A CHARACTERIZATION OF WARPED PRODUCT PSEUDO-SLANT SUBMANIFOLDS IN NEARLY COSYMPLECTIC MANIFOLDS

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1 Journal of Mathematical Sciences: Advances and Applications Volume 46, 017, Pages 1-15 Available at DOI: A CHARACTERIATION OF WARPED PRODUCT PSEUDO-SLANT SUBMANIFOLDS IN NEARLY COSYMPLECTIC MANIFOLDS Department of Mathematics Faculty of Science King Abdulaziz University P. O. Box 8003 Jeddah 1589 Saudi Arabia ralghefari@kau.edu.sa falleh@hotmail.com Abstract Recently, the second author studied warped product pseudo-slant submanifolds of nearly cosymplectic manifolds and obtained the general inequality for the squared norm of the second fundamental form. In this paper, we obtain the necessary and sufficient conditions that a pseudo-slant submanifold to be a locally warped product manifold. We also provide a non-trivial example of warped product pseudo-slant submanifolds of almost contact metric manifolds. 1. Introduction Almost contact manifolds with Killing structures tensors were defined in [3] as nearly cosymplectic manifolds. Later, these manifolds were studied by Blair and Showers from the topological point of view [5]. 010 Mathematics Subject Classification: 53C40, 53C4, 53C15. Keywords and phrases: slant submanifold, pseudo-slant submanifold, nearly cosymplectic manifold, warped products. Received May 10, Scientific Advances Publishers

2 An almost contact metric structure ( ϕ, ξ, η, g) satisfying ( ϕ) 0 is called a nearly cosymplectic structure. If we consider S 5 as a totally 6 geodesic hypersurface of a six dimensional sphere S, then it is known 5 that S has a non cosymplectic nearly cosymplectic structure. A normal nearly cosymplectic manifold is cosymplectic (see [4]). On the other hand, pseudo-slant submanifolds of almost contact metric manifold were introduced by Carriazo [7] under the name antislant submanifolds. Later, the warped products of these submanifolds of Kaehler manifolds were studied by Sahin [11] under the name of warped product hemi-slant submanifolds. In this paper, we study pseudo-slant submanifolds and their warped products in a nearly cosymplectic manifold. We characterise that a pseudo-slant submanifold of a nearly cosymplectic manifold is locally a warped product submanifold under the obtained condition. Finally, we construct an example of warped product pseudo-slant submanifolds of an almost contact metric manifold.. Preliminaries A ( n +1) -dimensional C manifold M is said to have an almost contact structure if there exist on M a tensor field ϕ of type ( 1, 1), a vector field ξ and a 1-form η satisfying [5] ϕ I + η ξ, ϕξ 0, η ϕ 0, η( ξ) 1. (.1) There always exists a Riemannian metric g on an almost contact manifold M satisfying the following compatibility condition: η ( ) g(, ξ), g( ϕ, ϕy ) g(, Y ) η( ) η( Y ), (.) where and Y are vector fields on M [5]. An almost contact structure ( ϕ, ξ, η) is said to be nearly cosymplectic if ϕ is Killing, i.e., if ( ϕ) Y + ( Y ϕ) 0, (.3)

3 or any A CHARACTERIATION OF WARPED PRODUCT 3, Y tangent to M, where denotes the Riemannian connection of the metric g. It is well known that an almost contact metric manifold is cosymplectic if and only if ϕ vanishes identically, i.e., ( ϕ) Y 0 and ξ 0. On a nearly cosymplectic manifold the structure vector field ξ is Killing (see [9]), that is, g( Y ξ, ) + g( ξ, Y ) 0, (.4) for any Y, 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. Denote by F ( M ) the algebra of smooth functions on M and by Γ ( TM ) the F ( M )-module of smooth sections of TM over M. Then the Gauss and Weingarten formulas are given by ( a) Y Y + h(, Y ); ( b) N A + N, (.5) for each, Y Γ( TM ) 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 g( h(, Y ), N ) g( A, Y ), where g denotes the N Riemannian metric on M as well as induced on M. For any Γ( TM ), we write ϕ T + F, (.6) where T is the tangential component and F is the normal component of ϕ. Similarly, if any N Γ( T M ), then ϕ N decompose as ϕ N BN + CN, (.7) N

4 4 where BN and CN are the tangential and the normal components of ϕ N, respectively. A submanifold M of an almost contact metric manifold M is said to be invariant if F is identically zero, that is, ϕ Γ( TM ) and antiinvariant if T is identically zero, that is, ϕ Γ( T M ), for any Γ( TM ). Now, if we denote by parts of ( ϕ ) Y, then P and Q Y the tangential and normal Y for all, Y Γ( TM ). ( ϕ) Y P Y + Q Y, (.8) We refer to [13] for the properties of P and Q, which we enlist here for later use: ( ) ( i) P W P W + W, ( ii) Q W Q W + Q W; p 1 + Y PY + Y ( ) ( i) P ( Y + W ) P Y + P W, ( ii) Q ( Y + W ) Q Y + Q W; p ( )( i) g( P Y, W ) g( Y, P ( ii) g( Q Y, N ) g( Y, P N ); p 3 ( ) P ϕy + Q ϕy ϕ( P Y + Q Y ), p 4 for all, Y, W Γ( TM ) and N Γ( T M ). Also, if M is a submanifold of a nearly cosymplectic manifold M, then from (.3) and (.8), we have ( a ) P Y + P 0; ( b) Q Y + Q 0, (.9) for any, Y Γ( TM ). Y As a generalisation of invariant and anti-invariant submanifolds Cabrerizo et al. [6] introduced the idea of slant submanifolds of almost contact metric manifolds in a similar way of Chen s characterization of slant submanifolds of almost Hermitian manifolds [8]. A submanifold M Y Y

5 A CHARACTERIATION OF WARPED PRODUCT 5 of an almost contact metric manifold M is said to be a slant submanifold if for each non zero vector tangent to M at x the angle θ between ϕ and T x M is constant for all T x M ξx and for any x M [6]. Obviously if θ 0, M is invariant and if θ π, M is an antiinvariant submanifold. A slant submanifold is said to be proper slant if it is neither invariant nor anti-invariant submanifold. They proved the following characterization for a slant submanifold of an almost contact metric manifold. Theorem.1 ([6]). 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 [ 0, 1] λ such that T λ( I + η ξ). Furthermore, if θ is slant angle of M, then λ cos θ. Following relations are straightforward consequences of the above theorem: g( T, TY ) cos θ( g(, Y ) η( Y ) η( )), (.10) g( F, FY ) sin θ( g(, Y ) η( Y ) η( )), (.11) for all, Y Γ( TM ). We, also have the following chacterization for a slant submanifold M of an almost contact metric manifold M. Theorem. ([15]). Let M be a proper slant submanifold of an almost contact metric manifold M, such that ξ Γ( TM ). Then ( a) BF sin θ( + η( ) ξ) ; ( b) CF FT, for any Γ( TM ).

6 6 3. Pseudo-Slant Submanifolds and their Warped Products In [7], Carriazo introduced a new class of submanifolds known as pseudo-slant submanifolds of almost contact metric manifolds which are the generalizations of contact CR and slant submanifolds. He defined these submanifolds under the name anti-slant submanifolds as follows: A submanifold M of an almost contact metric manifold M is said to be anti-slant (or pseudo-slant) if there exist two orthogonal distributions D 1 and D satisfying: (i) TM D D ; 1 ξ (ii) D 1 is an anti-invariant, i.e., ϕd 1 M; T π (iii) D is a slant distribution with slant angle θ. From the above definition it is clear that if θ 0, then M is a contact CR-submanifold and if we consider the dimensions of D 1 and D by q and p, respectively. Then, if q 0 (respectively, p 0 ), then M is proper slant (respectively, anti-invariant). Also, if q 0 and θ 0, then M is an invariant submanifold of M. If ν is an invariant normal subbundle of the normal bundle a pseudo-slant submanifold M, then we have T M of T M ϕd1 FD ν. Now, we have the following results for later use on a pseudo-slant submanifold M of a nearly cosymplectic manifold M. Theorem 3.1. Let M be a pseudo-slant submanifold of a nearly cosymplectic manifold M, then g( Y, ) sec θg( A TY A, ) + tan θη( Y ) g( ξ ) ϕ FTY, sec θg( Q Y, ϕ ) sec θg( Q, FY ), (3.1) for any, Y Γ( D ξ ) and any Γ( ). D 1

7 A CHARACTERIATION OF WARPED PRODUCT 7 Proof. For any, Y Γ( D ξ ) and Γ( ), we have g( Y, ) g( ϕ Y, ϕ ) D 1 g( ϕy, ϕ ) g(( ϕ) Y, ϕ ). Using (.6) and (.8), we get g( Y, ) g( TY, ϕ ) + g( FY, ϕ ) g( Q Y, F ). Then from (.) and (.5) (a), we obtain g( Y, ) g( h(, TY ) ϕ ) g( ϕ FY, ) g( Q Y, F ). Using the covariant differentiation property of ϕ, we derive g( Y, ) g( h(, TY ) ϕ ) + g(( ϕ) FY, ) g( ϕfy ), g( Q Y, F ) g( A TY ) g( FY ) g( ϕ, + P, BFY ), g( CFY, ) g( Q Y, F ). From the property ( p 3 ) (ii) and the Theorem., we obtain g( Y, ) g( Aϕ TY, ) g( Q, FY ) + sin θg( Y ), η( Y ) sin θg( ξ, ) + g( FTY, ) g( Q Y, F ). Then by (.4) and (.5), we arrive at cos g( Y, ) g( A TY, ) g( A, ) + η( Y ) sin θg( ξ ) θ ϕ FTY, g( Q, FY ) g( Q Y, F ). Thus, the proof follows from the last relation by using the symmetry of the shape operator A. We have the following corollary as a consequence of the above theorem.

8 8 Corollary 3.1. Let M be a pseudo-slant submanifold of a nearly cosymplectic manifold M, then the slant distribution D ξ defines a totally geodesic foliation on M if and only if g( A Aϕ TY, ) sin θη( Y ) g( ξ, ) and U ν, FTY Q for any, Y Γ( D ξ ), any Γ( D 1 ) and any U Γ( TM ). Theorem 3.. Let M be a pseudo-slant submanifold of a nearly cosymplectic manifold M, then cos θg ([, W ], ) g( A W A, T ) cos θη( ) g( ξ W ) ϕ ϕw, for any, W Γ( D 1 ) and any Γ( D ξ ). g( P W, T ), (3.) Proof. For any, W Γ( D 1 ) and Γ( D ξ ), we have g W, ([, W ], ) g( W, ) g( ) g( ϕ W, ϕ ) g( ϕ, ϕ ) + η( ) g( W, ξ) η( ) g(, ξ). W Then by the covariant derivative property of ϕ, we get g ([, W ], ) g( ϕw, ϕ ) g(( ϕ) W, ϕ ) g( ϕ, ϕ ) W W + g(( W ϕ), ϕ ) η( ) g( ξ, W ) + η( ) g( W ξ, ). By the orthogonality of vector fields and using (.4), (.6), and (.8), we obtain g, ([, W ], ) g( ϕw, ϕ ) g( P W, T ) g( Q W F ) g( ϕ, ϕ ) + g( P, T ) + g( Q F ) W W W, η ( ) g(, W ). ξ Again using (.6), (.9), (.), (.5) and the covariant derivative of ϕ, we derive

9 g A CHARACTERIATION OF WARPED PRODUCT 9, ([, W ], ) g( ϕw, h(, T )) + g( ϕf, W ) g(( ϕ) F W ) g( P W, T ) g( Q W, F ) + g( ϕ, h( W T )), g( ϕf, ) + g(( ϕ) F, ) η( ) g( ξ, W ). W W g( A W A, T ) + g( BF, W ) + g( CF W ) ϕ ϕw, g( P F, W ) g( P W, T ) g( Q W F ), g( BF, ) g( CF, ) + g( P F ) W W W, η( ) g( ξ, W ). From the property ( p 3 ) (ii) and the Theorem., we arrive at g ϕ ϕw, ([, W ], ) g( A W A, T ) sin θg(, W ) + g( Q W F ) + sin θη( ) g( ξ, W ) g( FT, W ) g( Q F ) W, g( P W, T ) g( Q W, F ) + sin θg( ) W, sin θη( ) g( ξ, ) + g( FT, ) η( ) g( ξ, W ). W Using the orthogonality of distributions and (.4), (.5), we obtain g ϕ ϕw, ([, W ], ) g( A W A, T ) + sin θg( W ) + sin W θη( ) g( ξ, W ) + g( A, W ) g( P W T ) FT, sin θg(, ) g( A W, ) η( ) g( ξ, W ). W FT By the symmetry of the shape operator A, we derive cos θg ([, W ], ) g( A W A, T ) cos θη( ) g( ξ W ) ϕ ϕw, g( PW, T ), which is (3.), hence the proof is complete.

10 10 In [1], Al-Solamy obtained the following results which are useful to prove the characterization of warped products. He also derived an inequality for the second fundamental form of the warped product pseudo-slant submanifold of the type M M f M of a nearly cosymplectic manifold M such that the structure vector field ξ tangent to M θ. θ Lemma 3.1 ([1]). Let M θ M M f be a warped product pseudoslant submanifold of a nearly cosymplectic manifold M, then (i) ξ ln f 0; (ii) g ( h(, Y ), ϕ ) g( h(, ), FY ) + g( h( Y, ), F ), for any, Y Γ( TM ) and Γ( TM ). θ Lemma 3. ([1]). Let M θ M M f be a warped product pseudoslant submanifold of a nearly cosymplectic manifold M, where M and M θ are anti-invariant and proper slant submanifolds of M, respectively. Then, g( h(, FT ) g( h( T, ), ϕw ) + g( h( T, ϕ ) cos θ( ln f ) g(, for any Γ( TM ) and W Γ( TM ). θ, Now, we are able to prove the main result of this section. Theorem 3.3. Let M be a pseudo-slant submanifold of a nearly cosymplectic manifold M such that the normal component of ( U ϕ)v lies in the invariant normal subbundle of M, i.e., Q UV Γ( ν), for any U, V Γ( TM ) and the tangential component of ( ϕ)w lies in the anti-

11 A CHARACTERIATION OF WARPED PRODUCT 11 invariant distribution, i.e., P W Γ( D 1 ), for any, W Γ( D1 ). Then M is locally a mixed geodesic warped product submanifold if and only if Aϕ 0, AFT cos θ( µ ), and g( ξ, U ) 0, (3.3) for any Γ( D ξ ), Γ( D 1 ) and U Γ( TM ), where µ is a smooth function on M such that W ( µ ) 0, for any W Γ( ). Proof. If M is a mixed geodesic warped product pseudo-slant submanifold of the form M M f M of a nearly cosymplectic θ manifold M, then first two parts of (3.3) directly follow from Lemma 3.1 (ii) and Lemma 3.. Also, g( ξ, U ) ξ ln fg(, U ) 0 (using Lemma 3.1 (i)), for any Γ( TM ) and any U Γ( TM ). Conversely, if M is a pseudo-slant submanifold with the given conditions such that (3.3) holds, then from (3.1), we have g( Y, ) g( A, TY ) g( A, ) + sin θη( Y ) g( ξ ) cos θ ϕ FTY, D 1 g( Q Y, ϕ ) g( Q, FY ). Then from (3.3) and by the given fact that Q UV Γ( ν), for any U, V Γ( TM ), we get cos θg( Y, ) 0, since M is proper pseudoslant then θ π thus the leaves of D ξ are totally geodesic in M. On the other hand, from (3.), we have cos θ g ([, W ], ) g( A T, W ) g( A T ) ϕ ϕw, cos θη( ) g( ξ, W ) g( PW, T ), for any, W Γ( D 1 ) and any Γ( D ξ ). Using (3.3) and the fact that Q UV Γ( ν), for any U, V Γ( TM ) and P W Γ( D 1 ), for any we get cos θ ([, W ], ) 0., W Γ( D1 ), g Since M is proper slant then cos θ 0 thus we conclude that D 1 is integrable. If we consider a

12 1 leaf M of D 1 in M with second fundamental form ĥ of for any, W Γ( D 1 ) and Γ( D ξ ), we have g( hˆ (, ) g( W, ) g( W, ). From (.), we obtain g( hˆ (, ) g( ϕ W, ϕ ) + η( ) g( W, ξ) M in M, then g( ϕw, ϕ ) g(( ϕ) W, ϕ ) η( ) g( ξ, W ). Again using (.), (.6), (.8) and the last term of (3.3), we arrive at g( hˆ (, ) g( ϕw, T ) g( ϕw, F ) g( P W, T ) g( Q W, F ). Then by the hypothesis of the theorem, the last two terms of the right hand side are identically zero. Thus, on using (.), (.5) and the covariant derivative property of ϕ, we derive g( hˆ (, ) g( ϕw, h(, T )) + g( ϕf, W ) g(( ϕ) F W ), g( A T, ) + g( BF, W ) + g( CF W ) ϕ W, g( P F, W ). Hence, the first term of the right and side is identically zero by using the first part of (3.3), then from the property ( p 3 ) (ii) and Theorem., we obtain g( h ˆ (, ) sin θg(, W ) + sin θη( ) g( ξ W ), g( FT, W ) + g( Q W, F ). By the hypothesis of the theorem, the second and the last terms in the right hand side vanish identically. Then, we get g( h ˆ (, ) sin θg( W, ) + g( A, W ). FT

13 A CHARACTERIATION OF WARPED PRODUCT 13 Then, from the second part of (3.3), we derive cos ( ˆ θ g h(, ) cos θ( µ ) g(, or h ˆ(, W ) µ g(, which shows that M is a totally umbilical submanifold of M with mean curvature ˆ H µ, where µ is the gradient of the function µ. Furthermore, W ( µ ) 0, for any W Γ( D 1 ) implies that Ĥ is parallel mean curvature vector. Then by a result of Hiepko [10], we conclude that M is locally a warped product manifold, which completes the proof. 9 Example 3.1. Consider a submanifold of R such that the Cartesian 9 coordinates of R are ( x 1, x, x3, x4, y1, y, y3, y4, t ) and the almost contact structure is given by ϕ x i y i,, ϕ ϕ 0, 1 i, j y j x j t 4. If 9 λi + µ j + ν is a vector field of R, then ϕ xi y j t µ j x j λ + and ϕ ( ) λi µ j + ν. Also, we can see x y t i that g (, ) λi + µ j + ν and g ( ϕ, ϕ ) λi + µ j, where g is the 9 Euclidean metric tensor of R. Then, we have g ( ϕ, ϕ ) g(, ) η( ) ξ, where ξ and hence ( ϕ, ξ, η, g) is an almost contact structure on t 9 R. Now consider a submanifold M of j 9 R defined by the immersion t t χ ( u, v, w, s, t) ( u cos θ, u sin θ, e w, 0, v cos θ, v sin θ, 0, e s, t ), for any θ 0, π. Then, the tangent space TM of M is spanned by the 4 following vectors: i y i

14 14 1 cos θ x 1 + sin θ x, cos θ y 1 + sin θ y, 3 t e x3 t, 4 e, 5 y 4. t Thus, we have ϕ 1 cos θ y 1 sin θ y, ϕ cos θ x 1 + sin θ x, 3 t t ϕ e, ϕ4 e, ϕ5 y 3 x4 Clearly, the vector fields ϕ 3 and ϕ 4 are orthogonal to TM. Then the anti-invariant and proper slant distributions of M, respectively, are D 1 Span{ 3, 4 } and D Span{ 1, } with slant angle 0. θ 1 cos 1 g( 1, ϕ ) 1 g(, ϕ ) cos 1 θ such that ξ 5 is tangent to M. 1 ϕ ϕ1 9 Hence, M is a proper pseudo-slant submanifold of R. Furthermore, it is easy to se that D 1 and D are integrable. We denote the integral manifolds of D 1 and D by tensor g of M is M and M, respectively. Then the metric θ g du + dv + dt t + e t ( dw + ds ) g + e g. 1 Thus M is a warped product pseudo-slant submanifold of 9 R with warping function t e f such that ξ is tangent to M. 5 θ References [1] F. R. Al-Solamy, An inequality for warped product pseudo-slant submanifolds of nearly cosymplectic manifolds, J. Inequal. Appl. 306 (015), 1-9. [] R. L. Bishop and B. O Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), [3] D. E. Blair, Almost contact manifolds with Killing structure tensors I, Pac. J. Math. 39 (1971), 85-9.

15 A CHARACTERIATION OF WARPED PRODUCT 15 [4] D. E. Blair and K. Yano, Affine almost contact manifolds and f-manifolds with affine killing structure tensors, Kodai Math. Sem. Rep. 3 (1971), [5] D. E. Blair and D. K. Showers, Almost contact manifolds with Killing structures tensors II, J. Differ. Geom. 9 (1974), [6] J. L. Cabrerizo, A. Carriazo, L. M. Fernandez and M. Fernandez, Slant submanifolds in Sasakian manifolds, Glasgow Math. J. 4 (000), [7] A. Carriazo, New Developments in Slant Submanifolds Theory, Narosa Publishing House, New Delhi, India, 00. [8] B. -Y. Chen, Slant immersions, Bull. Austral. Math. Soc. 41 (1990), [9] H. Endo, On the curvature tensor of nearly cosymplectic manifolds of constant ϕ -sectional curvature, Ann. St. Univ. Iasi. Tom. LI f (005), [10] S. Hiepko, Eine inner kennzeichungder verzerrten produkte, Math. Ann. 41 (1979), [11] B. Sahin, Warped product submanifolds of Kaehler manifolds with a slant factor, Ann. Pol. Math. 95 (009), [1] S. Uddin, S. H. Kon, M. A. Khan and Khushwant Singh, Warped product semiinvariant submanifolds of nearly cosymplectic manifolds, Math. Probl. Eng. 011 (011). [13] S. Uddin, A. Mustafa, B. R. Wong and C. Ozel, A geometric inequality for warped product semi-slant submanifolds of nearly cosymplectic manifolds, Rev. Dela Union Math. Argentina 55 (014), [14] S. Uddin and F. R. Al-Solamy, Warped product pseudo-slant submanifolds of cosymplectic manifolds, An. Ştiinţ. Univ. Al. I. Cuza Iaşi Mat. (N. S.) Tome LIII f 3 (016), [15] S. Uddin and F. R. Al-Solamy, Warped product pseudo-slant immersions in Sasakian manifolds, Publ. Math. Debrecen 91() (017), g