An Inequality for Warped Product Semi-Invariant Submanifolds of a Normal Paracontact Metric Manifold

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1 Filomat 31:19 (2017), Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: An Inequality for Warped Product Semi-Invariant Submanifolds of a Normal Paracontact Metric Manifold Mehmet Atçeken a, Süleyman Dirik b, Ümit Yıldırım c a Gaziosmanpasa University, Faculty of Arts Sciences, Department of Mathematics, Tokat/Turkey b Amasya University, Faculty of Arts Sciences, Department of Statistics, Amasya/ Turkey c Gaziosmanpasa University, Faculty of Arts Sciences, Department of Mathematics, Tokat/Turkey Abstract. The aim of this paper is to study the warped product semi-invariant submanifolds in a normal paracontact metric space form. We obtain some characterization new geometric obstructions for the warped product type M f M T. We establish a general inequality among the trace of the induced tensor, laplace operator, the squared norms of the second fundamental form warping function. These inequalities are discussed we obtain some new results. 1. Introduction The geometric inequalities of warped product submanifolds have been studied since B-Y. Chen introduced the notion of a CR-warped product submanifold in a Kaehler manifold established inequalities for the fundamental form in terms of warping function[3]. In a natural way, warped products appeared in differential geometry generalizing the class of Riemannian product manifolds to much larger one, called warped product manifolds, which are applied in general relativity to model the stard space time. Recently, Uddin, et al [6, 9] obtained some inequalities of warped product submanifolds in cosymplectic nearly trans-sasakian manifolds. They obtained an inequality for the length of the second fundamental form of the warped product submanifold a nearly cosymplectic manifold in terms of warping function, discussed this inequality found some new results. In [], authors obtained a characterization for warped product submanifolds in terms of warping function shape operator gave an inequality for squared norm of the second fundamental form. Motivated by the studies of the above authors, in this paper, we extend this idea into a normal paracontact metric manifold, which has not been attempted so far, derive the geometric inequalities of non-trivial warped product semi-invariant submanifold obtain an inequality involving the trace of the induced tensor warping function Mathematics Subject Classification. Primary 53C0, Secondary 53C2 Keywords. Paracontact Metric Manifold, Warped Product, Riemannian Product Space Form. Received:11 January 2017; Accepted: 23 September 2017 Communicated by Mića Stanković addresses: mehmet.atceken382@gmail.com (Mehmet Atçeken), suleyman.dirik@amasya.edu.tr (Süleyman Dirik), umit.yildirim@gop.edu.tr (Ümit Yıldırım)

2 M. Atçeken et al. / Filomat 31:19 (2017), Preliminaries Let M be an n-dimensional almost contact metric manifold with structure tensors (φ, ξ, η, ), where φ is (1,1)-type tensor field, ξ is a vector field, η is dual of ξ is also Riemannian metric tensor on M. If we have φ 2 X = X η(x)ξ, φξ = 0, η(φx) = 0, η(ξ) = 1, (1) (φx, φy) = (X, Y) η(x)η(y), η(x) = (X, ξ), (2) for any vector fields X, Y on M, then M is called almost paracontact metric manifold. An almost paracontact metric manifold M is said to be normal if ( X φ)y = (X, Y)ξ η(y)x + 2η(X)η(Y)ξ, (3) for any vector fields on M, where denotes the Riemannian connection on M[7]. (3) implies that X ξ = φx ( X η)y = (φx, Y). () On the other h, if a normal paracontact metric manifold M has a constant-c, denoted by M(c), then its the Riemannian curvature tensor R is given by R(X, Y)Z = 1 (c + 3){ (Y, Z)X (X, Z)Y} + 1 (c 1){η(X)η(Z)Y η(y)η(z)x + (X, Z)η(Y)ξ (Y, Z)η(X)ξ + (φy, Z)φX (φx, Z)φY 2 (φx, Y)φZ}, (5) for any vector fields X, Y, Z on M[7]. Now, let M be an isometrically immersed submanifold in a normal paracontact metric manifold M denote by the same symbol the Riemannian metric induced on M. Let Γ(TM) Γ(T M) be the differentiable vector fields set tangent normal to M, respectively. Also we denote by induced connections on Γ(TM) Γ(T M), respectively. Then the Gauss Weingarten formulas are given by X Y = X Y + h(x, Y) (6) X V = A V X + X V, (7) for any X, Y Γ(TM) V Γ(T M), where h A V are the second fundamental form shape operatory for the immersed of M into M, respectively. They are related as (h(x, Y), V) = (A V X, Y). (8) By R, we denote the Riemannian curvature tensor of, then we have R(X, Y)Z = R(X, Y)Z + A h(x,z) Y A h(y,z) X + ( X h)(y, Z) ( Y h)(x, Z), (9) where the covariant derivative of h is defined by ( X h)(y, Z) = X h(y, Z) h( XY, Z) h(y, X Z), (10) for any X, Y, Z Γ(TM).

3 M. Atçeken et al. / Filomat 31:19 (2017), Let M be an immersed submanifold of a normal paracontact metric manifold M. For any X Γ(TM), we can set φx = TX + NX, (11) where TX NX denote the tangential normal components of φx, respectively. In the same way, for any V Γ(T M), we can write φv = BV + CV, (12) where BV(resp. CV) are the tangential(resp. normal) components of φv. The squared norm trace of T at p M are, respectively, defined by T 2 = 2 (Te i, e j ), trace(t) = i, (Te i, e i ), (13) where {e 1, e 2,..., e n } is an orthonormal basis of the tangent space Γ(TM). Definition 2.1. A submanifold M of a normal paracontact metric manifold M is said to be semi-invariant submanifold if there exist two orthogonal distributions D D T such that i.) TM = D D T, ii.) D is anti-invariant distribution under φ, i.e., φ(d ) T M, iii.)d T is an invariant distribution φ, i.e., φ(d T ) TM. Next, let us suppose that M be a semi-invariant submanifold of a normal paracontact metric manifold M, then the normal bundle T M can be decomposed as follow as; T M = φ(d ) µ, (1) where µ is an invariant subbundle of T M. For a differentiable function f on M, the gradient Hessian form are, respectively, defined by X f = ( f, X), f = rad f, (15) H f (X, Y) = X(Y f ) ( X Y) f = ( X rad f, Y), (16) for any X, Y Γ(TM). As a consequence, we have f 2 = (e i ( f )) 2. The laplacian of f is defined by f = {( ei e i ) f e i (e i f )} = = H ln f (e i, e i ). ( ei rad f, e i ) From the integration on the manifolds theory, for M is a compact, orientable Riemannian manifold without boundary, we have f dv = 0, (19) M where dv denote the volume element of M[8]. (17) (18)

4 3. Warped Product Manifolds M. Atçeken et al. / Filomat 31:19 (2017), Bishop O Neill defined the notion of warped product manifolds to construct examples of Riemannian manifolds with a negative curvature. These manifolds are naturel generalizations of Riemannian product manifolds. Let (M 1, 1 ) (M 2, 2 ) be two Riemannian manifolds f be a positive defined differentiable function on M 1. Consider the product manifold M 1 M 2 with its canonical projections π 1 : M 1 M 2 M 1, π 2 : M 1 M 2 M 2. The warped product M = M 1 M 2 is the product manifold M 1 M 2 equipped with the Riemannian structure such that X 2 = π 1 (X) 2 + f 2 (π 1 (p)) π 2 (X) 2, (20) for any X Γ(TM), where is the st for the tangent map. So we have = π ( f π 1 ) 2 π 2 2. The function f is called the warping function on M[2]. Next we will give the following Lemma for later use. Lemma 3.1. Let M = M 1 f M 2 be a warped product manifold. We have i.) X Y Γ(TM 1 ) ii.) Z X = X Z = X(ln f )Z iii.) Z W = W (Z, W) ln f, Z for any X, Y Γ(TM 1 ) Z, W Γ(TM 2 ), where denote the Riemannian connections on M M 2, respectively. We note that a warped product manifold M = M 1 f M 2 is characterized by the fact that M 1 M 2 are totally geodesic totally umbilical submanifolds of M, respectively. If warped function f is constant, then warped product manifold is said to be Riemannian product.. Warped Product Semi-Invariant Submanifolds of A Normal Paracontact Metric Manifold In this section, we establish warped product semi-invariant submanifolds which are form M = M f M T, where M M T are anti-invariant invariant submanifolds of M, respectively. Furthermore, the co-vector field ξ is tangent to M. Otherwise, the warping function f is constant. Next, we will give an example for the method presented in this paper is effective. Example.1. Let M be a submanifold of R 7 with coordinates (x 1, x 2, x 3, y 1, y 2, y 3, t) given by x 1 = u, x 2 = u cos θ, x 3 = u sin θ, y 1 = u cos α, y 2 = usinα, y 3 = u, t = 2s, where u, θ, α s denote the arbitrary parameters. It is easy to check that the tangent bundle of M is spanned by the vectors e 1 = + cos θ + sin θ + cos α + sin α x 1 x 2 x 3 y 1 y 2 y 3 e 2 = u sin θ + u cos θ, e 3 = u sin α + u cos α x 2 x 3 y 1 y 2 e = ξ = 2 t. Now, we define the almost paracontact metric structure φ of R 7 by φ( ) =, φ( ) = φ( x i x i y i y i t ) = 0, η = 1 dt, 1 i n, 2

5 M. Atçeken et al. / Filomat 31:19 (2017), then we have φ 2 X = X η(x)ξ, (φx, φy) = (X, Y) η(x)η(y), φξ = 0 η(ξ) = 1. On the other h, with respect to the almost paracontact metric structure φ of R 7, the φγ(tm) becomes φe 1 = + cos θ + sin θ cos α sin α + x 1 x 2 x 3 y 1 y 2 y 3 φe 2 = e 2, φe 3 = e 3 φξ = 0. Since φe 1 is orthogonal to M, φe 2 φe 3 are tangent to M, Γ(TM ) Γ(TM T ) can be choosen subspace Γ(TM ) = sp{e 1, e } Γ(TM T ) = sp{e 1, e 2 }. Furthermore, the metric tensor of M is given by = (du 2 + dt 2 ) + u 2 (dθ 2 + dα 2 ) = M u 2 MT. Thus M is a -dimensional warped product semi-invariant submanifold of R 7 with warping function f = u 2. Lemma.2. Let M = M f M T be a semi-invariant submanifold of a normal paracontact metric manifold M such that ξ Γ(TM ). Then we have (h(x, Y), φu) = (X, Y)η(U) (TX, Y)U ln f, (21) (h(u, V), φw) = (h(u, W), φv), U, V, W ξ (22) (h(u, X), φv) = 0, (23) for any X, Y Γ(TM T ) U, V, W Γ(TM ). Proof. By using (3), (6) Lemma 3.1, we have (h(x, Y), φu) = ( X Y, φu) = (φ X Y, U) = ( X φy ( X φ)y, U) = ( X TY, U) (( X φ)y, U) = (X, TY)U ln f + (X, Y)η(U), which gives us (21). In the same way, we have (h(u, V), φw) = ( U V, φw) = ( U φw, V) = (( U φ)w + φ U W, V) = ( (U, W)ξ η(w)u + 2η(W)η(U)ξ, V) ( U W, φv) = (h(u, W), φv) (h(u, X), φv) = ( U X, φv) = (φ U X, V) Thus the proof is complete. = ( U φx ( U φ)x, V) = ( U φx, V) ( (U, X)ξ η(x)u + 2η(X)η(U)ξ, V) = U ln f (φx, V) = 0. Lemma.3. Let M = M f M T be a warped product semi-invariant submanifold of a normal paracontact metric manifold M. Then we have (h(φx, U), φh(x, U)) = h(x, U) 2, (2) for any X Γ(TM T ) U Γ(TM ).

6 M. Atçeken et al. / Filomat 31:19 (2017), Proof. Making use of (3),(6) from Lemma 3.1, we have (h(φx, U), φh(u, X)) = ( U φx U φx, φh(x, U)) = (( U φ)x + φ U X, φh(u, X)) (U ln f φx, φh(x, U)) = ( (X, U)ξ + η(x)u 2η(X)η(U)ξ, φh(u, X)) + ( U X, h(x, U)) = (h(x, U), h(x, U)), for any X Γ(TM T ) U Γ(TM ). Theorem.. Let M = M f M T be a warped product semi-invariant submanifold of a normal paracontact metric manifold M such that c 1. Then we have { } 1 h(x, U) 2 = (X, X) (c 1) (φu, φu) + Hln f (U, U) (U ln f ) 2 + (X, TX)η(U)U ln f, (25) for any X Γ(TM T ) U Γ(TM ). Proof. By using (9), (10) taking into account of Lemma 3.1, we have ( R(U, X)φX, φu) = (( U h)(x, φx) ( X h)(u, φx), φu) = ( U h(x, φx) h( U X, φx) h(x, U φx), φu) ( X h(u, φx) h( X U, φx) h(u, X φx), φu). By virtue of (21) (23), we obtain ( R(U, X)φX, φu) = U (h(x, φx), φu) ( U φu, h(x, φx)) (h( U X, φx), φu) (h( U φx, X), φu) X (h(u, φx), φu) + (h(u, φx), X φu) + U ln f (h(x, φx), φu) + (h( X φx, U), φu) = U[ (X, TX)η(U) (X, X)U ln f ] (φ U U, h(φx, X)) U ln f (h(x, φx), φu) + (h(u, φx), φ X U) + (h( X φx, U), φu) U ln f (h(φx, X), φu).

7 M. Atçeken et al. / Filomat 31:19 (2017), Also considering Lemma.2 M is totally geodesic in M, we reach ( R(U, X)φX, φu) = (X, TX)η( U U) (X, X)U 2 (ln f ) (φ U U, h(x, φx)) U ln f { (X, TX)η(U) (X, X)U ln f } + h(x, U) 2 + (h( XφX (X, φx) ln f, U), φu) = (X, TX)η( U U) (X, X)U 2 (ln f ) (X, TX)η( U U) + (X, X)( U U) ln f (X, TX)U ln f η(u) + (X, X)(U ln f ) 2 + (h( XφX, U), φu) (X, TX) (h( ln f, U), φu) + h(x, U) 2 = (X, X){( U U U 2 (ln f )) ln f } (TX, X)U ln f η(u) + (X, X)(U ln f ) 2 + h(x, U) 2 = (X, X)H ln f (U, U) (TX, X)U ln f η(u) + (X, X)(U ln f ) 2 + h(x, U) 2. On the other h, from (5), we conclude ( R(U, X)φX, φu) = 1 (c 1) (X, X) (φu, φu), (26) which proves our assertion. Now, let {e 1, e 2,...e p, e p+1 = ξ, e 1, e 2,..., e q } be an orthonormal basis of Γ(TM) such that e i, 1 i q + 1, are tangent to M e j, 1 j q, are tangent to M T. Substituting (25) into X = e j U = e i, for 1 i p j q, we obtain q h(e i, e j ) 2 1 = q (c 1)p + H ln f (e i, e i ) + (e i ln f ) 2 + q (Te j, e j )ξ ln f. (27) By means of (6) taking account of M = M f M T being warped product semi-invariant submanifold, we have ξ ln f (X, X) = (TX, X), which implies that ξ ln f = 1 q tr(t). Thus by using (18), (27) becomes q { } 1 h(e i, e j ) 2 = q (c 1)p + rad ln f 2 ln f + 1 q tr2 (T). (28) From the (28), we have the following Theorems. Theorem.5. Let M be a warped product semi-invariant submanifold of a normal paracontact metric manifold M(c) such that c 1. The squared of norm of the second fundamental form h satisfies the condition { } 1 h 2 q (c 1)p + rad ln f 2 ln f + 1 q tr2 (T). (29)

8 M. Atçeken et al. / Filomat 31:19 (2017), Theorem.6. Let M be a compact orientable warped product semi-invariant submanifold of a normal paracontact metric manifold M(c) such that c 1. M is a semi-invariant Riemannian product if only if the second fundamental form h of M satisfies q h(e i, e j ) 2 1 (c 1)pq + 1 q (tr2 (T)). (30) Proof. From (19) (28), we conclude M{ rad ln f 2 1 q q h(e i, e j ) 2 + ( 1 q tr(t))2 }dv = Vol(M) 1 (c 1)pq. (31) Here if (30) is satisfied, the we can derive rad ln f is constant. The converse is obvious. This proves our assertion. Acknowledgment: The authors sincerely thank the referee for the corrections comments in the revision of this paper. This work is supported by Scientific Research Project in Gaziosmanpasa University ( ). References [1] M. Atçeken, Contact CR-Warped Product Submanifolds of Cosymplectic Space Forms, Collect. Math. 62(1)(2011),17-26 doi: /s z [2] R. L. Bishop B. O Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 15(1969), 1-9. [3] B-Y. Chen, Geometry of warped product CR-submanifolds in Kaehler Manifolds, Monatsh. Math. 133 (2001) [] M. A. Khan A. Ishan, Semi-slant warped product submanifolds of a trans-sasakian manifold, Annals of the University of Cralova, mathematics computer science series. 1(1) (201), [5] M. A. Khan, S. Uddin R. Sachdeva, Semi-invariant warped product submanifolds of cosymplectic manifolds, J. inequal Appl. 2012, Article ID 19(2012). [6] A. Mustafa, S. Uddin W. R. Wong, Generalized inequalities on warped product submanifolds in nearly trans-sasakian anifolds, J. inequal Appl. 201, Article ID 36(201). [7] H. B. Pey A. Kumar, Anti-invariant submanifolds of almost paracontact manifolds, Indian J. pure appl. math. 16(6)( 1985), [8] P. Rao, Structures on complex manifolds, Lambert academic publishing [9] S. Uddin V. A. Khan, An inequalities for contact CR-warped product submanifolds of cosymplectic manifolds, J. inequal Appl Article ID 30(2012). [10] S. Uddin F. R. Al-Solamy, Warped product pseudo-slant immersions in Sasakian manifolds, Pub. Math. Debrecen 91(2) (2017), 1-1. [11] S. Uddin, Geometry of warped product semi-slant submanifolds of Kenmotsu manifolds, Bull. Math. Sci. (2017), doi: /s [12] S. Uddin, B.-Y. Chen F. R. Al-Solamy, Warped product bi-slant immersions in Kaehler manifolds, Mediterr. J. Math. (2017) 1: 95. doi: /s [13] S. Uddin, M. F. Naghi F. R. Al-Solamy, Another class of warped product submanifoldsof Kenmotsu manifolds, RACSAM (2017), doi: /s