A Note on Quasi bi-slant submanifolds of cosymplectic manifolds
aa r X i v : . [ m a t h . G M ] M a r A Note on Quasi bi-slant submanifolds ofcosymplectic manifolds
Mehmet Akif Akyol and Selahattin Beyendi Abstract.
The main purpose of the present paper is to define and studythe notion of quasi bi-slant submanifolds of almost contact metric man-ifolds. We mainly concerned with quasi bi-slant submanifolds of cosym-plectic manifolds as a generalization of slant, semi-slant, hemi-slant,bi-slant and quasi hemi-slant submanifolds. First, we give non-trivialexamples in order to demostrate the method presented in this paper iseffective and investigate the geometry of distributions. Moreover, Westudy these types of submanifolds with parallel canonical structures.
Mathematics Subject Classification (2010).
Primary 53C15, 53B20.
Keywords.
Slant submanifold, bi-slant submanifold, quasi bi-slant sub-manifold, cosymplectic manifold.
1. Introduction
Study of submanifolds theory has shown an increasing development in imageprocessing, computer design, economic modeling as well as in mathematicalphysics and in mechanics. In this manner, B-Y. Chen [6] initiated the notionof slant submanifold as a generalization of both holomorphic (invariant) andtotally real submanifold (anti-invariant) of an almost Hermitian manifold.Inspried by B-Y. Chen’s paper, many geometers have studied this notionin the different kind of structures: (see [22], [23]). Many consequent resultson slant submanifolds are collected in his book [5]. After this notion, as ageneralization of semi-slant submanifold which was defined by N. Papaghiuc[19] (see also [8]). A. Carriazo [3] and [4] introduced the notion of bi-slantsubmanifold under the name anti-slant submanifold. However, B. S¸ahin calledthese submanifolds hemi-slant submanifolds in [21]. (See also [9] and [10], [20],[24]).Furthermore, the submanifolds of a cosymplectic manifold have beenstudied by many geometers: See [11], [12], [13], [14], [15], [16], [18]. Taking intoaccount of the above studies, we are motivated to fill a gap in the literature by M. A. Akyol and S. Beyendigiving the notion of quasi bi-slant submanifolds in which the tangent bundleconsist of one invariant and two slant distributions and the Reeb vector field.In this paper, as a generalization of slant, semi-slant, hemi-slant, bi-slant andquasi hemi-slant submanifolds, we introduce quasi bi-slant submanifolds andinvestigate the geometry of distributions in detail.The paper is organized as follows: In section 2, we recall basic formu-las and definitions for a cosymplectic manifold and their submanifolds. Insection 3, we introduce the notion of quasi bi-slant submanifolds, giving annon-tirivial example and obtain some basic results for the next sections. Insection 4, we give some necessary and sufficient conditions for the geometryof distributions. Finally, we study these types of submanifolds with parallelcanonical structures.
2. Preliminaries
In this section, we give the definition of cosymplectic manifold and somebackground on submanifolds theory.A (2 m + 1)-dimensional C ∞ -manifold M said to have an almost contactstructure if there exist on M a tensor field ϕ of type (1,1), a vector field ξ and 1-form η satisfying: ϕ = − I + η ⊗ ξ, ϕξ = 0 , ηoϕ = 0 , η ( ξ ) = 1 . (2.1)There always exists a Riemannian metric g on an almost contact manifold M satisfying the following conditions g ( ϕX, ϕY ) = g ( X, Y ) − η ( X ) η ( Y ) , η ( X ) = g ( X, ξ ) (2.2)where
X, Y are vector fields on M. An almost contact structure ( ϕ, ξ, η ) is said to be normal if the almostcomplex structure J on the product manifold M × R is given by J ( X, f ddt ) = ( ϕX − f ξ, η ( X ) ddt ) , where f is a C ∞ -function on M × R has no torsion i.e., J is integrable. Thecondition for normality in terms of ϕ, ξ and η is [ ϕ, ϕ ] + 2 dη ⊗ ξ = 0 on M, where [ ϕ, ϕ ] is the Nijenhuis tensor of ϕ. Finally, the fundamental two-formΦ is defined Φ(
X, ϕY ) = g ( X, ϕY ) . An almost contact metric structure ( ϕ, ξ, η, g ) is said to be cosymplectic,if it is normal and both Φ and η are closed ([1], [2], [16]), and the structureequation of a cosymplectic manifold is given by( ∇ X ϕ ) Y = 0 (2.3)for any X, Y tangent to M, where ∇ denotes the Riemannian connection ofthe metric g on M. Moreover, for cosymplectic manifold ∇ X ξ = 0 . (2.4) Note on Quasi bi-slant submanifolds of cosymplectic manifolds 3 Example. ([17]) R n +1 with Cartesian coordinates ( x i , y i , z )( i = 1 , ..., n ) andits usual contact form η = dz and ξ = ∂∂z , here ξ is the characteristic vector field and its Riemannian metric g and tensorfield ϕ are given by g = n X i =1 (( dx i ) + ( dy i ) ) + ( dz ) , ϕ = δ ij − δ ij , i = 1 , ..., n. This gives a cosymplectic manifold on R n +1 . The vector fields e i = ∂∂y i ,e n + i = ∂∂x i , ξ form a ϕ -basis for the cosymplectic structure. On the otherhand, it can be shown that R n +1 ( ϕ, ξ, η, g ) is a cosymplectic manifold.Let M be a Riemannian manifold isometrically immersed in ¯ M and in-duced Riemannian metric on M is denoted by the same symbol g throughoutthis paper. Let A and h denote the shape operator and second fundamentalform, respectively, of immersion of M into ¯ M . The Gauss and Weingartenformulas of M into ¯ M are given by [6]¯ ∇ X Y = ∇ X Y + h ( X, Y ) (2.5)and ¯ ∇ X V = −A V X + ∇ ⊥ X V, (2.6)for any vector fields X, Y ∈ Γ( T M ) and V ∈ Γ( T ⊥ M ), where ∇ is theinduced connection on M and ∇ ⊥ represents the connection on the normalbundle T ⊥ M of M and A V is the shape operator of M with respect to normalvector V ∈ Γ( T ⊥ M ). Moreover, A V and h are related by g ( h ( X, Y ) , V ) = g ( A V X, Y ) (2.7)for any vector fields
X, Y ∈ Γ( T M ) and V ∈ Γ( T ⊥ M ).If h ( X, Y ) = 0 for all
X, Y ∈ Γ( T M ), then M is said to be totallygeodesic.
3. Quasi bi-slant submanifolds of cosmyplectic manifolds
In this section, we define the concept of quasi bi-slant submanifolds of cosym-plectic manifolds, giving a non-trivial exmaple and obtain some related resultsfor later use.
Definition 3.1.
A submanifold M of cosymplectic manifolds ( ¯ M , ϕ, ξ, η, ¯ g ) iscalled quasi bi-slant if there exists four orthogonal distributions D , D and D of M, at the point p ∈ M such that(i) T M = D ⊕ D ⊕ D ⊕ < ξ > (ii) The distribution D is invariant, i.e. ϕ D = D . (iii) ϕ D ⊥ D and ϕ D ⊥ D ;(iv) The distributions D , D are slant with slant angle θ , θ , respectively. M. A. Akyol and S. BeyendiTaking the dimension of distributions D , D and D are m , m and m , respectively. One can easily see the following cases: • If m = 0 and m = m = 0 , then M is a invariant submanifold. • If m = m = 0 and θ = π then M is an anti-invariant submanifold. • If m = 0 , m = m = 0 , θ = 0 and θ = π then M is a semi-invariantsubmanifold. • If m = m = 0 and 0 < θ < π then M is a slant submanifold. • If m = 0 , m = m = 0 , θ = 0 and 0 < θ < π then M is a semi-slantsubmanifold. • If m = 0 , m = m = 0 , θ = π and 0 < θ < π then M is a hemi-slantsubmanifold. • If m = 0 , m = m = 0 , and θ and θ are different from either 0 and π , then M is a bi-slant submanifold.If m = m = m = 0 and θ , θ = 0 , π , then M is called a proper quasibi-slant submanifold . Remark . In this paper, we assume that M is proper quasi bi-slant sub-manifold of a cosymplectic manifold ¯ M .
Now, we present an example of proper quasi bi-slant submanifold in R . Example.
We will use the canonical contact structure ϕ defined by ϕ ( x , y , ..., x n , y n , z ) = ( y , − x , ..., y n , − x n , . Thus we have ϕ ( ∂x i ) = ∂y i , ϕ ( ∂y j ) = − ∂x j and ϕ ( ∂z ) = 0 , ≤ i, j ≤ ∂x i = ∂∂x i . For any pair of real numbers θ , θ satisfying 0 < θ , θ < π , let us consider submanifold M θ ,θ of R defined by π θ ,θ ( u, s, w, k, t, r, z ) = ( u, s cos θ , , s sin θ , ω, k cos θ , , k sin θ , t, r, z ) . If we take e = ∂x , e = cos θ ∂y + sin θ ∂y ,e = ∂x , e = cos θ ∂y + sin θ ∂y ,e = ∂x , e = ∂y , e = ξ = ∂z then the restriction of e , ..., e to M forms an orthonormal frame of thetangent bundle T M.
Obviously, we get ϕe = ∂y , ϕe = − cos θ ∂x − sin θ ∂x , ϕe = ∂y ϕe = − cos θ ∂x − sin θ ∂x , ϕe = ∂y , ϕe = − ∂x . Let us put D = Span { e , e } , D = Span { e , e } , and D = Span { e , e } . Then obviously D , D and D , satisfy the definition of quasi bi-slant sub-manifold M θ ,θ defined by π θ ,θ is a proper quasi bi-slant submanifold of R with θ , θ as its bi-slant angles. Note on Quasi bi-slant submanifolds of cosymplectic manifolds 5Let M be a quasi bi-slant submanifold of a cosymplectic manifold ¯ M .
Then, for any X ∈ Γ( T M ) , we have X = P X + Q X + R X + η ( X ) ξ (3.1)where P , Q and R denotes the projections on the distributions D , D and D , recpectively. ϕX = T X + F X, (3.2)where T X and F X are tangential and normal components on M. Makingnow use of (3.1) and (3.2), we get immediately ϕX = T P X + T Q X + FQ X + T R X + FR X, (3.3)here since ϕ D = D , we have FP X = 0 . Thus we get ϕ ( T M ) =
D ⊕ T D ⊕ T D (3.4)and T ⊥ M = FD ⊕ FD ⊕ µ (3.5)where µ is the orthogonal complement of FD ⊕ FD in T ⊥ M and it isinvariant with recpect to ϕ. Also, for any Z ∈ T ⊥ M, we have ϕZ = B Z + C Z, (3.6)where B Z ∈ Γ( T M ) and C Z ∈ Γ( T ⊥ M ) . Taking into account of the condition (iii) in Definition (3.1), (3.2) and(3.6), we obtain the followings:
Lemma 3.3.
Let M be a quasi bi-slant submanifold of a cosymplectic manifold ¯ M .
Then, we have (a) T D ⊂ D , (b) T D ⊂ D , (c) BFD = D , (d) BFD = D . With the help of (3.2) and (3.6), we obtain the following Lemma.
Lemma 3.4.
Let M be a quasi bi-slant submanifold of a cosymplectic manifold ¯ M .
Then, we have (a) T U = − cos θ U , (b) T U = − cos θ U , (c) BF U = − sin θ U , (d) BF U = − sin θ U , (e) T U + BF U = − U , (f ) T U + BF U = − U , (g) FT U + CF U = 0 , (h) FT U + CF U = 0 , for any U ∈ D and U ∈ D . By using (2.3), Definition (3.1), (3.2) and (3.6), we obtain the followingLemma.
Lemma 3.5.
Let M be a quasi bi-slant submanifold of a cosymplectic manifold ¯ M .
Then, we have (i) T i U i = − cos θ i U i , (ii) g ( T i U i , T i V i ) = (cos θ i ) g ( U i , V i ) , (iii) g ( F i U i , F i V i ) = (sin θ i ) g ( U i , V i ) for any i = 1 , , U , V ∈ Γ( D ) and U , V ∈ Γ( D ) . M. A. Akyol and S. BeyendiWe need the following lemma for later use.
Lemma 3.6.
Let M be a quasi bi-slant submanifold of a cosymplectic manifold ¯ M , then for any Z , Z ∈ Γ( T M ) , we have the following ∇ Z T Z − T ∇ Z Z = A F Z Z + B h ( Z , Z ) (3.7) and ∇ ⊥ Z F Z − F∇ Z Z = C h ( Z , Z ) − h ( Z , T Z ) . (3.8) Proof.
Since ¯ M is a cosmyplectic manifold, we have that( ¯ ∇ Z ϕ ) Z = 0which implies that ¯ ∇ Z ϕZ − ϕ ¯ ∇ Z Z = 0 . By using (2.5) and (3.2), we get¯ ∇ Z T Z + ¯ ∇ Z F Z − ϕ ( ∇ Z Z + h ( Z , Z )) = 0 . Taking into account of (2.5), (2.6), (3.2) and (3.6), we obtain ∇ Z T Z + h ( Z , T Z ) − A F Z Z + ∇ ⊥ Z F Z − T ∇ Z Z − F∇ Z Z − B hZ , Z − C h ( Z , Z ) = 0 . Comparing the tangential and normal components, we have the requiredresults. (cid:3)
In a similar way, we have:
Lemma 3.7.
Let M be a quasi bi-slant submanifold of a cosymplectic manifold ¯ M , then we have the following ∇ Z B W − B∇ ⊥ Z W = A C W Z − T A W Z (3.9) and ∇ ⊥ Z C W − C∇ ⊥ Z W = −FA W Z − h ( Z , B W ) (3.10) for any Z ∈ Γ( T M ) and W ∈ Γ( T ⊥ M ) .
4. Integrability and Totally geodesic foliations
In this section we give some necessary and sufficient condition for the inte-grability of the distributions.First, we have the following theorem:
Theorem 4.1.
Let M be a quasi bi-slant submanifolds of ¯ M .
The invariantdistribution D is integrable if and only if g ( T ( ∇ X T Y − ∇ Y T X ) , Z ) = g ( h ( X, T Y ) − h ( Y, T X ) , ϕQZ + ϕRZ ) for any X, Y ∈ Γ( D ) and Z ∈ Γ( D ⊕ D ) . Note on Quasi bi-slant submanifolds of cosymplectic manifolds 7
Proof.
The distribution D is integrable on M if and only if g ([ X, Y ] , ξ ) = 0 and g ([ X, Y ] , Z ) = 0for any X, Y ∈ Γ( D ), Z ∈ Γ( D ⊕ D ) and ξ ∈ Γ( T M ) . Since M is acosymplectic manifold, we immediately have g ([ X, Y ] , ξ ) = 0 . Thus D isintegrable if and only if g ([ X, Y ] , Z ) = 0 . Now, for any
X, Y ∈ D and Z = Q Z + R Z ∈ Γ( D ⊕ D ) , by using (2.2), (2.5), we obtain g ([ X, Y ] , Z ) = g ( ϕ ¯ ∇ X Y, ϕZ ) − η ( ¯ ∇ X Y ) η ( Z ) − g ( ϕ ¯ ∇ Y X, ϕZ ) + η ( ¯ ∇ Y X ) η ( Z ) . Now, using (2.4), (3.2) and F Y = 0 for any Y ∈ Γ( D ) , we have g ([ X, Y ] , Z ) = g ( ¯ ∇ X ϕY, ϕZ ) − g ( ¯ ∇ Y ϕX, ϕZ )= g ( ¯ ∇ X T Y, ϕZ ) − g ( ¯ ∇ Y T X, ϕZ ) . Taking into account of (2.5) and (3.3) in the above equation, we get g ([ X, Y ] , Z ) = − g ( ϕ ∇ X T Y, Z ) + g ( h ( X, T Y ) , ϕZ )+ g ( ϕ ∇ Y T X, Z ) − g ( h ( Y, T X ) , ϕZ ) . Now again taking into account the equation (3.2), we obtain g ([ X, Y ] , Z ) = g ( T ( ∇ Y T X − ∇ X T Y ) , Z )+ g ( h ( X, T Y ) − h ( Y, T X ) , ϕQZ + ϕRZ )which completes the proof. (cid:3) For the slant distrubition D , we have: Theorem 4.2.
Let M be a quasi bi-slant submanifolds of ¯ M .
The slant distri-bution D is integrable if and only if g ( ∇ ⊥ U F V + ∇ ⊥ V F U , FR Z ) = g ( A FT V U − A FT U V , Z )+ g ( A F V U + A F U V , T Z ) for any U , V ∈ Γ( D ) , Z ∈ Γ( D ⊕ D ) . Proof.
The distribution D is integrable on M if and only if g ([ U , V ] , ξ ) = 0 and g ([ U , V ] , Z ) = 0for any U , V ∈ Γ( D ), Z ∈ Γ( D ⊕ D ) and ξ ∈ Γ( T M ) . The first case istrivial. Thus D is integrable if and only if g ([ U , V ] , Z ) = 0 . Now, for any U , V ∈ D and Z = P Z + R Z ∈ Γ( D ⊕ D ) , by using (2.2), (2.5), we obtain g ([ U , V ] , Z ) = − g ( ¯ ∇ U ϕ T V , Z ) − g ( ¯ ∇ U F V , ϕZ )+ g ( ¯ ∇ V ϕ T U , Z ) − g ( ¯ ∇ V F U , ϕZ )Taking into account the equation lemma (3.5) (i) in the above equation, weget g ([ U , V ] , Z ) = cos θ g ([ U , V ] , Z ) − g ( ¯ ∇ U FT V − ¯ ∇ V FT U , Z ) − g ( ¯ ∇ U F V + ¯ ∇ V F U , ϕ P Z + ϕ R Z ) . M. A. Akyol and S. BeyendiNow, using (2.6) and (3.3), we obtain g ([ U , V ] , Z ) = cos θ g ([ U , V ] , Z ) + g ( A FT V U − A FT U V , Z )+ g ( A F V U + A F U V , T Z ) − g ( ∇ ⊥ U F V + ∇ ⊥ V F U , FR Z )or sin θ g ([ U , V ] , Z ) = g ( A FT V U − A FT U V , Z )+ g ( A F V U + A F U V , T Z ) − g ( ∇ ⊥ U F V + ∇ ⊥ V F U , FR Z )which gives the assertion. (cid:3) In a similar way, we obtain the following case for the slant distribution D . Theorem 4.3.
Let M be a quasi bi-slant submanifolds of ¯ M .
The slant distri-bution D is integrable if and only if T ( ∇ U T V − A F V U ) ∈ Γ( D ) , B ( h ( U , T V ) + ∇ ⊥ U F V ) ∈ Γ( T M ) ⊥ and g ( A F Z V − ∇ V T Z, T U ) = g ( h ( V , T Z ) + ∇ ⊥ V F Z, F U ) for any U , V ∈ Γ( D ) , Z = P Z + Q Z ∈ Γ( D ⊕ D ) and W ∈ Γ( T M ) ⊥ . Theorem 4.4.
Let M be a quasi bi-slant submanifolds of ¯ M .
The invariantdistribution D defines totally geodesic foliation on M if and only if g ( ∇ X T Y, T Z ) = − g ( h ( X, T Y ) , F Z ) (4.1) and F∇ X T Y + C h ( X, T Y ) ∈ Γ( T M ) (4.2) for any
X, Y ∈ Γ( D ) , Z = Q Z + R Z ∈ Γ( D ⊕ D ) and W ∈ Γ( T M ) ⊥ . Proof.
The distribution D defines a totaly geodesic foliation on M if and onlyif g ( ¯ ∇ X Y, ξ ) = 0 , g ( ¯ ∇ X Y, Z ) = 0 and g ( ¯ ∇ X Y, W ) = 0 for any
X, Y ∈ Γ( D ) ,Z = Q Z + R Z ∈ Γ( D ⊕ D ) and W ∈ Γ( T M ) ⊥ . Then by using (2.2) and(2.4), we obtain g ( ¯ ∇ X Y, ξ ) = Xg ( Y, ξ ) − g ( Y, ¯ ∇ X ξ ) = − g ( Y, ¯ ∇ X ξ ) = 0 . (4.3)On the other hand, using (2.2), we find g ( ¯ ∇ X Y, Z ) = g ( ¯ ∇ X ϕY, ϕZ ) = g ( ¯ ∇ X T Y, ϕZ ) Note on Quasi bi-slant submanifolds of cosymplectic manifolds 9here we have used F Y = 0 for any Y ∈ Γ( D ) . Now, by using (3.3) and (2.5),we have g ( ¯ ∇ X Y, Z ) = g ( ∇ X T Y + h ( X, T Y ) , ϕ Q Z + ϕ R Z )= g ( ∇ X T Y + h ( X, T Y ) , T Q Z + FQ Z + T R Z + FR Z ))= g ( ∇ X T Y, T Q Z + T R Z ) + g ( h ( X, T Y ) , FQ Z + FR Z ))= g ( ∇ X T Y, T Z ) + g ( h ( X, T Y ) , F Z ) (4.4)for any X, Y ∈ Γ( D ) and Z = Q Z + R Z ∈ Γ( D ⊕ D ) . Now, for any
X, Y ∈ Γ( D ) and W ∈ Γ( T M ) ⊥ , we have g ( ¯ ∇ X Y, W ) = − g ( ϕ ¯ ∇ X ϕY , W ) = − g ( ϕ ( ∇ X T Y + h ( X, T Y )) , W ))= − g ( T ∇ X T Y + F∇ X T Y + B h ( X, T Y ) + C h ( X, T Y ) , W ))= − g ( F∇ X T Y + C h ( X, T Y ) , W ) (4.5)Thus proof follows (4.3), (4.4) and (4.5). (cid:3) Theorem 4.5.
Let M be a quasi bi-slant submanifolds of ¯ M .
The slant distri-bution D defines totally geodesic foliation on M if and only if g ( A FT V U , Z ) − g ( A F V U , T P Z )= g ( A F V U , T R Z ) − g ( ∇ ⊥ U F V , FR Z ) (4.6) and FA F V U − ∇ ⊥ U FT V − C∇ ⊥ U F V ∈ Γ( T M ) (4.7) for any
X, Y ∈ Γ( D ) , Z = Q Z + R Z ∈ Γ( D ⊕ D ) and W ∈ Γ( T M ) ⊥ . Proof.
The distribution D defines a totaly geodesic foliation on M if andonly if g ( ¯ ∇ U V , ξ ) = 0, g ( ¯ ∇ U V , Z ) = 0 and g ( ¯ ∇ U V , W ) = 0, for any U , V ∈ Γ( D ) , Z = P Z + R Z ∈ Γ( D ⊕ D ) and W ∈ Γ( T M ) ⊥ . Since M is a cosymplectic manifold, we immediately have g ( ¯ ∇ U V , ξ ) = 0 . Now, forany U , V ∈ Γ( D ) , and Z = P Z + R Z ∈ Γ( D ⊕ D ) , by using (2.2) and(2.4), we obtain g ( ¯ ∇ U V , Z ) = − g ( ¯ ∇ U ϕ T V , Z ) + g ( ¯ ∇ U F V , ϕ P Z + ϕ R Z ) . Now, by using lemma (3.5) (i), we get g ( ¯ ∇ U V , Z ) = cos θ g ( ¯ ∇ U V , Z ) − g ( −A FT V U + ∇ ⊥ U FT V , Z )+ g ( −A F V U + ∇ ⊥ U F V , T P Z )+ g ( −A F V U + ∇ ⊥ U F V , T R Z + FR Z )or sin θ g ( ¯ ∇ U V , Z ) = g ( A FT V U , Z ) − g ( A F V U , T P Z ) − g ( A F V U , T R Z ) + g ( ∇ ⊥ U F V , FR Z ) . (4.8)0 M. A. Akyol and S. BeyendiNow, for any U , V ∈ Γ( D ) and W ∈ Γ( T M ) ⊥ , we have g ( ¯ ∇ U V , W ) = − g ( ¯ ∇ U ϕ T V , W ) − g ( ϕ ( ¯ ∇ U F V ) , W ) − g ( ϕ ( −A F V U + ∇ ⊥ U F V ) , W )= cos θ g ( ¯ ∇ U V , W ) − g ( −A FT V U + ∇ ⊥ U FT V , W ) − g ( −T A F V U − FA F V U + B∇ ⊥ U F V + C∇ ⊥ U F V , W )orsin θ g ( ¯ ∇ U V , W ) = − g ( ∇ ⊥ U FT V , W ) + g ( FA F V U − C∇ ⊥ U F V , W )= g ( FA F V U − ∇ ⊥ U FT V − C∇ ⊥ U F V , W ) (4.9)Thus proof follows (4.8) and (4.9). (cid:3) Theorem 4.6.
Let M be a quasi bi-slant submanifolds of ¯ M .
The slant distri-bution D defines totally geodesic foliation on M if and only if T ( ∇ U T V − A F V U ) ∈ Γ( D ) , (4.10) B ( h ( U , T V ) + ∇ ⊥ U F V ) ∈ Γ( T M ) ⊥ (4.11) and g ( ∇ ⊥ U FT V − FA V U , W ) = g ( ∇ ⊥ U F V , C W ) (4.12) for any U , V ∈ Γ( D ) , Z = P Z + Q Z ∈ Γ( D ⊕ D ) and W ∈ Γ( T M ) ⊥ . From theorem (4.4), (4.5) and (4.6), we have the following decomposi-tion theorem:
Theorem 4.7.
Let M be a proper quasi bi-slant submanifolds of a cosmyplecticmanifold ¯ M .
Then M is a local product Riemannian manifold of the form M D × M D × M D , where M D , M D and M D are leaves of D , D and D , recpectively, if and only if the conditions (4.1) , (4.2) , (4.6) , (4.7) , (4.10) , (4.11) and (4.12) hold.
5. Quasi bi-slant submanifolds with parallel canonicalstructures
In this section, we obtain some results for the quasi bi-slant submanifolds withparallel canonical structure. Let M be a proper quasi bi-slant submanifold ofa cosymplectic manifold ¯ M .
Then we define( ¯ ∇ Z T ) Z = ∇ Z T Z − T ∇ Z Z (5.1)( ¯ ∇ Z F ) Z = ∇ ⊥ Z F Z − F∇ Z Z (5.2)( ¯ ∇ Z B ) W = ∇ Z B W − B∇ ⊥ Z W (5.3)( ¯ ∇ Z C ) W = ∇ ⊥ Z C W − C∇ ⊥ Z W (5.4)where Z , Z ∈ Γ( T M ) and W ∈ Γ( T M ) ⊥ .Then, the endomorphism T ( resp. F ) and the endomorphism B ( resp. C ) are Note on Quasi bi-slant submanifolds of cosymplectic manifolds 11parallel if ¯ ∇T ≡ resp. ¯ ∇F ≡
0) and ¯ ∇B ≡ resp. ¯ ∇C ≡ , respectively.Taking into account of (3.7), (3.8), (3.9), (3.10) and (5.1)-(5.4), we have thefollowing lemma. Lemma 5.1.
Let M be a quasi bi-slant submanifold of a cosymplectic manifold ¯ M . Then for any Z , Z ∈ Γ( T M ) and W ∈ Γ( T M ) ⊥ we obtain ( ¯ ∇ Z T ) Z = A F Z Z + B h ( Z , Z ) (5.5)( ¯ ∇ Z F ) Z = C h ( Z , Z ) − h ( Z , T Z ) (5.6)( ¯ ∇ Z B ) W = A C W Z + T A W Z (5.7)( ¯ ∇ Z C ) W = −FA W Z − h ( Z , B W ) . (5.8)First, we have the following theorem: Theorem 5.2.
Let M be a quasi bi-slant submanifold of a cosymplectic man-ifold ¯ M . Then, T is parallel if and only if the invariant distribution D istotally geodesic.Proof. For any
X, Y ∈ Γ( D ) , from (5.5), we have( ¯ ∇ X T ) Y = B h ( X, Y ) (5.9)here we have used A F Y X = 0 since F Y = 0 for any Y ∈ Γ( D ). Thus, ourassertion comes from (5.9). (cid:3) Theorem 5.3.
Let M be a quasi bi-slant submanifold of a cosymplectic man-ifold ¯ M . Then if F is parallel if and only if g ( A C V Z , Z ) = − g ( A V Z , T Z ) (5.10) for any Z , Z ∈ Γ( T M ) and V ∈ Γ( T M ) ⊥ . Proof.
Assume that F is parallel. Now, from (5.6), we have( ¯ ∇ Z F ) Z = C h ( Z , Z ) − h ( Z , T Z ) . (5.11)Now, taking inner product with V ∈ Γ( T M ) ⊥ in the above equation andusing (2.5), we obtain g (( ¯ ∇ Z F ) Z , V ) = g ( C h ( Z , Z ) − h ( Z , T Z ) , V )= g ( C h ( Z , Z ) , V ) − g ( h ( Z , T Z ) , V )= − g ( h ( Z , Z ) , ϕV ) − g ( ¯ ∇ Z T Z , V )= − g ( A C V Z , Z ) + g ( T Z , ¯ ∇ Z V )= − g ( A C V Z , Z ) + g ( T Z , −A V Z )which gives the assertion. (cid:3) Theorem 5.4.
Let M be a quasi bi-slant submanifold of a cosymplectic man-ifold ¯ M . Then F is parallel if and only if B is parallel. Proof.
By using (2.5), (5.6) and (5.7), we get g (( ¯ ∇ Z F ) Z , W ) = g ( C h ( Z , Z ) , W ) − g ( h ( Z , T Z ) , W )= − g ( h ( Z , Z ) , C W ) − g ( A W Z , T Z )= − g ( A C W Z , Z ) + g ( T A W Z , Z )= − g ( A C W Z − T A W Z , Z )= − g (( ¯ ∇ Z B ) W , Z )for any Z , Z ∈ Γ( T M ) and W ∈ Γ( T M ) ⊥ . This proves our assertion. (cid:3) Finally, we mention another non-trivial example of quasi bi-slant sub-manifold of a cosymplectic manifold.
Example.
Let M be a submanifold of R defined by x ( u, v, t, r, s, k, z ) = ( u, v, t, √ r, √ r, , s, k cos α, k sin α, , z ) . We can easily to see that the tangent bundle of M is spanned by the tangentvectors e = ∂∂x , e = ∂∂y , e = ∂∂x , e = 1 √ ∂∂y + 1 √ ∂∂x ,e = ∂∂x , e = cos α ∂∂y + sin α ∂∂x , e = ∂∂z = ξ. We define the almost contact structurev ϕ of R , by ϕ ( ∂∂x i ) = ∂∂y i , ϕ ( ∂∂y j ) = − ∂∂x j , ϕ ( ∂∂z ) = 0 , ≤ i, j ≤ . For any vector field Z = λ i ∂∂x i + µ j ∂∂y j + ν ∂∂z ∈ Γ( T R ) , then we have g ( Z, Z ) = λ i + µ j + ν , g ( ϕZ, ϕZ ) = λ i + µ j and ϕ Z = − λ i ∂∂x i − µ j ∂∂y j = − Z for any i, j = 1 , ..., . It follows that g ( ϕZ, ϕZ ) = g ( Z, Z ) − η ( Z ) . Thus( ϕ, ξ, η, g ) is an is an almost contact metric structure on R . Thus we have ϕe = ∂∂y , ϕe = ∂∂x , ϕe = ∂∂y , ϕe = − √ ∂∂x + 1 √ ∂∂y ,ϕe = ∂∂y , ϕe = − cos α ∂∂x + sin α ∂∂y , ϕe = 0 . By direct calculations, we obtain the distribution D = span { e , e } is an in-variant distribution, the distribution D = span { e , e } is a slant distributionwith slant angle θ = π and the distribution D = span { e , e } is also a slantdistribution with slant angle θ = α, < α < π . Thus M is a 7 − dimensionalproper quasi bi-slant submanifold of R with its usual almost contact metricstructure. Note on Quasi bi-slant submanifolds of cosymplectic manifolds 13 References [1] D. E. Blair,
Contact manifolds in Riemannian geometry , Lecture Notes in Math-ematic Springer-Verlag, New York, Vol. 509 (1976).[2] D. E. Blair,
The theory of quasi-Sasakian structure , J. Differential Geom. , no.3-4, 331-345, 1967.[3] A. Carriazo, New developments in slant submanifolds theory , Narasa PublishingHause New Delhi, India, 2002.[4] A. Carriazo, Bi-slant immersions. In: Proceeding of the ICRAMS 2000, Kharag-pur, pp. 8897 (2000).[5] B. Y. Chen,
Geometry of slant submanifolds , Katholieke Universiteit Leuven,Leuven, Belgium, View at Zentralblatt Math., 1990.[6] B. Y. Chen,
Slant immersions , Bull. Austral. Math. Soc., 41 (1990), 135-147.[7] J. L. Cabrerizo, A. Carriazo, L. M. Fernandez, M. Fernandez,
Slant submanifoldsin Sasakian manifolds , Glasgow Math. J., 42 (2000), 125-138.[8] J. L. Cabrerizo, A. Carriazo, L. M. Fernandez and M. Fernandez,
Semi-slantsubmanifolds of a Sasakian manifold , Geom. Dedic. 78(2), 183199 (1999).[9] S. Dirik and M. At¸ceken,
On the geometry of pseudo-slant submanifolds of acosymplectic manifold , International electronic journal of geometry, 9(1), (2016),45-56.[10] S. Dirik and M. Atceken,
Contact pseudo-slant submanifolds of a cosymplecticmanifold , New trends in Mathematical Sciences, 6(4), (2018), 154-164.[11] R. S. Gupta, S. K. Haider and A. Sharfuddin,
Slant submanifolds in cosym-plectic manifolds , In Colloquium Mathematicum Vol. 105, (2016), 207-219.[12] M. A. Khan,
Totally umbilical hemi slant submanifolds of Cosymplectic mani-folds , Mathematica Aeterna 3(8) (2013): 645-653.[13] U. K. Kim,
On anti-invariant submanifolds of cosymplectic manifolds , Bulletinof the Korean Mathematical Society, 21(1), (1984), 35-37.[14] M. A. Lone, M. S. Lone and M. H. Shahid,
Hemi-slant submanifolds of cosym-plectic manifolds , Cogent Mathematics 3(1) (2016): 1204143.[15] A. Lotta,
Slant submanifolds in contact geometry , Bulletin Mathematical So-ciety Roumanie, 39 (1996), 183-198.[16] G. D. Ludden,
Submanifolds of cosymplectic manifolds , Journal of DifferentialGeometry, 4, (1970), 237244.[17] Olzsak Z.,
On almost cosymplectic manifolds , Kodai Math J. , 239-250, 1981.[18] S. Uddin, C. Ozel, V. A. Khan, A classication of a totally umbilical slant sub-manifold of cosymplectic manifolds , Hindawi puplishing corporation abstractapplied analysis, article ID 716967, 8 pages (2012).[19] N. Papaghuic,
Semi-slant submanifolds of a Kaehlarian manifold , An. St. Univ.Al. I. Cuza. Univ. Iasi, 40 (2009), 55-61.[20] R. Prasad, S. K. Verma and S. Kumar,
Quasi hemi-slant submanifolds ofsasakian manifolds , Journal of Mathematical and Computational Science, 10(2),(2020), 418-435.[21] B. S¸ahin,
Warped product submanifolds of a Kaehler manifold with a slantfactor , Annales Polonici Mathematici, 95, (2009), 107226. [22] B. S¸ahin,
Slant submanifolds of an almost product Riemannian manifold , Jour-nal of the Korean Mathematical Society, 43(4), (2006), 717-732.[23] B. S¸ahin and S. Kele¸s,
Slant submanifolds of Kaehler product manifolds , Turk-ish Journal of Mathematics, 31 (2007), 6577.[24] H. M. Ta¸stan and F. ¨Ozdemir,
The geometry of hemi-slant submanifolds ofa locally product Riemannian manifold , Turkish Journal of Mathematics, 39(2015), 268284.Mehmet Akif Akyol Bingol UniversityFaculty of Arts and Sciences,Department of Mathematics12000, Bing¨ol, Turkeye-mail: [email protected]
Selahattin Beyendi ˙Inonu UniversityFaculty of Education44000, Malatya, Turkeye-mail:˙Inonu UniversityFaculty of Education44000, Malatya, Turkeye-mail: