aa r X i v : . [ m a t h . G M ] S e p Conformal Bi-slant Submersions
Sezin Aykurt Sepet
Kır¸sehir Ahi Evran University, Department of Mathematics, Kır¸sehir, [email protected]
October 1, 2020
Abstract
We study conformal bi-slant submersions from almost Hermitianmanifolds onto Riemannian manifolds as a generalized of conformalanti-invariant, conformal semi-invariant, conformal semi-slant, confor-mal slant and conformal hemi-slant submersions. We investigated theintegrability of distributions and obtain necessary and sufficient con-ditions for the maps to have totally geodesic fibers. Also we studiedthe total geodesicity of such maps.
Keywords :Bi-slant submersion, conformal bi-slant submersion,almost Hermitian manifold. : 53C15, 53C43
In complex geometry, as a generalization of holomorphic and totally realimmersions, slant immersions were defined by Chen [11]. Cabrerizo et al[10] defined bi-slant submanifolds in almost contact metric manifolds. In[30] Uddin et al. studied warped product bi-slant immersions in Kaehlermanifolds. They proved that there do not exist any warped product bi-slantsubmanifolds of Kaehler manifolds other than hemi-slant warped productsand CR-warped products.The theory of Riemannian submersions as an analogue of isemetric im-mersions was initiated by O’Neill [20] and Gray[14]. The Riemannian sub-mersions are important in physics owing to applications in the Yang-Millstheory, Kaluza-Klein theory, robotic theory, supergravity and superstringtheories. In Kaluza-Klein theory, the general solution of a recent model1s given in point of harmonic maps satisfying Einstein equations (see [8,9, 16, 12, 32, 17, 19]). Altafini [5] expressed some applications of submer-sions in the theory of robotics and S¸ahin [24] also investigated some appli-cations of Riemannian submersions on redundant robotic chains. On theother hand Riemannian submersions are very useful in studying the geome-try of Riemannian manifolds equipped with differentiable structures. In [31]Watson introduced the notion of almost Hermitian submersions between al-most complex manifolds. He investigated some geometric properties betweenbase manifold and total manifold as well as fibers. S¸ahin [25] introducedanti-invariant Riemannian submersions from almost Hermitian manifolds.He showed that such maps have some geometric properties. Also he stud-ied slant submersions from almost Hermitian manifolds onto a Riemannianmanifolds [27]. Recently, considering different conditions on Riemanniansubmersions many studies have been done (see [6, 21, 22, 23, 26, 28, 29]).As a special horizontally conformal maps which were introduced inde-pendently by Fuglede and Ishihara, horizontally conformal submersions aredefined as follows ( M , g ) and ( M , g ) are Riemannian manifolds of dimen-sion m and m , respectively. A smooth submersion f : ( M , g ) → ( M , g )is called a horizontally conformal submersion if there is a positive function λ such that λ g ( X , Y ) = g ( f ∗ X , f ∗ Y )for all X , Y ∈ Γ (cid:16) (ker f ∗ ) ⊥ (cid:17) . Here a horizontally conformal submersion f is called horizontally homothetic if the gradλ is vertical i.e. H ( gradλ ) = 0 . We denote by V and H the projections on the vertical distributions ( kerf ∗ )and horizontal distributions ( kerf ∗ ) ⊥ . It can be said that Riemannian sub-mersion is a special horizontally conformal submersion with λ = 1. Re-cently, Akyol and S¸ahin introduced conformal anti-invariant submersions[2], conformal semi-invariant submersion[3], conformal slant submersion [4]and conformal semi-slant submersions[1]. Also the geometry of conformalsubmersions have been studied by several authors [15, 18].In section 2 we review basic formulas and definitions needed for this pa-per. In section 3, we define the new conformal bi-slant submersion fromalmost Hermitian manifolds onto Riemannian manifolds and present a ex-ample. We investigate the geometry of the horizontal distribution and thevertical distribution. Finally we obtain necessary and sufficient conditionsfor a conformal bi-slant submersion to be totally geodesic.2 Preliminaries
Let ( M , g , J ) be an almost Hermitian manifold. Then this means that M admits a tensor field J of type (1 ,
1) on M which satisfy J = − I, g ( J E , J E ) = g ( E , E ) (2.1)for E , E ∈ Γ( T M ). An almost Hermitian manifold M is called Kaehle-rian manifold if ( ∇ E J ) E = 0 , E , E ∈ Γ (
T M )where ∇ is the operator of Levi-Civita covariant differentiation.Now, we will give some definitions and theorems about the concept of(horizontally) conformal submersions. Definition 2.1.
Let ( M , g ) and ( M , g ) are two Riemannian manifoldswith the dimension m and m , respectively. A smooth map f : ( M , g ) → ( M , g ) is called horizontally weakly conformal or semi conformal at q ∈ M if, eitheri. df q = 0 , orii. df q is surjective and there exists a number Ω( q ) = 0 satisfying g ( df q X, df q Y ) = Ω( q ) g ( X, Y ) for X, Y ∈ Γ (ker( df )) ⊥ .Here the number Ω( q ) is called the square dilation. Its square root λ ( q ) = p Ω( q ) is called the dilation. The map f is called horizontally weakly confor-mal or semi-conformal on M if it is horizontally weakly conformal at everypoint of M . it is said to be a conformal submersion if f has no criticalpoint. Let f : M → M be a submersion. A vector field X on M is called abasic vector field if X ∈ Γ (cid:16) (ker f ∗ ) ⊥ (cid:17) and f -related with a vector field X on M i.e f ∗ ( X q ) = X f ( q ) for q ∈ M .The two (1 ,
2) tensor fields T and A on M are given by the formulas T ( E , E ) = T E E = H∇ V E V E + V∇ V E H E (2.2) A ( E , E ) = A E E = V∇ H E H E + H∇ H E V E (2.3)3or E , E ∈ Γ (
T M ) [13].Note that a Riemannian submersion f : M −→ M has totally geodesicfibers if and only if T vanishes identically.Considering the equations (2.3) and (2.4), one can write ∇ U U = T U U + ¯ ∇ U U (2.4) ∇ U X = H∇ U X + T U X (2.5) ∇ X U = A X U + V∇ X U (2.6) ∇ X X = H∇ X X + A X X (2.7)for X , X ∈ Γ (cid:16) (ker f ∗ ) ⊥ (cid:17) and U , U ∈ Γ (ker f ∗ ), where ¯ ∇ U U = V∇ U U .Then we easily seen that T U and A X are skew-symmetric i.e g ( A X E , E ) = − g ( E , A X E ) and g ( T U E , E ) = − g ( E , T U E ) for any E , E ∈ Γ (
T M ). For the special case where f as the horizontal, the followingProposition be given: Proposition 1.
Let f : ( M , g ) → ( M , g ) be a horizontally conformalsubmersion with dilation λ and X , X ∈ Γ (cid:16) (ker f ∗ ) ⊥ (cid:17) , then A X X = 12 (cid:18) V [ X , X ] − λ g ( X , X ) grad V (cid:18) λ (cid:19)(cid:19) (2.8)Let f : ( M , g ) → ( M , g ) be a smooth map between ( M , g ) and( M , g ) Riemannian manifolds. Then the second fundamental form of f isgiven by ( ∇ f ∗ ) ( E , E ) = ∇ fE f ∗ ( E ) − f ∗ (cid:0) ¯ ∇ E E (cid:1) (2.9)for any E , E ∈ Γ (
T M ). It is known that the second fundamental form f is symmetric [7]. Lemma 2.1.
Suppose that f : M → M is a horizontally conformal sub-mersion. Then for X , X ∈ Γ (cid:16) (ker f ∗ ) ⊥ (cid:17) and U , U ∈ Γ (ker f ∗ ) we havei. ( ∇ f ∗ ) ( X , X ) = X (ln λ ) f ∗ X + X (ln λ ) f ∗ X − g ( X , X ) f ∗ ( ∇ ln λ ) ii. ( ∇ f ∗ ) ( U , U ) = − f ∗ ( T U U ) iii. ( ∇ f ∗ ) ( X , U ) = − f ∗ (cid:0) ¯ ∇ X U (cid:1) = − f ∗ ( A X V ) . f is called a totally geodesic map if ( ∇ f ∗ ) ( E , E ) = 0for E , E ∈ Γ( T M ) [7].We assume that g is a Riemannian metric tensor on the manifold M = M × M and the canonical foliations D M and D M intersect verticallyeverywhere. Then g is the metric tensor of a usual product of Riemannianmanifold if and only if D M and D M are totally geodesic foliations. Definition 3.1.
Let ( M , g , J ) be an almost Hermitian manifold and ( M , g ) a Riemannian manifold. A horizontal conformal submersion f : M −→ M is called a conformal bi-slant submersion if D and ¯ D are slant distributionswith the slant angles θ and ¯ θ , respectively, such that ker f ∗ = D ⊕ ¯ D . f iscalled proper if its slant angles satisfy θ, ¯ θ = 0 , π . We now give a example of a proper conformal bi-slant submersion.
Example 1.
We consider the compatible almost complex structure J ω on R such that J ω = (cos ω ) J + (sin ω ) J , < ω ≤ π where J ( x , x , x , x , x , x , x , x ) = ( − x , x , − x , x , − x , x , − x , x ) J ( x , x , x , x , x , x , x , x ) = ( − x , x , x , − x , − x , x , x , − x ) Consider a submersion f : R → R defined by f ( x , x , x , x , x , x , x , x ) = π (cid:18) x − x √ , x , x − x √ , x (cid:19) Then it follows that D = span { U = 1 √ (cid:18) ∂∂x + ∂∂x (cid:19) , U = ∂∂x } ¯ D = span { U = 1 √ (cid:18) ∂∂x + ∂∂x (cid:19) , U = ∂∂x } Thus f is conformal bi-slant submersion with θ and ¯ θ such that cos θ = √ cos ω and cos ¯ θ = √ sin ω . f is a conformal bi-slant submersion from a almost Her-mitian manifold ( M , g , J ) onto a Riemannian manifold ( M , g ). For U ∈ Γ (ker f ∗ ), we have U = αU + βU (3.1)where αU ∈ Γ ( D ) and βU ∈ Γ ( D ).Also, for U ∈ Γ (ker f ∗ ), we write J U = ξU + ηU (3.2)where ξU ∈ Γ (ker f ∗ ) and ηU ∈ Γ (ker f ∗ ) ⊥ .For X ∈ Γ (cid:16) (ker f ∗ ) ⊥ (cid:17) , we have J X = B X + C X (3.3)where B X ∈ Γ (ker f ∗ ) and C X ∈ Γ (cid:16) (ker f ∗ ) ⊥ (cid:17) .The horizontal distribution (ker f ∗ ) ⊥ is decompesed as(ker f ∗ ) ⊥ = ηD ⊕ ηD ⊕ µ where µ is the complementary distribution to ηD ⊕ ηD in (ker f ∗ ) ⊥ .Considering Definition 3.1 we can give the following result that we willuse throughout the article. Theorem 3.1.
Suppose that f is a conformal bi-slant submersion from analmost Hermitian manifold ( M , g , J ) onto a Riemannian manifold ( M , g ) .Then we havei) ξ U = − (cid:0) cos θ (cid:1) U for U ∈ Γ ( D ) ii) ξ V = − (cid:0) cos ¯ θ (cid:1) V for V ∈ Γ (cid:0) ¯ D (cid:1) Proof.
The proof of this theorem is similar to slant immersions [11].
Theorem 3.2.
Suppose that f is a proper conformal bi-slant submersionfrom a Kaehlerian manifold ( M , g , J ) onto a Riemannian manifold ( M , g ) with slant functions θ, ¯ θ . Theni) the distribution D is integrable if and only if λ − g ( ∇ f ∗ ( U , ηU ) , f ∗ ηV ) = g ( T U ηξU − T U ηξU , V )+ g ( T U ηU − T U ηU , ξV )+ λ − g ( ∇ f ∗ ( U , ηU ) , f ∗ ηV ) . i) the distribution ¯ D is integrable if and only if λ − g ( ∇ f ∗ ( V , ηV ) , f ∗ ηU ) = g ( T V ηξV − T V ηξV , U )+ g ( T V ηV − T V ηV , ξU )+ λ − g ( ∇ f ∗ ( V , ηV ) , f ∗ ηU ) . where U , U ∈ Γ ( D ) , V , V ∈ Γ (cid:0) ¯ D (cid:1) .Proof. i ) From U , U ∈ Γ ( D ) and V ∈ Γ (cid:0) ¯ D (cid:1) we have g ([ U , U ] , V ) = g ( ∇ U ξU , J V ) + g ( ∇ U ηU , J V ) − g ( ∇ U ξU , J V ) − g ( ∇ U ηU , J V ) . Considering Theorem 3.1 we arrivesin θg ([ U , U ] , V ) = − g ( ∇ U ηξU , V ) + g ( ∇ U ηU , J V )+ g ( ∇ U ηξU , V ) − g ( ∇ U ηU , J V ) . By using the equation (2.5) we obtainsin θg ([ U , U ] , V ) = g ( T U ηξU − T U ηξU , V ) + g ( T U ηU − T U ηU , ξV ) − λ − g ( ∇ f ∗ ( U , ηU ) , f ∗ ηV )+ λ − g ( ∇ f ∗ ( U , ηU ) , f ∗ ηV ) . The proof of ii ) can be made by applying similar calculations. Theorem 3.3.
Suppose that f is a proper conformal bi-slant submersionfrom a Kaehlerian manifold ( M , g , J ) onto a Riemannian manifold ( M , g ) with slant functions θ, ¯ θ . Then the distribution D defines a totally geodesicfoliation if and only if λ − g ( ∇ f ∗ ( ηU , U ) , f ∗ ηV ) = − g ( T U ηξU , V ) + g ( T U ηU , ξV ) . (3.4) and λ − g (cid:16) ∇ fX f ∗ ηU , f ∗ ηU (cid:17) = − sin θg ([ U , X ] , U ) + g ( A X ηξU , U )+ g ( grad (ln λ ) , X ) g ( ηU , ηU )+ g ( grad (ln λ ) , ηU ) g ( X , ηU )+ g ( grad (ln λ ) , ηU ) g ( X , ηU ) − g ( A X ηU , ξU ) (3.5) where U , U ∈ Γ ( D ) , V ∈ Γ (cid:0) ¯ D (cid:1) and X ∈ Γ (cid:16) (ker f ∗ ) ⊥ (cid:17) . roof. For U , U ∈ Γ ( D ) and V ∈ Γ (cid:0) ¯ D (cid:1) we have g ( ∇ U U , V ) = − g (cid:0) ∇ U ξ U , V (cid:1) − g ( ∇ U ηξU , V ) + g ( ∇ U ηU , J V ) . Thus we can writesin θg ( ∇ U U , V ) = − g ( T U ηξU , V ) + g ( T U ηU , ξV )+ g ( H∇ U ηU , ηV ) . Using (2.9) we obtainsin θg ( ∇ U U , V ) = − g ( T U ηξU , V ) + g ( T U ηU , ξV ) − λ − g ( ∇ f ∗ ( ηU , U ) , f ∗ ηV ) . which is first equation in Theorem 3.3.On the other hand any U , U ∈ Γ( D ) and X ∈ Γ (cid:16) (ker f ∗ ) ⊥ (cid:17) we canwrite g ( ∇ U U , X ) = − g ([ U , X ] , U ) − g ( ∇ X U , U )= − g ([ U , X ] , U ) + g ( ∇ X J ξU , U ) − g ( ∇ X ηU , J U ) . Using Theorem 3.1, we arrive following equation g ( ∇ U U , X ) = − g ([ U , X ] , U ) − cos θg ( ∇ X U , U )+ g ( ∇ X ηξU , U ) − g ( ∇ X ηU , J U )From (2.7) and Lemma 2.1 we havesin θg ( ∇ U U , X ) = − sin θg ([ U , X ] , U ) + g ( A X ηξU , U ) − g ( A X ηU , ξU ) − λ − g (cid:16) ∇ fX f ∗ ηU , f ∗ ηU (cid:17) + g ( grad (ln λ ) , X ) g ( ηU , ηU )+ g ( grad (ln λ ) , ηU ) g ( X , ηU )+ g ( grad (ln λ ) , ηU ) g ( X , ηU )This completes the proof. Theorem 3.4.
Suppose that f is a proper conformal bi-slant submersionfrom a Kaehlerian manifold ( M , g , J ) onto a Riemannian manifold ( M , g ) with slant functions θ, ¯ θ . Then the distribution ¯ D defines a totally geodesicfoliation if and only if λ − g ( ∇ f ∗ ( ηV , V ) , f ∗ ηU ) = − g ( T V ηξV , U ) + g ( T V ηV , ξU ) . (3.6) nd λ − g (cid:16) ∇ fX f ∗ ηV , f ∗ ηV (cid:17) = − sin ¯ θg ([ V , X ] , V ) + g ( A X ηξV , V )+ g ( grad (ln λ ) , X ) g ( ηV , ηV )+ g ( grad (ln λ ) , ηV ) g ( X , ηV )+ g ( grad (ln λ ) , ηV ) g ( X , ηV ) − g ( A X ηV , ξV ) (3.7) where U ∈ Γ ( D ) , V , V ∈ Γ (cid:0) ¯ D (cid:1) and X ∈ Γ (cid:16) (ker f ∗ ) ⊥ (cid:17) .Proof. The proof of this theorem is similar to the proof of Theorem 3.3.
Theorem 3.5.
Suppose that f is a proper conformal bi-slant submersionfrom a Kaehlerian manifold ( M , g , J ) onto a Riemannian manifold ( M , g ) with slant functions θ, ¯ θ .Then, the vertical distribution (ker f ∗ ) is a locallyproduct M D × M ¯ D if and only if the equations (3.4), (3.5), (3.6) and (3.7)are hold where M D and M ¯ D are integral manifolds of the distributions D and ¯ D , respectively. Theorem 3.6.
Suppose that f is a proper conformal bi-slant submersionfrom a Kaehlerian manifold ( M , g , J ) onto a Riemannian manifold ( M , g ) with slant functions θ, ¯ θ . Then the distribution (ker f ∗ ) ⊥ defines a totallygeodesic foliation if and only if λ − g (cid:16) ∇ fX f ∗ ηU , f ∗ CX (cid:17) = − g ( A X ηU , BX ) + λ − g (cid:16) ∇ fX f ∗ ηξU , f ∗ X (cid:17) − g ( grad ln λ, X ) g ( ηξU , X ) − g ( grad ln λ, ηξU ) g ( X , X )+ g ( X , ηξU ) g ( grad ln λ, X )+ g ( grad ln λ, ηU ) g ( X , CX ) − g ( X , ηU ) g ( grad ln λ, CX ) . (3.8) and λ − g (cid:16) ∇ fX f ∗ ηV , f ∗ CX (cid:17) = − g ( A X ηV , BX ) + λ − g (cid:16) ∇ fX f ∗ ηξV , f ∗ X (cid:17) − g ( grad ln λ, X ) g ( ηξV , X ) − g ( grad ln λ, ηξV ) g ( X , X )+ g ( X , ηξV ) g ( grad ln λ, X )+ g ( grad ln λ, ηV ) g ( X , CX ) − g ( X , ηV ) g ( grad ln λ, CX ) . (3.9)9 here X , X ∈ Γ (ker f ∗ ) ⊥ , U ∈ Γ ( D ) and V ∈ Γ (cid:0) ¯ D (cid:1) .Proof. For X , X ∈ Γ (ker π ∗ ) ⊥ and U ∈ Γ ( D ) we can write g ( ∇ X X , U ) = − g ( ∇ X ξU , J X ) − g ( ∇ X ηU , J X )From Theorem 3.1 we have g ( ∇ X X , U ) = − cos θg ( ∇ X U , X ) + g ( ∇ X ηξU , X ) − g ( ∇ X ηU , J X )By using the equation (2.7) we derivesin θg ( ∇ X X , U ) = g ( H∇ X ηξU , X ) − g ( H∇ X ηU , CX ) − g ( ∇ X ηU , BX ) . Then it follows from Lemma 2.1 thatsin θg ( ∇ X X , U ) = − g ( A X ηU , BX ) + λ − g (cid:16) ∇ fX f ∗ ηξU , f ∗ X (cid:17) − g ( grad ln λ, X ) g ( ηξU , X ) − g ( grad ln λ, ηξU ) g ( X , X )+ g ( X , ηξU ) g ( grad ln λ, X ) − λ − g (cid:16) ∇ fX f ∗ ηU , f ∗ CX (cid:17) + g ( grad ln λ, ηU ) g ( X , CX ) − g ( X , ηU ) g ( grad ln λ, CX ) . Thus we have the first desired equation. Similarly for X , X ∈ Γ (cid:16) (ker π ∗ ) ⊥ (cid:17) and V ∈ (cid:0) ¯ D (cid:1) we findsin ¯ θg ( ∇ X X , V ) = − g ( A X ηV , BX ) + λ − g (cid:16) ∇ fX f ∗ ηξV , f ∗ X (cid:17) − g ( grad ln λ, X ) g ( ηξV , X ) − g ( grad ln λ, ηξV ) g ( X , X )+ g ( X , ηξV ) g ( grad ln λ, X ) − λ − g (cid:16) ∇ fX f ∗ ηV , f ∗ CX (cid:17) + g ( grad ln λ, ηV ) g ( X , CX ) − g ( X , ηV ) g ( grad ln λ, CX ) . Hence the proof is completed. 10 heorem 3.7.
Suppose that f is a proper conformal bi-slant submersionfrom a Kaehlerian manifold ( M , g , J ) onto a Riemannian manifold ( M , g ) with slant functions θ, ¯ θ .Then the distribution (ker f ∗ ) defines a totally geodesicfoliation on M if and only if λ − g (cid:16) ∇ fX f ∗ ωU , f ∗ ωV (cid:17) = (cid:0) cos θ − cos ¯ θ (cid:1) g ( ∇ X QU , V ) − g ( A X V , ηξU ) − g ( A X ξV , ηU ) − sin θg ([ U , X ] , V ) − g ( X , ηU ) g ( grad ln λ, ηV )+ g ( grad ln λ, X ) g ( ηU , ηV )+ g ( grad ln λ, ηU ) g ( X , ηV ) (3.10) where X ∈ Γ (cid:16) (ker f ∗ ) ⊥ (cid:17) and U , V ∈ Γ (ker f ∗ ) .Proof. Given X ∈ Γ (cid:16) (ker f ∗ ) ⊥ (cid:17) and U , V ∈ (ker f ∗ ). Then we obtain g ( ∇ U V , X ) = − g ([ U , X ] , V ) + g ( J ∇ X ξU , V ) − g ( ∇ X ηU , JV ) By using Theorem 3.1 we have g ( ∇ U V , X ) = − g ([ U , X ] , V ) − cos θg ( ∇ X P U , V ) − cos ¯ θg ( ∇ X QU , V ) + g ( ∇ X ηξU , V ) − g ( ∇ X ωU , ξV ) − g ( ∇ X ωU , ηV ) . Then we arrivesin θg ( ∇ U V , X ) = (cid:0) cos θ − cos ¯ θ (cid:1) g ( ∇ X QU , V )+ g ( ∇ X ηξU , V ) − sin θg ([ U , X ] , V ) − g ( ∇ X ηU , ξV ) − g ( ∇ X ηU , ηV )From the equation (2.6) and Lemma 2.1 we obtainsin θg ( ∇ U V , X ) = (cid:0) cos θ − cos ¯ θ (cid:1) g ( ∇ X QU , V ) − g ( A X V , ηξU ) − sin θg ([ U , X ] , V ) − g ( A X ξV , ηU )+ g ( grad ln λ, X ) g ( ηU , ηV )+ g ( grad ln λ, ηU ) g ( X , ηV ) − g ( X , ηU ) g ( grad ln λ, ηV ) − λ − g (cid:16) ∇ fX f ∗ ηU , f ∗ ηV (cid:17) Using above equation the desired equality is achieved.11 heorem 3.8.
Suppose that f is a proper conformal bi-slant submersionfrom a Kaehlerian manifold ( M , g , J ) onto a Riemannian manifold ( M , g ) with slant functions θ, ¯ θ . Then, the total space M is a locally product M D × M ¯1 D × M f ∗ ) ⊥ if and only if the equations (3.4), (3.5), (3.6),(3.7), (3.8) and (3.9) are hold where M D , M D and M f ∗ ) ⊥ are integralmanifolds of the distributions D , ¯ D and (ker f ∗ ) ⊥ , respectively. Theorem 3.9.
Suppose that f is a proper conformal bi-slant submersionfrom a Kaehlerian manifold ( M , g , J ) onto a Riemannian manifold ( M , g ) with slant functions θ, ¯ θ . Then, the total space M is a locally product M f ∗ × M f ∗ ) ⊥ if and only if the equations (3.8), (3.9) and (3.10)arehold where M f ∗ and M f ∗ ) ⊥ are integral manifolds of the distributions ker f ∗ and (ker f ∗ ) ⊥ , respectively. Theorem 3.10.
Suppose that f is a proper conformal bi-slant submer-sion from a Kaehlerian manifold ( M , g , J ) onto a Riemannian manifold ( M , g ) with slant functions θ, ¯ θ . Then f is totally geodesic if and only if − λ − g (cid:16) ∇ fηV f ∗ ηU , f ∗ J CX (cid:17) = (cid:0) cos θ − cos ¯ θ (cid:1) g ( T U QV , X )+ λ − g ( ∇ f ∗ ( ξU , ηV ) , f ∗ J CX ) − g ( ηU , ηV ) g ( grad ln λ, J CX )+ λ − g ( ∇ f ∗ ( U , ηξV ) , f ∗ X ) − g ( T U ηV , BX ) and λ − g (cid:16) ∇ fX f ∗ ηU , f ∗ CX (cid:17) = (cid:0) cos θ − cos ¯ θ (cid:1) g ( A X QU , X )+ λ − g (cid:16) ∇ fX f ∗ ηξU , f ∗ X (cid:17) − g ( grad ln λ, X ) g ( ηξU , X ) − g ( grad ln λ, ηξU ) g ( X , X )+ g ( X , ηξU ) g ( grad ln λ, X )+ g ( grad ln λ, ηU ) g ( X , CX ) − g ( X , ηU ) g ( grad ln λ, CX )+ g ( A X BX , ηU ) . where X , X ∈ Γ (cid:16) (ker f ∗ ) ⊥ (cid:17) and U , V ∈ Γ (ker f ∗ ) . roof. Given U , V ∈ Γ (ker f ∗ ) and X ∈ Γ (cid:16) (ker f ∗ ) ⊥ (cid:17) Then we write λ − g ( ∇ f ∗ ( U , V ) , f ∗ X ) = − λ − g ( f ∗ ∇ U V , f ∗ X ) . From Theorem 3.1 we obtain (cid:0) sin θ (cid:1) λ − g ( ∇ f ∗ ( U , V ) , f ∗ X ) = (cid:0) cos θ − cos ¯ θ (cid:1) g ( ∇ U QV , X ) − g ( ∇ U ηV , J X ) + g ( ∇ U ηξV , X )Considering (2.4), (2.5) and Lemma 2.1 we find (cid:0) sin θ (cid:1) λ − g ( ∇ f ∗ ( U , V ) , f ∗ X ) = (cid:0) cos θ − cos ¯ θ (cid:1) g ( T U QV , X )+ λ − g ( ∇ f ∗ ( ξU , ηV ) , f ∗ J CX ) − g ( ηU , ηV ) g ( grad ln λ, J CX )+ λ − g (cid:16) ∇ fηV f ∗ ηU , f ∗ J CX (cid:17) + λ − g ( ∇ f ∗ ( U , ηξV ) , f ∗ X ) − g ( T U ηV , BX ) . Therefore we obtain the first equation of Theorem 3.6.On the other hand, for X , X ∈ Γ (cid:16) (ker f ∗ ) ⊥ (cid:17) and U ∈ Γ (ker f ∗ ) we canwrite (cid:0) sin θ (cid:1) λ − g ( ∇ f ∗ ( U , X ) , f ∗ X ) = (cid:0) cos θ − cos ¯ θ (cid:1) g ( ∇ X QU , X )+ g ( ∇ X ηU , BX ) − g ( ∇ X ηU , CX ) . By using the equation (2.6) and Lemma 2.1, we arrive (cid:0) sin θ (cid:1) λ − g ( ∇ f ∗ ( U , X ) , f ∗ X ) = (cid:0) cos θ − cos ¯ θ (cid:1) g ( A X QU , X )+ λ − g (cid:16) ∇ fX f ∗ ηξU , f ∗ X (cid:17) − g ( grad ln λ, X ) g ( ηξU , X ) − g ( grad ln λ, ηξU ) g ( X , X )+ g ( X , ηξU ) g ( grad ln λ, X ) − λ − g (cid:16) ∇ fX f ∗ ηU , f ∗ CX (cid:17) + g ( grad ln λ, ηU ) g ( X , CX ) − g ( X , ηU ) g ( grad ln λ, CX )+ g ( A X BX , ηU ) . This concludes the proof. 13 eferences [1] M.A. Akyol,
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