A general inequality for contact CR-warped product submanifolds in cosymplectic space forms
aa r X i v : . [ m a t h . DG ] J u l Accepted Manuscript: Comptes rendus de l’Academie bulgare des Sciences
A general inequality for contact CR-warped productsubmanifolds in cosymplectic space forms
Falleh R. Al-Solamy and Siraj Uddin
Dedicated to the MSU Distinguished Professor Bang-Yen Chenon the occasion of his 73rd birthday, in gratitude for his guidance and friendship
Abstract
B.-Y. Chen initiated the study of warped product submanifolds in hisfundamental seminal papers [6, 7, 8]. In this paper, we study contactCR-warped product submanifolds of cosymplectic space forms and provean optimal inequality by using Gauss and Codazzi equations. In addition,we obtain two geometric inequalities for contact CR-warped product sub-manifolds with a compact invariant factor.
Key words:
Warped products, contact CR-submanifolds, contact CR-warped product, compact manifold, geometric inequality, Dirichlet energy,cosymplectic space form.
Let ( M , g ) and ( M , g ) be two Riemannian manifolds and f : M → (0 , ∞ )and π : M × M → M , π : M × M → M , the projections map givenby π ( p, q ) = p and π ( p, q ) = q for any ( p, q ) ∈ M × M . Then, the warpedproduct M = M × f M is the product manifold M × M equipped with theRiemannian structure such that g ( X, Y ) = g ( π ∗ X, π ∗ Y ) + ( f ◦ π ) g ( π ∗ X, π ∗ Y ) (1.1)for any X, Y tangent to M , where ∗ is the symbol for the tangent maps. Thefunction f is called the warping function of M . In particular a warped productmanifold is said to be trivial or Riemannian product manifold if the warpingfunction is constant.Let M = M × f M be a warped product. For a vector field X tangent to M ,the lift of X on M = M × f M is the tangent vector field ˜ X on M = M × f M whose value at each ( p, q ) is the lift X p to ( p, q ). Thus, the lift of X is theunique vector field on M = M × f M that is π M -related to X and π M -relatedto the zero vector field on M . The set of all such lifts of vector fields on M is L ( M ). Similarly, we denote by L ( M ) the lifts of vector fields fromvector fields tangent to M .Then for unit vector fields X, Y ∈ L ( M ) and Z ∈ L ( M ), we have ∇ X Z = ∇ Z X = X (ln f ) Z, g ( ∇ X Y, Z ) = 0 (1.2)which implies that ([19], page 210) K ( X ∧ Z ) = 1 f (cid:8) ( ∇ X X ) f − X f (cid:9) . (1.3)If we choose the local orthonormal frame e , · · · , e n such that e , · · · , e n aretangent to M and e n +1 , · · · , e n are tangent to M , then we have∆ ff = n X i =1 K ( e i ∧ e j ) (1.4)for each j = n + 1 , · · · , n . For the most up-to-date survey on warped productmanifolds and submanifolds, we refer to B.-Y. Chen’s books [11, 13] and hissurvey article [12].Recently, M.-I. Munteanu established an inequality in [18] for the squarednorm of the second fundamental form of a contact CR-warped product subman-ifold in Sasakian space form along a similar line of B.-Y. Chen [9, 10]. Further,a similar inequality has been obtained for contact CR-warped products in Ken-motsu space forms by Arslan et al. in [1]. On the other hand, warped productsubmanifolds of cosymplectic manifolds were studied in [15], [16] and [21, 22, 23].Motivated by these work done in this spirit, we establish in this paper thefollowing inequality. Theorem 1.1.
Let ˜ M ( c ) be a (2 m + 1) -dimensional cosymplectic space formwith constant sectional curvature c and M = M T × f M ⊥ be a warped productsubmanifold of ˜ M ( c ) . Then we have(i) The squared norm of the second fundamental form σ of M satisfies k σ k ≥ q (cid:16) k∇ (ln f ) k − ∆(ln f ) + pc (cid:17) , (1.5) where dim M T = 2 p + 1 , dim M ⊥ = q and ∇ (ln f ) is the gradient of ln f and ∆ is the Laplacian operator of M T .(ii) If the equality sign holds in (i), then M T is totally geodesic in ˜ M ( c ) and M ⊥ is a totally umbilical submanifold of ˜ M ( c ) . The paper is organized as follows: Section 2 is devoted to preliminaries. InSection 3, first we develop some basic results for later use and then we proveTheorem 1.1. In the last section, we prove two geometric inequalities as anapplication of Theorem 1.1 by considering compact invariant factor M T .2 Preliminaries
Let ˜ M be a (2 m + 1)-dimensional almost contact manifold with almost contactstructure ( ϕ, ξ, η ), i.e., a structure vector field ξ , a (1 ,
1) tensor field ϕ and a1-form η on ˜ M such that ϕ X = − X + η ( X ) ξ, η ( ξ ) = 1, for any vector field X on ˜ M [5]. There always exists a compatible Riemannian metric g satisfying g ( ϕX, ϕY ) = g ( X, Y ) − η ( X ) η ( Y ), for any vector field X, Y tangent to ˜ M . Thusthe manifold ˜ M is said to be almost contact metric manifold and ( ϕ, ξ, η, g ) isits almost contact metric structure. It is clear that η ( X ) = g ( X, ξ ). The funda-mental 2-form Φ on ˜ M is defined as Φ( X, Y ) = g ( X, ϕY ), for any vector fields
X, Y tangent to ˜ M . The manifold ˜ M is said to be almost cosymplectic if theforms η and Φ are closed, i.e., dη = 0 and d Φ = 0, where d is an exterior differ-ential operator. An almost cosyplectic and normal manifold is cosymplectic . Itis well known that an almost contact metric manifold ˜ M is cosymplectic if andonly if ˜ ∇ X ϕ vanishes identically, where ˜ ∇ is the Levi-Civita connection on ˜ M [17].A cosymplectic manifold ˜ M with constant ϕ -sectional curvature is called a cosymplectic space form and denoted by ˜ M ( c ). Then the Riemannian curvaturetensor ˜ R is given by˜ R ( X, Y ; Z, W ) = c (cid:8) g ( X, W ) g ( Y, Z ) − g ( X, Z ) g ( Y, W ) + g ( X, ϕW ) g ( Y, ϕZ ) − g ( X, ϕZ ) g ( Y, ϕW ) − g ( X, ϕY ) g ( Z, ϕW ) − g ( X, W ) η ( Y ) η ( Z ) + g ( X, Z ) η ( Y ) η ( W ) − g ( Y, Z ) η ( X ) η ( W ) + g ( Y, W ) η ( X ) η ( Z ) (cid:9) . (2.1)Let M be a n -dimensional Riemannian manifold isometrically immersed ina Riemannian manifold ˜ M . Then, the Gauss and Weingarten formulae arerespectively given by ˜ ∇ X Y = ∇ X Y + σ ( X, Y ) and ˜ ∇ X N = − A N X + ∇ ⊥ X N , forany X, Y tangent to M , where ∇ is the induced Riemannian connection on M , N is a vector field normal to M , σ is the second fundamental form of M , ∇ ⊥ is thenormal connection in the normal bundle T M ⊥ and A N is the shape operator ofthe second fundamental form. They are related as g ( A N X, Y ) = g ( σ ( X, Y ) , N ),where g denotes the Riemannian metric on ˜ M as well as the metric induced on M . Let M be an n -dimensional submanifold of an almost contact metric (2 m +1)-manifold ˜ M such that restricted to M , the vectors e , · · · , e n are tangent to M and hence e n +1 , · · · e m +1 are normal to M . Then, the mean curvaturevector ~H is defined by ~H = n trσ = n P ni,j =1 σ ( e i , e i ), where { e , · · · , e n } is alocal orthonormal frame of the tangent bundle T M of M . A submanifold M is called minimal in ˜ M if its mean curvature vector vanishes identically and M is totally geodesic in ˜ M , if σ ( X, Y ) = 0, for all
X, Y tangent to M . If σ ( X, Y ) = g ( X, Y ) H for all X, Y tangent to M , then M is totally umbilical submanifold of ˜ M . 3or any X tangent to M , we decompose ϕX as φX = P X + F X , where
P X and
F X are the tangential and normal components of ϕX , respectively. For asubmanifold M of an almost contact manifold ¯ M , if F is identically zero then M is invariant and if P is identically zero then M is anti-invariant .Let R and ˜ R denote the Riemannian curvature tensors of M and ˜ M , respec-tively. Then the equation of Gauss is given by R ( X, Y ; Z, W ) = ˜ R ( X, Y, Z, W ) + g ( σ ( X, W ) , σ ( Y, Z )) − g ( σ ( X, Z ) , σ ( Y, W )) , (2.2)for X, Y, Z, W tangent to M .For the second fundamental form σ , we define the covariant derivative ˜ ∇ σ by ( ˜ ∇ X σ )( Y, Z ) = ∇ ⊥ X σ ( Y, Z ) − σ ( ∇ X Y, Z ) − σ ( Y, ∇ X Z ) (2.3)for any X, Y, Z tangent to M .The equation of Codazzi is( ˜ R ( X, Y ) Z ) ⊥ = ( ˜ ∇ X σ )( Y, Z ) − ( ˜ ∇ Y σ )( X, Z ) , (2.4)where ( ˜ R ( X, Y ) Z ) ⊥ is the normal component of ( ˜ R ( X, Y ) Z ).Let M be a Riemannian p -manifold and e , · · · , e p be an orthonormal framefields on M . Then for a differentiable function ψ on M , the Laplacian ∆ ψ of ψ is defined by ∆ ψ = n X i =1 (cid:8) ( ˜ ∇ e i e i ) ψ − e i e i ψ (cid:9) . (2.5)The scalar curvature of M at a point p in M is given by τ ( p ) = X ≤ i Let M = M T × f M ⊥ be a contact CR-warped productsubmanifold of a cosymplectic space form ˜ M ( c ) such that ξ is tangential to M T .Then, for the unit normal vectors X ∈ L ( M T ) and Z ∈ L ( M ⊥ ), from (2.4), wehave ˜ R ( X, ϕX, Z, ϕZ ) = g (( ∇ ⊥ X σ )( ϕX, Z ) , ϕZ ) − g (( ∇ ⊥ ϕX σ )( X, Z ) , ϕZ ) . Then from (2.3), we derive˜ R ( X, ϕX, Z, ϕZ ) = g ( ∇ ⊥ X σ ( ϕX, Z ) − σ ( ∇ X ϕX, Z ) − σ ( ϕX, ∇ X Z ) , ϕZ ) − g ( ∇ ⊥ ϕX σ ( X, Z ) − σ ( ∇ ϕX X, Z ) − σ ( X, ∇ ϕX Z ) , ϕZ ) . (3.4)Now, we compute the following terms as follows g ( ∇ ⊥ X σ ( ϕX, Z ) , ϕZ ) = Xg ( σ ( ϕX, Z ) , ϕZ ) − g ( σ ( ϕX, Z ) , ∇ ⊥ X ϕZ )= Xg ( ˜ ∇ Z ϕX, ϕZ ) − g ( σ ( ϕX, Z ) , ˜ ∇ X ϕZ ) . Using the cosymplectic characteristic equation and the compatible metric prop-erty and the fact that ξ is tangent to M T , we derive g ( ∇ ⊥ X σ ( ϕX, Z ) , ϕZ ) = Xg ( ˜ ∇ Z X, Z ) − g ( σ ( ϕX, Z ) , ϕ ˜ ∇ X Z ) . Then by Gauss formula and the relation (1.2), we obtain g ( ∇ ⊥ X σ ( ϕX, Z ) , ϕZ ) = X ( X (ln f )) g ( Z, Z ) − X (ln f ) g ( σ ( ϕX, Z ) , ϕZ ) − g ( σ ( ϕX, Z ) , ϕσ ( X, Z )) . Then from (3.1) and (3.3) , we get g ( ∇ ⊥ X σ ( ϕX, Z ) , ϕZ ) = X (ln f ) g ( Z, Z ) + 2 X (ln f ) g ( ∇ X Z, Z ) − ( X (ln f )) g ( Z, Z ) − k σ ( X, Z ) k − ϕX (ln f ) g ( σ ( X, Z ) , ϕZ ) . Again using (1.2) and (3.1), we derive g ( ∇ ⊥ X σ ( ϕX, Z ) , ϕZ ) = X (ln f ) g ( Z, Z ) + ( X (ln f )) g ( Z, Z ) − k σ ( X, Z ) k + ( ϕX (ln f )) g ( Z, Z ) . (3.5)6ince M T is invariant and totally geodesic in M T × f M ⊥ [4, 6], then by thecosymlectic characteristic equation, we have g ( σ ( ∇ X ϕX, Z ) , ϕZ ) = g ( σ ( ϕ ∇ X X, Z ) , ϕZ ) . (3.6)Also, from (3.1), we have g ( h ( ϕX, Z ) , ϕW ) = X (ln f ) g ( Z, W ), thus with thehelp of this fact (3.6) becomes g ( σ ( ∇ X ϕX, Z ) , ϕZ ) = ( ∇ X X ln f ) g ( Z, Z ) . (3.7)Similarly, we obtain the following g ( σ ( ϕX, ∇ X Z ) , ϕZ ) = ( X ln f ) g ( σ ( ϕX, Z ) , ϕZ ) . Then from (3.1), we get g ( σ ( ϕX, ∇ X Z ) , ϕZ ) = ( X ln f ) g ( Z, Z ) . (3.8)Now, interchanging X by ϕX in (3.5), (3.7) and (3.8), then the following rela-tions hold respectively − g ( ∇ ⊥ ϕX σ ( X, Z ) , ϕZ ) = (( ϕX ) ln f ) g ( Z, Z ) + ( ϕX ln f ) g ( Z, Z ) − k σ ( ϕX, Z ) k + ( X ln f ) g ( Z, Z ) , (3.9) g ( σ ( ∇ ϕX X, Z ) , ϕZ ) = − ( ∇ ϕX ϕX ln f ) g ( Z, Z ) (3.10)and g ( σ ( X, ∇ ϕX Z ) , ϕZ ) = − ( ϕX ln f ) g ( Z, Z ) . (3.11)Now, we compute k σ ( ϕX, Z ) k = g ( σ ( ϕX, Z ) , σ ( ϕX, Z )) . Then from (3.3), we derive k σ ( ϕX, Z ) k = g ( ϕσ ( X, Z ) , ϕσ ( X, Z )) + 2( ϕX ln f ) g ( σ ( X, Z ) , ϕZ )+ ( X ln f ) g ( Z, Z ) + ( ϕX ln f ) g ( Z, Z ) . Then by the property of compatible metric and (3.1), we obtain k σ ( ϕX, Z ) k = k σ ( X, Z ) k − ( ϕX ln f ) g ( Z, Z ) + ( X ln f ) g ( Z, Z ) . (3.12)With the help of (3.12), the relation (3.9) becomes g ( ∇ ⊥ ϕX σ ( X, Z ) , ϕZ ) = − (( ϕX ) ln f ) g ( Z, Z ) + k σ ( X, Z ) k − ϕX ln f ) g ( Z, Z ) . (3.13)7hen from (3.4), (3.5), (3.7), (3.8) and (3.13), we derive˜ R ( X, ϕX, Z, ϕZ ) = ( X ln f ) g ( Z, Z ) − ( ∇ X X ln f ) g ( Z, Z )+ (( ϕX ) ln f ) g ( Z, Z ) − ( ∇ ϕX ϕX ln f ) g ( Z, Z ) − k σ ( X, Z ) k + 2( ϕX ln f ) g ( Z, Z ) . (3.14)Also, from (2.1) and (2.2), we have˜ R ( X, ϕX, Z, ϕZ ) = c (cid:0) η ( X ) − k X k (cid:1) g ( Z, Z ) , (3.15)for any unit tangent vector X ∈ L ( M T ) and Z ∈ L ( M ⊥ ). Then from (3.14) and(3.15), we get2 k σ ( X, Z ) k = ( X ln f ) g ( Z, Z ) − ( ∇ X X ln f ) g ( Z, Z ) + (( ϕX ) ln f ) g ( Z, Z ) − ( ∇ ϕX ϕX ln f ) g ( Z, Z ) + 2( ϕX ln f ) g ( Z, Z ) − c (cid:0) η ( X ) − k X k (cid:1) g ( Z, Z ) . (3.16)Now, consider the orthonormal frame fields of M T and M ⊥ as follows: { X , · · · , X p , X p +1 = ϕX , · · · , X p = ϕX p , X p +1 = ξ } and { Z , · · · , Z q } arethe frame fields of the tangent spaces of M T and M ⊥ , then summing over i = 1 , · · · , p + 1 and j = 1 , · · · , q in (3.16), thus we have2 p +1 X i =1 q X j =1 k σ ( X i , Z j ) k = − p +1 X i =1 q X j =1 (cid:0) ( ∇ X i X i (ln f )) − X i (ln f ) (cid:1) g ( Z j , Z j ) − p +1 X i =1 q X j =1 (cid:0) ( ∇ ϕX i ϕX i (ln f )) − ( ϕX i ) (ln f ) (cid:1) g ( Z j , Z j )+ 2 p +1 X i =1 q X j =1 ( ϕX i (ln f )) g ( Z j , Z j ) − p +1 X i =1 q X j =1 c (cid:0) η ( X i ) − k X i k (cid:1) g ( Z j , Z j ) . Then, from the definition of the gradient and (2.5), we derive2 p +1 X i =1 q X j =1 k σ ( X i , Z j ) k = − q ∆(ln f ) + 2 q k ϕ ∇ (ln f ) k + pqc or k σ k T ⊥ = − q ∆(ln f ) + q k∇ (ln f ) k + pqc k σ k ≥ k σ k T ⊥ only and left all otherterms in the right hand side in the inequality of that theorem. Then using (3.17)in this relation we get inequality (i) with equality sign holds if and only if σ ( L ( M T ) , L ( M T )) = σ ( L ( M ⊥ ) , L ( M ⊥ )) = 0 . (3.18)i.e., M is both M T and M ⊥ -totally geodesic. The equality case holds just likeTheorem 3.1. Hence, the proof is complete. For a warped product CR-submanifold M T × f M ⊥ of a cosymplectic space form,if the holomorphic submanifold M T is compact, then we have the following usefulresults. Theorem 4.1. Let M T × f M ⊥ be a warped product CR-submanifold of a cosym-plectic space form ˜ M ( c ) such that M T is a compact invariant submanifold of ˜ M ( c ) . Then(i) For any s ∈ M ⊥ , we have Z M T ×{ s } k σ k dV T ≥ pqc vol(M T ) , (4.1) where dV T and vol(M T ) are the volume element and the volume of M T ,respectively and p + 1 = dim M T , q = dim M ⊥ .(ii) The equality sign holds in (i) identically if and only if M is a Riemannianproduct of M T and M ⊥ , i.e., the warping function f is constant on M .Proof. For a warped product submanifold M T × f M ⊥ with compact M T , fromTheorem 1.1, we have Z M T ×{ s } k σ k dV T ≥ q Z M T ×{ s } (cid:16) k∇ (ln f ) k − ∆(ln f ) + pc (cid:17) dV T . (4.2)Since M T is compact, it follows from Hopf’s Lemma that Z M T ×{ s } k σ k dV T ≥ q Z M T ×{ s } (cid:0) k∇ (ln f ) k (cid:1) dV T + pqc vol(M T ) . (4.3)Thus, inequality (4.3) implies inequality (4.1), with the equality sign holding ifand only if (1) f is constant i.e., M is Riemannian product and (2) the equality k σ k = pqc holds identically. Hence, the theorem is proved completely.Next, let us assume f is non-constant. Then the minimal principle on λ yields (see [3] page 186, [10]) Z M T k∇ (ln f ) k dV T ≥ λ Z M T (ln f ) dV T (4.4)9ith equality holding if and only if ∆ ln f = λ ln f holds.Now, we give the following result. Theorem 4.2. Let M = M T × f M ⊥ be a warped product CR-submanifold of acosymplectic space form ˜ M ( c ) with compact M T . If the warping function f isnon-constant, then, for each s ∈ M ⊥ , we have Z M T ×{ s } k σ k dV T ≥ qλ Z M T (ln f ) dV T + pqc vol(M T ) (4.5) where dV T , λ and vol(M T ) are the volume element, the first positive eigenvalueof the Laplacian ∆ and the volume of M T , respectively.Moreover, the equality sign of (4.5) holds identically if and only if we have:(i) ∆ ln f = λ ln f (ii) M is both M T -totally geodesic and M ⊥ -totally geodesic.Proof. By combining (4.3) and (4.4), we get inequality (4.5). From the abovediscussion, we know that the equality sign of (4.5) holds identically if and onlyif we have (i) ∆ ln f = λ ln f and (ii) the warped product is both M T and M ⊥ totally geodesic.Another motivation of Theorem 1.1 is to give the expression of Dirichletenergy of the warping function in physics, which is defined of a function ψ on acompact manifold M as follows E ( ψ ) = 12 Z M k∇ ψ k dV (4.6)where ∇ ψ is the gradient of ψ and dV is the volume element.Now, we give the expression of Dirichlet energy of the warping function fora contact CR-warped product M T × f M ⊥ in a cosymplectic space form ˜ M ( c )with compact invariant submanifold M T . For any s ∈ M ⊥ , from Theorem 1.1,we have Z M T ×{ s } k σ k dV T ≥ q Z M T ×{ s } (cid:16) k∇ ln f k + pc (cid:17) dV T . (4.7)From (4.6) and (4.7), we find E (ln f ) ≤ q Z M T ×{ s } k σ k dV T − pc T )which is the Dirichlet energy E (ln f ) (0 ≤ E (ln f ) < ∞ ) of the warping function. Acknowledgements. 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Alqahtani, Chen type inequality for warped product im-mersions in cosymplectic space forms , J. Nonlinear Sci. Appl. (2016),2914–2921.[24] S. Uddin and F.R. Al-Solamy, Another proof of derived inequality for warpedproduct semi-invariant submanifolds of cosymplectic manifolds , J. Math.Anal. (4) (2016), 93–97.[25] S. Uddin, B.-Y. Chen and F.R. Al-Solamy, Warped product bi-slant im-mersions in Kaehler manifolds , Mediterr. J. Math., (2017) :95. doi:10.1007/s00009-017-0896-8.[26] S. Uddin and F.R. Al-Solamy, Warped product pseudo-slant immersions inSasakian manifolds , Pub. Math. Debrecen (2) (2017), 1-14.Author’s addresses:Falleh R. Al-Solamy:Department of Mathematics, Faculty of Science, King Abdulaziz University,21589 Jeddah, Saudi Arabia E-mail : [email protected] Siraj Uddin:Department of Mathematics, Faculty of Science, King Abdulaziz University,21589 Jeddah, Saudi Arabia E-mail : [email protected]@gmail.com