zero by Lemma 3.1(ii) for a mixed geodesic warped product submanifold. Also, we have no relation for the μ components with h and g(h(z, W), FW ), for any Z, W, W Ɣ(TN ) in terms of the warping function. Thus, we shall leave all positive terms except the fifth term, then we have h 2 = 2p i,j=1 p i,j=1 q g ( ) 2 h(e i, e j ), ẽ r q g ( h(e i, e j ), sec θfe ) 2 p r + i,j=1 q g ( h(e i, e j ), csc θ sec θfpe ) 2. r Using Lemma 3.2 for mixed geodesic warped products, we derive h 2 csc 2 θ p q ( Pe g(ei, e j ) 2 + cot 2 θ i,j=1 p q ( e g(ei, e j ) 2 i,j=1 = q csc 2 ( θ Pe q csc 2 θ q csc 2 θ ( Pe ln f ) 2 + q cot 2 θ 2p r=p+1 Since e ln f = ξ ln f =0,from(2.10), we derive ( Pe p ( e. h 2 q csc 2 θ P θ ln f 2 q csc 2 θ p + q cot 2 ( θ e. p g ( e r+p, P θ ln f ) 2 Then by Theorem 2.1,we obtain h 2 q cot 2 θ { θ ln f 2 g ( θ ln f, ξ ) 2} q csc 2 θ sec 2 θ p g ( Pe r, P θ ln f ) 2 + q cot 2 θ p ( e. Then from (2.9), (2.13), and the trigonometric identities, finally, we get h 2 q cot 2 θ θ ln f 2, which is inequality (i). If the equality holds in (i), then from the second and third remaining terms g ( h(x, Y ), FY ) =0, X, Y, Y Ɣ(TN θ ) h(x, Y ) Ɣ(ϕTN μ) (3.9)

9 Al-Solamy Journal of Inequalities and Applications (2015) 2015:306 Page 9 of 9 and g ( h(x, Y ), ζ ) =0, X, Y Ɣ(TN θ )andζ Ɣ(μ) h(x, Y ) Ɣ(ϕTN FTN θ ). (3.10) Then from (3.9)and(3.10), we get h(x, Y ) Ɣ(ϕTN ), X, Y Ɣ(TN θ ). (3.11) Similarly, from the remaining fourth and sixth terms, we conclude that g ( h(z, W), ϕw ) =0, Z, W, W Ɣ(TN ) h(z, W) Ɣ(FTN θ μ) (3.12) and g ( h(z, W), ζ ) =0, Z, W Ɣ(TN )andζ Ɣ(μ) h(z, W) Ɣ(ϕTN FTN θ ). (3.13) Then from (3.12)and(3.13), we get h(z, W) Ɣ(FTN θ ), Z, W Ɣ(TN ). (3.14) Thus (ii) follows from (3.11)and(3.14).This completes the proof of the theorem. Competing interests The author declares that he has no competing interests. Acknowledgements The author is thankful to Dr. Siraj Uddin for the discussion as regards the constructed frame which improved the inequality. Also he is thankful to the reviewers for their valuable suggestions, which really improved the quality of the manuscript. Received: 7 May 2015 Accepted: 15 September 2015 References 1. Blair, DE: Almost contact manifolds with Killing structure tensors I. Pac. J. Math. 39, (1971) 2. Blair, DE, Showers, DK: Almost contact manifolds with Killing structures tensors II. J. Differ. Geom. 9, (1974) 3. Blair, DE, Yano, K: Affine almost contact manifolds and f -manifolds with affine Killing structure tensors. Kodai Math. Semin. Rep. 23, (1971) 4. Carriazo, A: New Developments in Slant Submanifolds Theory. Narosa Publishing House, New Delhi (2002) 5. Sahin, B: Warped product submanifolds of Kaehler manifolds with a slant factor. Ann. Pol. Math. 95, (2009) 6. Uddin, S, Kon, SH, Khan, MA, Singh, K: Warped product semi-invariant submanifolds of nearly cosymplectic manifolds. Math. Probl. Eng. (2011). doi: /2011/ Uddin, S, Khan, KA: An inequality for contact CR-warped product submanifolds of nearly cosymplectic manifolds. J. Inequal. Appl. 2012, 304 (2012) 8. Uddin, S, Mustafa, A, Wong, BR, Ozel, C: A geometric inequality for warped product semi-slant submanifolds of nearly cosymplectic manifolds. Rev. Unión Mat. Argent. 55, (2014) 9. Endo, H: On the curvature tensor of nearly cosymplectic manifolds of constant ϕ-sectional curvature. An. Ştiinţ. Univ. Al.I. Cuza Iaşi, Mat. 51, (2005) 10. Cabrerizo, JL, Carriazo, A, Fernandez, LM, Fernandez, M: Slant submanifolds in Sasakian manifolds. Glasg. Math. J. 42, (2000) 11. Bishop, RL, O Neill, B: Manifolds of negative curvature. Trans. Am. Math. Soc. 145, 1-49 (1969)