Characterization of warped product submanifolds of Lorentzian concircular structure Manifolds
aa r X i v : . [ m a t h . DG ] M a r . CHARACTERIZATION OF WARPED PRODUCT SUBMANIFOLDSOF LORENTZIAN CONCIRCULAR STRUCTURE MANIFOLDS
SHYAMAL KUMAR HUI ∗ , LAURIAN-IOAN PISCORAN AND TANUMOY PAL Abstract.
Recently Hui et al. ([10], [11]) studied contact CR-warped productsubmanifolds and also warped product pseudo-slant submanifolds of a (
LCS ) n -manifold ¯ M . In this paper we have studied the characterization for both theseclasses of warped product submanifolds. It is also shown that there do not existsany proper warped product bi-slant submanifold of a ( LCS ) n -manifold. Althoughwe constructed an example of a bi-slant submanifold of ( LCS ) n -manifold. Introduction
As a generalization of Riemannian product manifold Bishop and O’Neill [6] intro-duced the notion of warped product manifold and later it was studied in ([1], [2], [14],[17], [28]-[30]). The existence or non-existence of warped product manifolds playssome important role in differential geometry as well as physics.As a generalization of LP-Sasakian manifold introduced independently by Mat-sumoto [15] and also by Mihai and Rosca [16], Shaikh [19] introduced the notion ofLorentzian concircular structure manifolds (briefly, (
LCS ) n -manifolds) with an ex-ample. Then Shaikh and Baishya ([21], [22]) investigated the applications of ( LCS ) n -manifolds to the general theory of relativity and cosmology. The ( LCS ) n -manifoldsare also studied in ([8], [20], [23]-[26]).Due to important applications in applied mathematics and theoretical physics, thegeometry of submanifolds has become a subject of growing interest. Analogous to al-most Hermitian manifolds, the invariant and anti-invariant subamnifolds are dependon the behaviour of almost contact metric structure φ . The study of the differ-ential geometry of a contact CR-submanifold as a generalization of invariant andanti-invariant subamnifold was introduced by Bejancu [5]. In this connection it ismentioned that different class of submanifolds of ( LCS ) n -manifolds are studied in([3], [4], [9], [10], [12], [13], [27]). Recently Hui et al. ([10], [11]) studied contact CR-warped product submanifolds and also warped product pseudo-slant submanifolds ofa ( LCS ) n -manifold ¯ M . In this paper we have studied the characterization for boththese classes of warped product submanifolds. An example of bi-slant submanifold of( LCS ) n -manifold is constructed. However, it is also shown that there do not existsany proper warped product bi-slant submanifold of a ( LCS ) n -manifold. Mathematics Subject Classification.
Key words and phrases. ( LCS ) n -manifold, CR- submanifold, pseudo slant submanifold, bi-slantsubmanifold, warped product submanifold ∗ Corresponding author. ∗ , L-.I. PISCORAN AND T. PAL Preliminaries
Let ¯ M be an n -dimensional Lorentzian manifold [18] admitting a unit timelikeconcircular vector field ξ , called the characteristic vector field of the manifold. Thenwe have(2.1) g ( ξ, ξ ) = − . Since ξ is a unit concircular vector field, it follows that there exists a non-zero 1-form η such that for(2.2) g ( X, ξ ) = η ( X ) , the equation of the following form holds [31](2.3) ( ¯ ∇ X η )( Y ) = α { g ( X, Y ) + η ( X ) η ( Y ) } , ( α = 0)(2.4) ¯ ∇ X ξ = α { X + η ( X ) ξ } , α = 0for all vector fields X , Y , where ¯ ∇ denotes the operator of covariant differentiationwith respect to the Lorentzian metric g and α is a non-zero scalar function satisfies(2.5) ¯ ∇ X α = ( Xα ) = dα ( X ) = ρη ( X ) ,ρ being a certain scalar function given by ρ = − ( ξα ). If we put(2.6) φX = 1 α ¯ ∇ X ξ, then from (2.4) and (2.6) we have(2.7) φX = X + η ( X ) ξ, (2.8) g ( φX, Y ) = g ( X, φY )from which it follows that φ is a symmetric (1,1) tensor and called the structuretensor of the manifold. Thus the Lorentzian manifold ¯ M together with the unittimelike concircular vector field ξ , its associated 1-form η and an (1,1) tensor field φ is said to be a Lorentzian concircular structure manifold (briefly, ( LCS ) n -manifold),[19]. Especially, if we take α = 1, then we can obtain the LP-Sasakian structure ofMatsumoto [15]. In a ( LCS ) n -manifold ( n >
2) ¯ M , the following relations hold [19]:(2.9) η ( ξ ) = − , φξ = 0 , η ( φX ) = 0 , g ( φX, φY ) = g ( X, Y ) + η ( X ) η ( Y ) , (2.10) φ X = X + η ( X ) ξ, (2.11) ( ¯ ∇ X φ ) Y = α { g ( X, Y ) ξ + 2 η ( X ) η ( Y ) ξ + η ( Y ) X } , (2.12) ( Xρ ) = dρ ( X ) = βη ( X )for all X, Y, Z ∈ Γ( T ¯ M ) and β = − ( ξρ ) is a scalar function.Let M be a submanifold of ¯ M with induced metric g . Also let ∇ and ∇ ⊥ are the HARACTERIZATION OF WARPED PRODUCT SUBMANIFOLDS OF (
LCS ) n -MANIFOLDS 3 induced connections on the tangent bundle T M and the normal bundle T ⊥ M of M respectively. Then the Gauss and Weingarten formulae are given by(2.13) ¯ ∇ X Y = ∇ X Y + h ( X, Y )and(2.14) ¯ ∇ X V = − A V X + ∇ ⊥ X V for all X, Y ∈ Γ( T M ) and V ∈ Γ( T ⊥ M ), where h and A V are second fundamentalform and the shape operator (corresponding to the normal vector field V ) respectivelyfor the immersion of M into ¯ M and they are related by g ( h ( X, Y ) , V ) = g ( A V X, Y ),for any
X, Y ∈ Γ( T M ) and V ∈ Γ( T ⊥ M ).For any X ∈ Γ( T M ) and V ∈ Γ( T ⊥ M ), we can write(a) φX = P X + QX, (b) φV = bV + cV (2.15)where φX, bV are the tangential components and QX, cV are the normal compo-nents.A submanifold M of a ( LCS ) n -manifold ¯ M is said to be invariant if φ ( T p M ) ⊆ T p M ,for every p ∈ M and anti-invariant if φT p M ⊆ T ⊥ p M , for every p ∈ M .A submanifold M of a ( LCS ) n -manifold ¯ M is said to be a CR-submnaifold if there isa differential distribution D : p → D p ⊆ T p M such that D is an invariant distributionand the orthogonal complementary distribution D ⊥ is anti-invariant.The normal space of a CR-submanifold M is decomposed as T ⊥ M = Q D ⊥ ⊕ ν , where ν is the invariant normal subbundle of M with respect to φ .A submanifold M of a ( LCS ) n -manifold ¯ M is said to be slant if for each non-zerovector X ∈ T p M the angle between φX and T p M is a constant, i.e. it does not dependon the choice of p ∈ M .A submanifold M of a ( LCS ) n -manifold ¯ M is said to be a pseudo-slant submanifoldif there exists a pair of orthogonal distributions D ⊥ and D θ such that(i) TM admits the orthogonal direct decomposition T M = D ⊥ ⊕ D θ ,(ii) The distribution D ⊥ is anti-invariant,(iii) The distribution D θ is slant with slant angle θ = 0 , π .From the definition it is clear that if θ = 0, then M is a CR-submanifold. We say thata pseudo-slant submanifold is proper if θ = 0 , π . The normal space of a pseudo-slantsubmanifold M is decomposed as T ⊥ M = Q D θ ⊕ φ D ⊥ ⊕ ν .On a slant submanifold M of a ( LCS ) n -manifold ¯ M , we have [4] P X = cos θ [ X + η ( X ) ξ ] , (2.16) Q X = sin θ [ X + η ( X ) ξ ] , (2.17)where θ is the slant angle of M in ¯ M .From (2.16) and (2.17), we get g ( P X, P Y ) = cos θ [ g ( X, Y ) + η ( X ) η ( Y )] , (2.18) g ( QX, QY ) = sin θ [ g ( X, Y ) + η ( X ) η ( Y )] , (2.19) S. K. HUI ∗ , L-.I. PISCORAN AND T. PAL for any X, Y ∈ Γ( T M ).Also for a slant submanifold from (2.15) and (2.16), we have(2.20) bQX = sin θ ( X + η ( X ) ξ ) and cQX = − QP X
For a Riemannian manifold ¯ M of dimension n and a smooth function f on ¯ M , ∇ f ,the gradient of f which is defined by(2.21) g ( ∇ f, X ) = X ( f )(or = ( Xf ))for any X ∈ Γ( T M ). Definition 2.1. [6] Let ( N , g ) and ( N , g ) be two Riemannian manifolds withRiemannian metric g and g respectively and f be a positive definite smooth functionon N . The warped product of N and N is the Riemannian manifold N × f N =( N × N , g ), where(2.22) g = g + f g . A warped product manifold N × f N is said to be trivial if the warping function f is constant. Proposition 2.1. [18] Let M = N × f N be a warped product manifold. Then ∇ U X = ∇ X U = ( X ln f ) U, for any X , Y ∈ Γ( T N ) and U ∈ Γ( T N ). Theorem 2.1 (Hiepko’s Theorem) . [7] Let D and D be two orthogonal complemen-tary distributions on a Riemannian manifold M . Suppose that D and D are bothinvolutive such that D is a totally geodesic foliation and D is a spherical foliation.Then M is locally isometric to a non trivial warped product M × f M , where M and M are integral manifolds of D and D , respectively. characterization for contact CR-warped product submanifolds In [11] it is shown that contact CR-warped product submanifolds of ¯ M of the form N ⊥ × f N T , where N T and N ⊥ are invariant and anti-invariant submanifolds of ¯ M respectively, exists if ξ ∈ Γ( T N ⊥ ) and does not exists if ξ ∈ Γ( T N T ). In this sectionwe find a characterization for a submanifolds M of ¯ M to be contact CR-warpedproduct of the form N ⊥ × f N T such that ξ ∈ Γ( T N ⊥ ). First we prove the followingLemma: Lemma 3.1.
Let M = N ⊥ × f N T be a warped product submanifold of ¯ M such that ξ ∈ Γ( T N ⊥ ) , then g ( h ( X, Y ) , φZ ) = − αη ( Z ) g ( X, Y ) − ( Z ln f ) g ( φX, Y )(3.1) for X, Y ∈ Γ( T N T ) and Z ∈ Γ( T N ⊥ ) . HARACTERIZATION OF WARPED PRODUCT SUBMANIFOLDS OF (
LCS ) n -MANIFOLDS 5 Proof.
For
X, Y ∈ Γ( T N T ) and Z, ξ ∈ Γ( T N ⊥ ) we have from (2.11) and (2.13) that g ( h ( X, Y ) , φZ ) = g ( ¯ ∇ X Y, φZ )= g ( ¯ ∇ X φY, Z ) − g (( ¯ ∇ X φ ) Y, Z )= − g ( φY, ∇ X Z ) − αg ( X, Y ) η ( Z ) . By virtue of Proposition 2.1 from the above relation we get (3.1). (cid:3)
Now interchanging X by φX and Y by φY , we get the following respective relations g ( h ( φX, Y ) , φZ ) = − αη ( Z ) g ( φX, Y ) − Z (ln f ) g ( X, Y ) , (3.2) g ( h ( X, φY ) , φZ ) = − αη ( Z ) g ( φX, Y ) − Z (ln f ) g ( X, Y ) , (3.3) g ( h ( φX, φY ) , φZ ) = − αη ( Z ) g ( X, Y ) − Z (ln f ) g ( φX, Y ) . (3.4) Corollary 3.1.
Let M = N ⊥ × f N T be a warped product submanifold of ¯ M suchthat ξ ∈ Γ( T N ⊥ ). Then g ( h ( φX, Y ) , φZ ) = g ( h ( X, φY ) , φZ )and g ( h ( φX, φY ) , φZ ) = g ( h ( X, Y ) , φZ )for X, Y ∈ Γ( T N T ) and Z ∈ Γ( T N ⊥ ).Now we have the following characterization theorem: Theorem 3.1.
Let M be a contact CR-submanifold of a ( LCS ) n -manifold ¯ M suchthat ξ is tangent to the anti-invariant distribution D ⊥ . Then M is locally a warpedproduct submanifold if and only if (3.5) A φZ X = − αη ( Z ) X − ( Zµ ) φX for any X ∈ Γ( D ) and Z ∈ Γ( D ⊥ ) and also for some smooth function µ on M suchthat ( Y µ ) = 0 for any Y ∈ D .Proof. If M be a contact CR-warped product submanifold, then for any X ∈ Γ( T M T )and Z, W ∈ Γ( T M ⊥ ), we have g ( A φZ X, W ) − g ( h ( X, W ) , φZ ) = g ( ¯ ∇ W X, φZ ) = g ( φ ¯ ∇ W X, Z ) . Using (2.11) in the above equation we get(3.6) g ( A φZ X, W ) = g ( ¯ ∇ W φX, Z ) . Then from (2.13) and Proposition 2.1 we have from (3.6) that g ( A φZ X, W ) = 0 andtherefore A φZ X has no component in Γ( T N ⊥ ). Hence by virtue of Lemma 3.1, therelation (3.5) follows.Conversely, let M be a contact CR-submanifold of ¯ M with the invariant and anti-invariant distributions D and D ⊥ such that the relation (3.5) holds. Then for any S. K. HUI ∗ , L-.I. PISCORAN AND T. PAL X ∈ Γ( D ) and Z, W ∈ Γ( D ⊥ ), and using (2.13) we have g ( ∇ Z W, φX ) = g ( ¯ ∇ Z φW, X ) − g (( ¯ ∇ Z φ ) W, X )= − g ( φW, ¯ ∇ Z X )= − g ( A φW X, Z ) . Using (3.5) in above relation we get g ( ∇ Z W, φX ) = 0 . Similarly, we get g ( ∇ W Z, φX ) = 0 . Thus we obtain(3.7) g ( ∇ Z W + ∇ W Z, φX ) = 0 , Which implies that ∇ Z W + ∇ W Z ∈ Γ( D ⊥ ), i.e., D ⊥ is integrable and its leaves aretotally geodesic in M . Again for any X, Y ∈ Γ( D ) and Z ∈ Γ( D ⊥ ), we get g ( ∇ X Y, Z ) = g ( ¯ ∇ X φY, φZ ) + η ( Z ) g ( Y, ¯ ∇ X ξ ) . (3.8)Using (2.6) in (3.8), we get(3.9) g ( ∇ X Y, Z ) = g ( h ( X, φY ) , φZ ) + αη ( Z ) g ( Y, φX ) . Interchanging X and Y in (3.9), we get(3.10) g ( ∇ Y X, Z ) = g ( h ( φX, Y ) , φZ ) + αη ( Z ) g ( X, φY ) . From (3.9) and (3.10), we have(3.11) g ([ X, Y ] , Z ) = g ( h ( X, φY ) , φZ ) − g ( h ( φX, Y ) , φZ ) . Using (3.5) in (3.11), we get g ([ X, Y ] , Z ) = 0 and therefore D is integrable on M .Let us consider a leaf N T of D in M and let h T be the second fundamental form of N T in M , then we have g ( h T ( X, Y ) , Z ) = g ( φ ¯ ∇ Y X, φZ ) − η ( Z ) g ( ¯ ∇ Y X, ξ )(3.12) = − g ( φX, ¯ ∇ Y φZ ) + η ( Z ) g ( X, ¯ ∇ Y ξ ) . Using (2.6) and (2.14), (3.12) yields(3.13) g ( h T ( X, Y ) , Z ) = g ( φX, A φZ Y ) + αη ( Z ) g ( X, φY ) . From (3.5) and (3.13), we obtain(3.14) g ( h T ( X, Y ) , Z ) = − ( Zµ ) g ( X, Y ) . Using (2.21) in (3.14), we get h T ( X, Y ) = − ( ∇ µ ) g ( X, Y ) , (3.15)where ∇ µ is the gradient of the function µ and therefore N T is totally umbilical in M with mean curvature −∇ µ . Moreover, the condition ( Y µ ) = 0, for any Y ∈ D implies that the leaves of D are extrinsic spheres in M , i.e., the integral manifold N T of D is umbilical and its mean curvature vector field is non zero and parallel along N T . Hence by Hiepko’s theorem M is locally a warped product N ⊥ × f N T , where HARACTERIZATION OF WARPED PRODUCT SUBMANIFOLDS OF (
LCS ) n -MANIFOLDS 7 N T and N ⊥ denote the integral manifolds of the distributions D and D ⊥ respectivelyand f is the warping function. Thus the theorem is proved completely. (cid:3) characterization for warped product pseudo slant submanifolds Recently Hui et al. [10] studied warped product pseudo-slant submanifolds of(
LCS ) n -manifolds. In this section we obtain a characterization for a submanifold M of ¯ M to be a warped product pseudo-slant submanifold of the form N θ × f N ⊥ , where N θ is a slant submanifold tangent to ξ and N ⊥ is an anti-invariant submanifolds of¯ M . Lemma 4.1.
Let M be a proper pseudo-slant submanifold of a ( LCS ) n -manifold ¯ M with anti-invariant and proper slant dfistributions D ⊥ and D θ , respectively such that ξ ∈ Γ( D θ ) . Then (4.1) g ( ∇ X Y, Z ) = sec θ [ g ( h ( X, P Y ) , φZ ) + g ( h ( X, Z ) , QP Y )] , for any X, Y ∈ Γ( D θ ) and Z ∈ Γ( D ⊥ ) .Proof. For any
X, Y ∈ Γ( D θ ) and Z ∈ Γ( D ⊥ ), we have g ( ∇ X Y, Z ) = g ( φ ¯ ∇ X Y, φZ )= g ( ¯ ∇ X φY, φZ ) − g (( ¯ ∇ X φ ) Y, Z )= g ( ¯ ∇ XP Y, φZ ) + g ( ¯ ∇ X QY, φZ )= g ( h ( X, P Y ) , φZ ) + g ( ¯ ∇ XφQY, Z ) − g (( ¯ ∇ X φ ) QY, φZ )= g ( h ( X, P Y ) , φZ ) + g ( ¯ ∇ X bQY, Z ) + g ( ¯ ∇ X cQY, Z ) . Using (2.20) in the above relation, we get g ( ∇ X Y, Z ) = g ( h ( X, P Y ) , φZ ) + sin θg ( ¯ ∇ X Y, Z ) − g ( ¯ ∇ X QP Y, Z ) . From which (4.1) follows. (cid:3)
Corollary 4.1.
Let M be a proper pseudo-slant submanifold of a ( LCS ) n -manifold¯ M with anti-invariant and proper slant distributions D ⊥ and D θ , respectively suchthat ξ ∈ Γ( D θ ). Then the distribution D θ defines a totally geodesic foliation if andonly if g ( h ( X, P Y ) , φZ ) + g ( h ( X, Z ) , QP Y ) = 0for every X, Y ∈ Γ( D θ ) and Z ∈ Γ( D ⊥ ). Lemma 4.2.
Let M = N θ × f N ⊥ be a warped product submanifold of a ( LCS ) n -manifold ¯ M , where N ⊥ and N θ are ant-invariant and proper slant submanifold of ¯ M such that ξ ∈ Γ( T N θ ) . Then g ( h ( X, Y ) , φZ ) + g ( h ( X, Z ) , QY ) = 0 , (4.2) g ( h ( Z, W ) , QX ) + g ( h ( X, Z ) , QW ) = ( φX ln f ) g ( Z, W ) , (4.3) g ( h ( Z, W ) , QP X ) + g ( h ( P X, Z ) , QW ) = cos θ [( X ln f ) + αη ( X )] g ( Z, W ) . (4.4) S. K. HUI ∗ , L-.I. PISCORAN AND T. PAL Proof.
For any
X, Y ∈ Γ( T N θ ) and Z ∈ Γ( T N ⊥ ), we have g ( h ( X, Y ) , φZ ) = g ( ¯ ∇ X Y, φZ )= g ( φ ¯ ∇ X Y, Z )= g ( ¯ ∇ X φY, Z )= g ( ¯ ∇ X P Y, Z ) + g ( ¯ ∇ X QY, Z ) . Using Proposition 2.1 in the above relation, we get(4.5) g ( h ( X, Y ) , φZ ) = ( X ln f ) g ( Z, P Y ) − g ( h ( X, Z ) , QY ) . Thus (4.2) follows from (4.5). Also, for any X ∈ Γ( T N θ ) and Z, W ∈ Γ( T N ⊥ ), wehave g ( h ( Z, W ) , QX ) = g ( ¯ ∇ Z W, φX ) − g ( ¯ ∇ Z W, P X )= g ( ¯ ∇ Z φW, X ) − g (( ¯ ∇ Z φ ) W, X ) − g ( ¯ ∇ Z W, P X )= − g ( h ( X, Z ) , φW ) + g ( W, ¯ ∇ Z P X ) . Using Proposition 2.1 in the above relation we get (4.3). Interchanging X by P X in(4.3) we get (5.4) (cid:3)
Now, we prove the following characterization theorem for warped product pseudo-slant submanifolds.
Theorem 4.1.
Let M be a proper pseudo-slant submanifold of a ( LCS ) n -manifold ¯ M with anti-invariant distribution D ⊥ and proper pseudo-slant distribution D θ , re-spectively such that ξ ∈ Γ( D θ ) . Then M is locally a mixed-geodesic warped productsubmanifold of the form N θ × f N ⊥ if and only if (4.6) A φZ X = 0 and A QP X Z = cos θ [( Xµ ) + αη ( X )] Z, for any X ∈ Γ( D θ ) , Z ∈ Γ( D ⊥ ) and for some function µ on M satisfying ( Zµ ) = 0 ,for any Z ∈ Γ( D ⊥ ) .Proof. Let M = N θ × f N ⊥ be a mixed geodesic warped product submanifold ofa ( LCS ) n -manifold ¯ M such that (4.6) holds. Then for any X, Y ∈ Γ( D θ ) and Z ∈ Γ( D ⊥ ), from (4.2) and (5.4), we get (4.6).Conversely, Let M is a proper pseudo-slant submanifold of a ( LCS ) n -manifold ¯ M such that (4.6) holds.Then for any for any X, Y ∈ Γ( D θ ) and for any Z ∈ Γ( D ⊥ ), from (4.1) and (4.6),we get g ( ∇ X Y, Z ) = 0 and hence the leaves of D θ are totally geodesic in M . HARACTERIZATION OF WARPED PRODUCT SUBMANIFOLDS OF (
LCS ) n -MANIFOLDS 9 Also, for any X ∈ Γ( D ) and Z, W ∈ Γ( D ⊥ ), we have g ([ Z, W ] , X ) = g ( ¯ ∇ Z W, X ) − g ( ¯ ∇ W Z, X )= g ( φ ¯ ∇ Z W, φX ) − g ( φ ¯ ∇ W Z, φX )= g ( ¯ ∇ Z φW, φX ) − g ( ¯ ∇ W φZ, φX )= − g ( φW, ¯ ∇ Z φX ) + g ( φZ, ¯ ∇ W φX )= − g ( φW, ¯ ∇ Z P X ) − g ( φW, ¯ ∇ Z QX ) + g ( φZ, ¯ ∇ W P X ) + g ( φZ, ¯ ∇ W QX )= − g ( φW, h ( Z, P X )) − g ( W, ¯ ∇ Z bQX ) − g ( W, ¯ ∇ Z cQX )+ g ( φZ, h ( W, P X )) + g ( Z, ¯ ∇ W bQX ) + g ( Z, ¯ ∇ W cQX ) . Using (2.20) in the above relation, we get g ([ Z, W ] , X ) = − g ( A φW P X, Z ) + g ( A φZ P X, W ) + sin θg ([ Z, W ] , X )(4.7) + g ( A QP X
Z, W ) − g ( A QP X
W, Z ) . Using (4.6) in (4.7), we get(4.8) cos θg ([ Z, W ] , X ) = 0 . Since D θ is proper pseudo-slant so, θ = 0 , π . Therefore, g ([ Z, W ] , X ) = 0 and hencethe anti-invariant distribution D ⊥ is integrable.Now, let h ⊥ be the second fundamental form of a leaf N ⊥ of D ⊥ in M . Then for any Z, W ∈ Γ( D ⊥ ) and X ∈ Γ( D θ ), we have g ( h ⊥ ( Z, W ) , X ) = g ( φ ¯ ∇ Z W, φX )= g ( ¯ ∇ Z φW, φX )= g ( ¯ ∇ Z φW, P X ) + g ( ¯ ∇ Z φW, QX )= − g ( A φW Z, P X ) − g ( W, ¯ ∇ Z φQX )= − g ( A φW P X, Z ) − g ( W, ¯ ∇ Z bQX ) − g ( W, ¯ ∇ Z cQX )= − g ( A φW P X, Z ) + sin θg ( ¯ ∇ Z W, X ) − g ( A QP X
W, Z ) . Therefore(4.9) cos θg ( h ⊥ ( Z, W ) , X ) = − g ( A φW P X, Z ) − g ( A QP X
W, Z ) . Using (4.6) in (4.9), we get(4.10) cos θg ( h ⊥ ( Z, W ) , X ) = − cos θ [( Xµ ) + αη ( X )] g ( Z, W ) . Thus, we get h ⊥ ( Z, W ) = − [ −→∇ ⊥ µ + αξ ] g ( Z, W ) , where −→∇ ⊥ µ is gradient of the function µ .Therefore N ⊥ is totally umbilical in M with the mean curvature H ⊥ = − ( −→∇ ⊥ µ + αξ ).Now, let D N be the normal connection of N ⊥ in M . Then for any Y ∈ Γ( D θ ) and Z ∈ Γ( D ⊥ ), we have g ( D NZ −→∇ ⊥ µ + αξ, Y ) = g ( ∇ Z −→∇ ⊥ µ, Y ) + αg ( ∇ Z ξ, Y ) . ∗ , L-.I. PISCORAN AND T. PAL Also, from (2.6) and (2.13) we get ∇ Z ξ = 0.Therefore, g ( D NZ −→∇ ⊥ µ + αξ, Y ) = g ( ∇ Z −→∇ ⊥ µ, Y ) = 0, since ( Zµ ) = 0 for every Z ∈ Γ( D ⊥ ) and hence the mean curvature of N ⊥ is parallel.Thus the leaves of the distribution D ⊥ are totally umbilical in M with non-vanishingparallel mean curvature vector H ⊥ , i.e. N ⊥ is an extrinsic sphere in M . Thereforeby Theorem 2.1, M is a warped product submanifold. (cid:3) warped product bi-slant submanifolds of ( LCS ) n -manifolds Definition 5.1.
A submanifold M of a ( LCS ) n -manifold ¯ M is said to be a bi-slantsubmanifold if there exists a pair of orthogonal distributions D and D of M suchthat( i ) T M = D ⊕ D ( ii ) φ D ⊥D and φ D ⊥D ( iii ) D , D are slant submanifolds with slant angles θ and θ , respectively.If we assume θ = 0 and θ = π , then M is a CR-submanifold and if θ = 0 and θ = 0 , π , then M is a semi-slant submanifold. Also, if θ = π and θ = 0 , π , then M is a pseudo-slant submanifold. A bi-slant submanifold M of a ( LCS ) n -manifold ¯ M issaid to be proper if the slant distributions D and D are of slant angles θ , θ = 0 , π .For a proper bi-slant submanifold M of a ( LCS ) n -manifold, the normal bundle of M is decomposed as T ⊥ M = Q D ⊕ Q D ⊕ ν, where ν is the invariant normal subbundle of M .Now we will construct a bi-slant submanifold of a ( LCS ) n -manifold. Example 5.1.
Consider the semi-Euclidean space R with the cartesian coordinates( x , y · · · , x , y , t ) and paracontact structure φ (cid:18) ∂∂x i (cid:19) = ∂∂y i , φ (cid:18) ∂∂y j (cid:19) = ∂∂x j , φ (cid:18) ∂∂t (cid:19) = 0 , ≤ i, j ≤ . It is clear that R is a Lorentzian metric manifold manifold with usual semi-Euclideanmetric tensor. For any θ , θ ∈ [0 , π ] let M be a submanifold of R defined by χ ( u, v, w, s, t ) = ( w + u cos θ , u sin θ , s + v cos θ , v sin θ , , , t ) . HARACTERIZATION OF WARPED PRODUCT SUBMANIFOLDS OF (
LCS ) n -MANIFOLDS 11 Then the tangent space of M is spanned by the following vectors Z = cos θ ∂∂x + sin θ ∂∂x ,Z = cos θ ∂∂y + sin θ ∂∂y ,Z = ∂∂x ,Z = ∂∂y ,Z = ∂∂t . Then we have φZ = cos θ ∂∂y + sin θ ∂∂y ,φZ = cos θ ∂∂x + sin θ ∂∂x ,φZ = ∂∂y ,φZ = ∂∂x ,φZ = 0 . We take D = Span { Z , Z } and D = Span { Z , Z } , then g ( Z , φZ ) = cos θ and g ( Z , φZ ) = cos θ . Thus the distributions D and D are slant with slant angles θ and θ respectively and hence M is a bi-slant submanifold. Lemma 5.1.
Let M be a proper bi-slant submanifold of a ( LCS ) n -manifold ¯ M withthe slant distributions D and D such that ξ ∈ Γ( D ) . Then cos θ g ( ∇ X X , Y ) = g ( ∇ X P X , P Y ) + g ( h ( X , P X ) , QY )(5.1) + g ( h ( X , Y ) , QP X ) , for any X ∈ Γ( D ) and X , Y ∈ Γ( D ) , where θ and θ are the slant angles ofslant distributions D and D respectively.Proof. For any X ∈ Γ( D ) and X , Y ∈ Γ( D ), we have g ( ∇ X X , Y ) = g ( φ ¯ ∇ X X , φY )= g ( ¯ ∇ X φX , φY )= g ( ¯ ∇ X P X , P Y ) + g ( ¯ ∇ X P X , QY ) + g ( ¯ ∇ − X QX , φY ) . = g ( ∇ X P X , P Y ) + g ( h ( X , P X ) , QY )+ g ( ¯ ∇ X bQX , Y ) + g ( ¯ ∇ X cQX , Y ) . ∗ , L-.I. PISCORAN AND T. PAL Using (2.20) in the above relation we get (5.1). (cid:3)
Theorem 5.1.
There does not exists a proper warped product bi-slant submanifold M = M × f M of ¯ M such that ξ ∈ Γ( T M ) .Proof. Let M = M × f M be a proper warped product bi-slant submanifold of ¯ M .Then for X ∈ Γ( T M ) and X , Y ∈ Γ( T M ), we have g ( h ( X , P X ) , QY ) = g ( ¯ ∇ X P X , φY ) + g ( P X ¯ ∇ X P Y )= cos θ g ( ¯ ∇ X X , Y ) − g ( h ( X , Y ) , QP X ) + g ( P X , ¯ ∇ X P Y ) . Using Proposition 2.1 in the above relation, we get(5.2) g ( h ( X , P X ) , QY ) + g ( h ( X , Y ) , QP X ) = 2 cos θ ( X ln f ) g ( X , Y ) . Again using Proposition 2.1 in (5.1), we get(5.3) g ( h ( X , P X ) , QY ) + g ( h ( X , Y ) , QP X ) = 0 . From (5.2) and (5.3), we get(5.4) cos θ ( X ln f ) = 0 . Since M is a proper warped product bi-slant submanifold so, θ = π . Therefore( X ln f ) = 0 for every X ∈ Γ( T M ) and hence M does not exists. (cid:3) References [1] Atceken, M.,
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