CUBO A Mathematical Journal Vol.20, No¯ 01, (79–94). March 2018 http: // dx. doi. org/ 10. 4067/ S0719-06462018000100079 Anti-invariant ξ⊥-Riemannian Submersions From Hyperbolic β-Kenmotsu Manifolds Mohd Danish Siddiqi Department of Mathematics, Faculty of Science, Jazan University, Jazan-Kingdom of Saudi Arabia. anallintegral@gmail.com, msiddiqi@jazanu.edu.sa Mehmet Akif Akyol Department of Mathematics, Faculty of Arts and Sciences, Bingöl University, 12000 Bingöl, Turkey, mehmetakifakyol@bingol.edu.tr ABSTRACT In this paper, we introduce anti-invariant ξ⊥-Riemannian submersions from Hyperbolic β-Kenmotsu Manifolds onto Riemannian manifolds. Necessary and sufficient condi- tions for a special anti-invariant ξ⊥-Riemannian submersion to be totally geodesic are studied. Moreover, we obtain decomposition theorems for the total manifold of such submersions. RESUMEN En este art́ıculo se introducen las submersiones ξ⊥-Riemannianas anti-invariantes desde variedades hiperbólicas β-Kenmotsu sobre variedades Riemannianas. Se estudian condi- ciones necesarias y suficientes para que ciertas submersiones ξ⊥-Riemannianas anti- invariantes especiales sean totalmente geodésicas. Más aún, se obtienen teoremas de descomposión para la variedad total de dichas submersiones. Keywords and Phrases: Riemannian submersion Anti-invariant ξ⊥-Riemannian submersions, Hyperbolic β-Kenmotsu Manifolds, Integrability Conditions. geometry. 2010 AMS Mathematics Subject Classification: 53C25, 53C20, 53C50, 53C40. http://dx.doi.org/10.4067/S0719-06462018000100079 Ignacio Castillo Ignacio Castillo Ignacio Castillo Ignacio Castillo 80 Mohd Danish Siddiqi and Mehmet Akif Akyol CUBO 20, 1 (2018) 1 Introduction The geometry of Riemannian submersions between Riemannian manifolds has been intensively studied and sevral results has been pulished (see O’Neill [7] and Gray [4]). In [11] Waston defined almost Hermitian submersion between almost Hermitian manifolds and in most cases he show that the base manifold and each fiber has the same kind of structure as the total space. He also show that the vertical and horizontal distributions are invariant. On the other hand, the geometry of anti-invariant Riemannian submersions is different from the geometry of almost Hermitian sub- mersions. For example, since every holomorphic map between Kahler manifolds is harmonic [2], it follows that any holomorphic submersion between Kahler manifolds is harmonic. However, this result is not valid for anti-invariant Riemannian submersions, which was first studied by Sahin in [8]. Similarly, Ianus and Pastore [5] shows φ-holomorphic maps between contact manifolds are harmonic. This implies that any contact submersion is harmonic. However, this result is not valid for anti-invariant Riemannian submersions. In [1], Chinea defined almost contact Riemannian sub- mersion between almost contact metric manifolds. In [6], Lee studied the vertical and horizontal distribution are φ-invariant. Moreover, the characteristic vector field ξ is horizontal. We note that only φ-holomorphic submersions have been consider on an almost contact manifolds [3]. It was 1976, Upadhyay and Dube [10] introduced the notion of almost hyperbolic contact (f, g, η, ξ)- structure. Some properties of CR-submanifolds of trans hyperbolic Sasakian manifold were studied in [9]. In this paper, we consider a Riemannian submersion from a Hyperbolic β-Kenmotsu Mani- folds under the assumption that the fibers are anti-invariant with respect to the tensor field of type (1, 1) of almost hyperbolic contact manifold. This assumption implies that the horizontal distribu- tion is not invariant under the action of tensor field of the total manifold of such submersions. In other words, almost hyperbolic contact are useful for describing the geometry of base manifolds, anti-invariant submersion are however served to determine the geometry of total manifold. The paper is organized as follows: In Section 2, we present the basic information needed for this paper. In Section 3, we give the definition of anti-invariant ξ⊥-Riemannian submersions. We also introduce a special anti-invariant ξ⊥-Riemannian submersions and obtain necessary and suf- ficient conditions for such submersions to be totally geodesic or harmonic. In Section 4, we give decomposition theorems by using the existence of anti-invariant ξ⊥-Riemannian submersions and observe that such submersions put some restrictions on the geometry of the total manifold. 2 Preliminaries In this section, we define almost hyperbolic contact manifolds, recall the notion of Riemannian submersion between Riemannian manifolds and give a brife review of basic facts if Riemannian submersion. Let M be an almost hyperbolic contact metric manifold with an almost hyperbolic contact metric structure (φ, ξ, η, gM), where φ is a (1, 1) tensor field, ξ is a vector field, η is a 1-form and CUBO 20, 1 (2018) Anti-invariant ξ⊥-Riemannian Submersions . . . 81 gM is a compatible Riemannian metric on M such that φ2 = I − η ⊗ ξ, φξ = 0, η ◦ φ = 0, η(ξ) = −1, (2.1) gM(φX, φY) = −gM(X, Y) − η(X)η(Y) (2.2) gM(X, φY) = −gM(φX, Y), gM(X, ξ) = η(X) (2.3) An almost hyperbolic contact metric structure (φ, ξ, η, gM) on M is called trans-hyperbolic Sasakian [9] if and only if (∇Xφ)Y = α(g(X, Y)ξ − η(Y)φX) + β(g(φX, Y) − η(Y)φX) (2.4) for all X, Y tangent to M, α and β are smooth functions on M and we say that the trans-hyperbolic Sasakian structure of type (α, β). From the above condition it follows that ∇Xξ = −α(φX) + β(X − η(X)ξ), (2.5) (∇Xη)Y = −αg(φX, Y) + βg(φX, φY), (2.6) where ∇ is the Riemannian connection of Levi-Civita covariant differentiation. More generally one has the notion of a hyperbolic β-Kenmotsu structure which be defined by (∇Xφ)Y = β(g(φX, Y)ξ − η(Y)φX), (2.7) where β is non-zero smooth function. Also we have ∇Xξ = β[X − η(X)ξ]. (2.8) Thus α = 0 and therefore a trans-hyperbolic Sasakian structure of type (0, β) with a non-zero constant is always hyperbolic β-Kenmotsu manifold. Let (Mm, gM) and (N n, gN) be Riemannian manifolds, where dimM = m, dimN = N and m > n. A Riemannian submersion F : M → N is a map from M onto N satisfying the following axioms: (1) (S1) F has maximal rank (2) (S2) The differential F∗ preserves the lengths of horizontal vectors. For each q ∈ N, F−1(q) is an (m − n)-dimensional submanifold of M. The submanifold F−1(q) are called fibers. A vector field on M is called vertical if it is always tangent to fibers. A vector field on M is called horizontal if it is always orthogonal to fibers. A vector field X on M is called basic if X is horizontal and F-related to a vector field X∗ on N, i.e., F∗Xp = X∗F(p) for all p ∈ M. Note that we denote the projection morphisms on the distributions kerF∗ and (kerF∗) by V and H, respectively. We recall the following lemma from O’Neill [7]. 82 Mohd Danish Siddiqi and Mehmet Akif Akyol CUBO 20, 1 (2018) Lemma 2.1. Let F : M → N be a Riemannian submersion between Riemannian manifolds and X, Y be basic vector fields of M. Then (1) (1) gM(X, Y) = gN(X∗, Y∗) ◦ F. (2) (2) the horizontal part [X, Y]H of [X, Y] is a basic vector field and corresponds to [X∗, Y∗], i.e., F∗([X, Y]) = [X∗, Y∗]. (3) (3) [V, X] is vertical for any vector field V of kerF∗. (4) (4) ((∇)MX Y)H is the basic vector field corresponding to ∇NX∗Y∗. The geometry of Riemannian submersion is characterized by O’Neill’s tensor T and A defined for vector fields E, F on M by AEF = H∇HEVF + V∇HEHF (2.9) TEF = H∇VEVF + V∇VEHF (2.10) where ∇ is the Levi-Civita connection of gM. It is easy to see that a Riemannian submersion F : M → N has totally geodesic fibers if and only if T vanishes identically. For any E ∈ (TM), TC = TVC and A is horizontal, A = AHE. We note that the tensor T and A satisfy TUW = TWU, U, W ∈ (kerF∗) (2.11) AXY = −AYX = 1 2 V[X, Y], X, Y ∈ (kerF∗)⊥ (2.12) On the other hand, from (2.6) and (2,7), we have ∇VW = TVW + ∇̄VW (2.13) ∇VX = H∇VX + TVX (2.14) ∇XV = AXV + V∇XV (2.15) ∇XY = H∇XY + AXV (2.16) for X, Y ∈ (kerF∗)⊥ and V, W ∈ (kerF∗), where ∇̄VW = V∇VW. If X is basic then H∇VX = AXV. Finally, we recall the notion of harmonic maps between Riemannian manifolds. Let (M, gM) and (N, gN) be Riemannian manifolds and supposed that φ : M → N is a smooth map. Then the differential φ∗ of φ can be viewed a section of the bundle Hom(TM, φ −1TN) → M, where φ−1TN is the pullback bundle which has fibers (φ−1TN)p = Tφ(p)N, p ∈ M. Hom(TM, φ−1TN) CUBO 20, 1 (2018) Anti-invariant ξ⊥-Riemannian Submersions . . . 83 has a connection ∇ induced from the Levi-Civita connection ∇M and the pullback connection ∇φ. Then the second fundamental form of φ is given by (∇φ∗)(X, Y) = ∇φ X φ ∗ (Y) − φ ∗ (∇MX Y) (2.17) for X, Y ∈ TM. It is known that the second fundamental form is symmetric. A smooth map φ : (M, gM) → (N, gN) is said to be harmonic if trace(∇φ∗) = 0. On the other hand, the tensor field of φ is the section τ(φ) of (φ−1TN) defined by τ(φ) = divφ∗ = m∑ i=1 (∇φ∗)(ei, ei), (2.18) where {e1, .....em} is the orthogonal frame on M. Then it follows that φ is harmonic if and only if τ(φ) = 0 (see [7]). 3 Anti-invariant ξ⊥- Riemannian Submersions In this section, we define anti-invariant ξ⊥- Riemannian submersion from hyperbolic β-Kenmotsu manifold onto a Riemannian manifold and investigate the integrability of distributions and obtain a necessary and sufficient condition for such submersions to be totally geodesic map. We also investigate the harmonicity of a special Riemannian submersion. Definition 3.1. Let (M, gM, φ, ξ, η) be a hyperbolic β-Kenmotsu manifold and (N, gN) a Rie- mannian manifold. Suppose that there exists a Riemannian submersion F : M → N such that ξ is normal to kerF∗ and kerF∗ is anti-invariant with respect to φ, ie., φ(kerF∗) ⊂ (kerF∗)⊥. Then we say that F is an anti-invariant ξ⊥-Riemannian submersion. Now, we assume that F : (M, gM, φ, ξ, η) → (N, gN) is an anti-invariant ξ⊥-Riemannian submersion. First of all, from Definition 3.1, we have (kerF∗) ⊥ ∩ (kerF∗) 6= 0. We denote the complementary orthogonal distribution to φ(kerF∗) in (kerF∗) ⊥ by µ. Then we have (kerF∗) ⊥ = φ(kerF∗) ⊕ µ, (3.1) where φ(µ) ⊂ µ. Hence µ contains ξ. Thus, for X ∈ (kerF∗)⊥, we have φX = BX + CX, (3.2) where BX ∈ (kerF∗) and CX ∈ (µ). On the other hand, since F∗(kerF∗)⊥ = TN and F is a Riemannian submersion, using (3.2), we have gN (F∗φV, F∗φCX) = 0 for any X ∈ (kerF∗)⊥ and V ∈ (kerF∗), which implies TN = F∗(φ((kerF∗)) ⊕ F∗(µ). 84 Mohd Danish Siddiqi and Mehmet Akif Akyol CUBO 20, 1 (2018) Example 3.2. Let us consider a 5-dimensional manifold M̄ = { (x1, x2, x3, x4, z) ∈ R5 : z 6= 0 } , where (x1, x2, x3, x4, z) are standard coordinates in R 5. We choose the vector fields E1 = e −z ∂ ∂x1 , E2 = e −z ∂ ∂x2 , E3 = e −z ∂ ∂x3 , E4 = e −z ∂ ∂x4 , E5 = e −z ∂ ∂x1 , which are linearly independent at each point of M̄. We define g by g = e2zG, where G is the Euclidean metric on R5. Hence {E1, E2, E3, E4, E5} is an orthonormal basis of M̄. We consider an 1-form η defined by η = ezdz, η(X) = g(X, E5), ∀X ∈ TM̄. We defined the (1, 1) tensor field φ by φ { 2∑ i=2 ( xi ∂ ∂xi + xi+2 ∂ ∂xi+2 + z ∂ ∂z ) } = 2∑ i=2 ( xi ∂ ∂xi+2 − xi+2 ∂ ∂xi ) . Thus, we have φ(E1) = E3, φ(E2) = E4, φ(E3) = −E1, φ(E4) = −E2, φ(E5) = 0. The linear property of g and φ yields that η(E5) = −1, φ 2(X) = X − η(X)E5 g(φX, φY) = −g(X, Y) − η(X)η(Y), for any vector fields X, Y on M̄. Thus, M̄ (φ, ξ, η, g) defines an almost hyperbolic contact metric manifold with ξ = E5. Moreover, let ∇̄ be the Levi-Civita connection with respect to metric g. Then we have [E1, E2] = 0. Similarly [E1, ξ] = e −zE1, [E2, ξ] = e −zE2, [E3, ξ] = e −zE3, [E4, ξ] = e −zE4, [Ei, Ej] = 0, 1 ≤ i 6=≤ 4. The Riemannian connection ∇̄ of the metric g is given by 2g(∇̄XY, Z) = Xg(Y, Z) + Yg(Z, X) − Zg(X, Y) − g(X, [Y, Z]) − g(Y, [X, Z]) + g(Z, [X, Y]), By Koszul’s formula, we obtain the following equations ∇̄E1E1 = −e−zξ, ∇̄E2E2 = −e−zξ, ∇̄E3E3 = −e−zξ, ∇̄E4E4 = −e−zξ, ∇̄ξξ = 0, ∇̄ξEi = 0, ∇̄Eiξ = e−zEi, 1 ≤ i ≤ 4 and ∇̄EiEi = 0 for all 1 ≤ i, j ≤ 4. Thus, we see that M is a trans-hyperbolic Sasakian manifold of type (0, e−z), which is hyperbolic β-Kenmotsu manifold. Here α = 0 and β = e−z. Now, we define (1, 1) tensor field as follows φ(x1, x2, x3, x4, z) = (−x3, −x4, x1, x3, z). Now, we can give the following example. CUBO 20, 1 (2018) Anti-invariant ξ⊥-Riemannian Submersions . . . 85 Example 3.3. Let (M1, g1 = e 2zG, φ, ξ, η) be an almost Hyperbolic contact manifolds and M2 be R 3. The Riemannian metric tensor field g2 is defined by g2 = e 2z(dy1 ⊗ dy1 +dy2 ⊗ dy2 +dy3 ⊗ dy3) on M2. Let φ be a submersion defined by φ : R5 −→ R3 (x1, x2, x3, x4, z) ( x1 + x3√ 2 , z, x1 + x2√ 2 ) Then it follows that kerφ∗ = span {V1 = ∂x1 − ∂x3, V2 = ∂x2 − ∂x2} and (kerφ∗) ⊥ = span {X1 = ∂x1 + ∂x3, X2 = ∂x2 + ∂x2, X3 = z = ξ} Hence we have φV1 = X1 and φV2 = X2. It means that φ(kerφ) ⊂ (kerφ)⊥. A straight computations, we get φ∗X1 = ∂y1, φ∗X2 = ∂y3 and φ∗X3 = ∂y2. Hence, we have g1(Xi, Xi) = g2(φ∗Xi, φ∗Xi), for i = 1, 2, 3. Thus φ is a anti-invariant ξ⊥ Riemannian submersion. Lemma 3.4. Let F be an anti-invariant ξ⊥-Riemannian submersion from a hyperbolic β-Kenmotsu manifold (M, gM, φ, ξ, η) onto a Riemannian manifold (N, gN). Then we have gM(CY, φV) = 0, (3.3) gM(∇XCY, φV) = −gM(CY, φAXV) (3.4) for X, Y ∈ ((kerF∗)⊥) and V ∈ (kerF∗). Proof. For Y ∈ ((kerF∗)⊥) and V ∈ (kerF∗), using (2.2), we have gM(CY, φV) = gM(φY − BY, φV) = gM(φY, φV) = −gM(Y, V) − η(Y)η(V) = −gM(Y, V) = 0 since BY ∈ (kerF∗) and φV, ξ ∈ ((kerF∗)⊥). Differentiating (3.3) with respect to X, we get gM(∇XCY, φV) = − gM(CY, ∇XφV) =gM(CY, (∇Xφ)V) − gM(CY, φ(∇XV)) = − gM(CY, φ(∇XV)) = − gM(CY, φAXV) − gM(CY, φν∇XV) = − gM(CY, φAXV) due to φν∇XV ∈ (kerF∗)). Our assertion is complete. 86 Mohd Danish Siddiqi and Mehmet Akif Akyol CUBO 20, 1 (2018) We study the integrability of the distribution (kerF∗) ⊥ and then we investigate the geometry of leaves of kerF∗ and (kerF∗) ⊥. We note it is known that the distribution (kerF∗) is integrable. Theorem 3.5. Let F be an anti-invaraint ξ⊥-Riemannian submersion from a hyperbolic β-Kenmotsu manifold (M, gM, φ, ξ, η) onto a Riemannian manifold (N, gN). The followings are equivalent. (1) (kerF∗) ⊥ is integrable, (2) gN((∇F∗)(Y, BX), F∗φV) = gN((∇F∗)(X, BY), F∗φV) +gM(CY, φAXV) − gM(CX, φAYV) +βη(Y)gM(X, V) − βη(X)gM(Y, V), (3) gM(AXBY − AYBY, φV) = gM(CY, φAXV) − gM(CX, φAYV) +βη(Y)gM(X, V) − βη(X)gM(Y, V). for X, Y ∈ (kerF∗)⊥ and V ∈ (kerF∗). Proof. For Y ∈ (kerF∗)⊥ and V ∈ (kerF∗), from Definition 3.1, φV ∈ (kerF∗)⊥ and φY ∈ (kerF∗)⊕ µ. Using (2.2) and (2.4), we note that for X ∈ (kerF∗)⊥, gM(∇XY, V) = gM(∇XφY, φV) − βη(Y)gM(X, V) (3.5) −(α + β)η(X)η(Y)η(V). Therefore, from (3.5), we get gM([X, Y], V) = gM(∇XφY, φV) − gM(∇YφX, φV) = βη(X)gM(Y, V) − βη(Y)gM(X, V) = gM(∇XBY, φV) + gM(∇XCY, φV) −gM(∇YBX, φV) − gM(∇YCX, φV) −βη(Y)gM(X, V) + βη(X)gM(Y, V). Since F is a Riemannian submersion, we obtain gM([X, Y], V) = gN(F∗∇XBY, F∗φV) + gM(∇XCY, φV) −gN(F∗∇YBX, F∗φV) − gM(∇YCX, φV) −βη(Y)gM(X, V) + βη(X)gM(Y, V). CUBO 20, 1 (2018) Anti-invariant ξ⊥-Riemannian Submersions . . . 87 Thus, from (2.15) and (3.4), we have gM([X, Y], V) = gN(−(∇F∗(X, BY) + (∇F∗)(Y, BX), F∗φV) −gM(CY, φAXV + gM(CX, φAYV) −βη(Y)gM(X, V) + βη(X)gM(Y, V). which proves (1) ⇐⇒ (2). On the other hand, using (2.14), we obtain (∇F∗)(Y, BX) − (∇F∗)(X, BY) = −F∗(∇YBX − ∇XBY) = −F∗(AYBX − AXBY), which shows that (2) ⇐⇒ (3) Corollary 3.6. Let F be an anti-invaraint ξ⊥-Riemannian submersion from a hyperbolic β- Kenmotsu manifold (M, gM, φ, ξ, η) onto a Riemannian manifold (N, gN) with (kerF∗) ⊥ = φ(kerF∗)⊕ < ξ >. Then the following are equivalent: (1) (kerF∗) ⊥ is integrable (2) (∇F∗)(X, φY) + βη(X)F∗Y = (∇F∗)(Y, φX) + βη(Y)F∗X (3) AXφY + βη(X)Y = AYφX + βη(Y)X, for X, Y ∈ (kerF∗)⊥. Theorem 3.7. Let F be an anti-invariant ξ⊥-Riemannian submersion from a hyperbolic β-Kenmotsu manifold (M, gM, φ, ξ, η) onto a Riemannian manifold (N, gN). The following are equivalent: (1) (kerF∗) ⊥ defines a totally geodesic foliation on M. (2) gM(AXBY, φV) = gM(CY, φAXY) − βη(X)gM(X, V) − βη(X)gM(Y, V), (3) gN((∇F∗)(Y, φX), F∗φV) = gM(CY, φAXV) − βη(X)gM(X, V) − βη(X)gM(Y, V), for X, Y ∈ (kerF∗) ⊥ and V ∈ (kerF∗). Proof. For X, Y ∈ (kerF∗)⊥ and V ∈ (kerF∗), from (3.5), we have gM(∇XY, V) = gM(AXBY, φV) + gM(∇XCY, φV) − βη(Y)gM(X, V) − βη(X)η(Y)η(V) Then from (3.4), we have gM(∇XY, V) = gM(AXBY, φV) + gM(CY, φAXV) − βη(Y)gM(X, V) − βη(X)η(Y)η(V) which shows (1) ⇐⇒ (2). On the other hand, from (2.12) and (2.14), we have gM(AXBY, φV) = gN(−(∇F∗)(X, BY), F∗φV), which proves (2) ⇐⇒ (3). 88 Mohd Danish Siddiqi and Mehmet Akif Akyol CUBO 20, 1 (2018) Corollary 3.8. Let F be an anti-invariant ξ⊥-Riemannian submersion from a hyperbolic β- Kenmotsu manifold (M, gM, φ, ξ, η) onto a Riemannian manifold (N, gN) with (kerF∗) ⊥ = φ(kerF∗)⊕ < ξ >. Then the following are equivalent: (1) (kerF∗) ⊥ defines a totally geodesic folition on M (2) AXφY = βη(Y)X − (α + β)η(X)Y (3) (∇F∗)(Y, φX) = βη(Y)F∗X − β)η(X)F∗Y for X, Y ∈ (kerF∗)⊥. Theorem 3.9. Let F be an anti-invariant ξ⊥-Riemannian submersion from a hyperbolic β-Kenmotsu manifold (M, gM, φ, ξ, η) onto a Riemannian manifold (N, gN). The following are equivalent: (1) kerF∗ defines a totally geodesic folition on M (2) −gN(∇F∗)(V, φX, F∗φW) = 0 (3) TVBX + ACXV ∈ (µ), for X, ∈ (kerF∗)⊥ and V, W ∈ (kerF∗) Proof. For X, ∈ (kerF∗)⊥ and V, W ∈ (kerF∗), gM(W, ξ) = 0 implies that from (2.4) gM(∇VW, ξ) = −gM(W, ∇Vξ) = gM(W, β(V − η(V)ξ)) = 0. Thus we have gM(∇VW, X) = −gM(φ∇VW, φX) − η((∇VW)η(X) = −gM(φ∇V W, φX) = −gM(∇VφW, φX) + gM((∇V φ)W, φX) = gM(φW, ∇V φX). Since F is Riemannian submersion, we have gM(∇VW, X) = gN(F∗φW, F∗∇VφX) = −gN(F∗φW, (∇F∗)(VφX)), which proves (1) ⇐⇒ (2). By direct calculation, we derive −gN(F∗φW, (∇F∗)(VφX)) = gM(φW, ∇VφX) = gM(φW, ∇V BX + ∇VCX) = gM(φW, ∇VBX + [V, CX] + ∇CXV). CUBO 20, 1 (2018) Anti-invariant ξ⊥-Riemannian Submersions . . . 89 Since [V, CX] ∈ (kerF∗), from (2.10) and (2.12), we obtain −gN(F∗φW, (∇F∗)(VφX)) = gM(φW, TVBX + ACXV), which proves (2) ⇐⇒ (3). As an analouge of a Lagrangian Riemannian submersion in [11], we have a similar result; Corollary 3.10. Let F be an anti-invaraint ξ⊥-Riemannian submersion from a hyperbolic β- Kenmotsu manifold (M, gM, φ, ξ, η) onto a Riemannian manifold (N, gN) with (kerF∗) ⊥ = φ(kerF∗)⊕ < ξ >. Then the following are equivalent: (1) (kerF∗) ⊥ defines a totally geodesic folition on M (2) −(∇F∗)(V, φX) = 0 (3) TVφW = 0, X, ∈ (kerF∗)⊥ and V, W ∈ (kerF∗). Proof. From Theorem 3.6, it is enough to show (2) ⇐⇒ (3). Using (2.14) and (2.11), we have −gN(F∗φW, (∇F∗)(VφX)) = gM(∇VφW, φX) = gM(TVφW, φX). Since TVφW ∈ (kerF∗), the proof is complete. We note that a differentiable map F between two Riemannian manifolds is called totally geodesic if ∇F∗ = 0. For the special Riemannian submersion, we have the following characteriza- tion. Theorem 3.11. Let F be an anti-invariant ξ⊥-Riemannian submersion from a hyperbolic β- Kenmotsu manifold (M, gM, φ, ξ, η) onto a Riemannian manifold (N, gN) with (kerF∗) ⊥ = φ(kerF∗)⊕ < ξ >. Then F is a totally geodesic map if and only if TVφW = 0, V, W ∈ (kerF∗) (3.6) and AXφW = 0, X ∈ (kerF⊥∗ ). (3.7) Proof. First of all, we recall that the second fundamental form of a Riemannian submersion satisfies (∇F∗)(X, Y) = 0 ∀ X, Y ∈ (kerF⊥∗ ). (3.8) For V, W ∈ (kerF∗), we get 90 Mohd Danish Siddiqi and Mehmet Akif Akyol CUBO 20, 1 (2018) (∇F∗)(X, Y) = F∗(φTVφW). (3.9) On the other hand, from (2.1), (2.2) and (2.14), we get (∇F∗)(X, W) = F∗(φAXφW), X ∈ (kerF⊥∗ ). (3.10) Therefore, F is totally geodesic if and only if φ(TVφW) = 0 ∀ V, W ∈ (kerF⊥∗ ). (3.11) and φ(AXφW) = 0 ∀ X ∈ (kerF⊥∗ ). (3.12) From (2.2), (2.6) and (2.7), we have TVφW = 0 ∀ V, W ∈ (kerF∗). (3.13) and AXφW = 0 ∀ X ∈ (kerF⊥∗ ). From (2.4), F is totally geodesic if and only the equation (3.6) and (3.7) hold Finally, in this section, we give a necessary and sufficient condition for a special Riemannian submersion to be harmonic as an analouge of Lagrangian Riemannian submersion in [11]. Theorem 3.12. Let F be an anti-invaraint ξ⊥-Riemannian submersion from a hyperbolic β- Kenmotsu manifold (M, gM, φ, ξ, η) onto a Riemannian manifold (N, gN) with (kerF∗) ⊥ = φ(kerF∗)⊕ < ξ >. Then F is harmonic if and only if Trace(φTV ) = 0 for V ∈ (kerF∗). Proof. From [5], we know that F is harmonic if and only if F has minimal fibers. Thus F is harmonic if and only if ∑m1 i=1 Teiei = 0. On the other hand, from (2.4), (2.11) and (2.10), we have TVφW = φTVW (3.14) due to ξ ∈ (kerF⊥ ∗ ) for any V, W ∈ (kerF∗). Using (3.14), we get m1∑ i=1 gM(Teiφei, V) = m1∑ i=1 gM(φTeiφei, V) = − m1∑ i=1 gM(Teiei, φV) for any V ∈ (kerF∗). Thus skew-symmetric T implies that m1∑ i=1 gM(φTeiφei, V) = − m1∑ i=1 gM(Teiei, φV). Using (2.8) and (2.2), we have m1∑ i=1 gM(ei, φTVei) = − m1∑ i=1 gM(φei, TVei) = − m1∑ i=1 gM(Teiei, φV) which shows our assertion. CUBO 20, 1 (2018) Anti-invariant ξ⊥-Riemannian Submersions . . . 91 4 Decomposition theorems In this section, we obtain decomposition theorems by using the existence of anti-invariant ξ⊥- Riemannian submersions. First, we recall the following. Theorem 4.1. [10] Let g be a Riemannian metric on the manifold B = M × N and assume that the canonical foliations DM and DN intersect perpendicular every where. Then g is the metric tensor of (1) (i) a twisted product M ×f N if and only if DM is totally geodesic foliation and DN is a totally umbilical foliation. (2) (ii) a warped product M ×f N if and only if DM is totally geodesic foliation and DN is a spheric foliation, i.e., it is umbilical and its mean curvature vector field is parallel. (3) (iii) a usual product of Riemannian manifold if and only if DM and DN are totally geodesic foliations. Our first decomposition theorem for anti-invariant ξ⊥-Riemannian submersion comes from Theo- rem 3.4 and 3.6 in terms of the second fundamental forms of such submersions. Theorem 4.2. Let F be an anti-invariant ξ⊥-Riemannian submersion from a hyperbolic β-Kenmotsu manifold (M, gM, φ, ξ, η) on to a Riemannian manifold (N, gN). Then M is locally product man- ifold if and only if −gN((∇F∗)(Y, φX), F∗φV) = gM(CY, φAXV) − βη(Y)gM(X, V) and −gN((∇F∗)(V, φX), F∗φW) = 0 for X, Y ∈ (kerF⊥ ∗ ) and V, W ∈ (kerF∗). From Corollary 3.5 and 3.7, we have the following decomposition theorem: Theorem 4.3. Let F be an anti-invariant ξ⊥-Riemannian submersion from a hyperbolic β-Kenmotsu manifold (M, gM, φ, ξ, η) on to a Riemannian manifold (N, gN) with (kerF ⊥ ∗ )⊕ < ξ >. Then M is a locally product manifold if and only if AXφY = (α+β)η(Y)X and TVφW = 0, for X, Y ∈ (kerF⊥∗ ) and V, W ∈ (kerF∗). Next we obtain a decomposition theorem which is related to the notion of a twisted product manifold. Theorem 4.4. Let F be an anti-invariant ξ⊥-Riemannian submersion from a hyperbolic β-Kenmotsu manifold (M, gM, φ, ξ, η) on to a Riemannian manifold (N, gN) with (kerF ⊥ ∗ )⊕ < ξ >. Then M is locally twisted product manifold of the form MkerF⊥ ∗ ×f MkerF∗ if and only if 92 Mohd Danish Siddiqi and Mehmet Akif Akyol CUBO 20, 1 (2018) TVφX = −gM(X, TVV) ‖V‖−2 − βη(Y)gM(φX, φV). and AXφY = βη(Y)X for X, Y ∈ (kerF⊥ ∗ ) and V ∈ (kerF∗), where M(kerF⊥ ∗ ) and M(kerF∗) are integrable manifolds of the distributions (kerF⊥ ∗ ) and (kerF∗). Proof. For X ∈ (kerF⊥ ∗ ) and V ∈ (kerF∗), from (2.4) and (2.11), we obtain gM(∇VW, X) = gM(TVφW, φX) = −gM(φW, TVφX) Since TV is skew-symmetric. This implies that kerF∗ is totally umbilical if and only if TVφX − βη(V)gM(φX, φV) = −X(λ)φV, where λ is a function on M. By direct computation, TVφX = −gM(X, TVV) ‖V‖−2 − βη(Y)gM(φX, φV). Then the proof follows from Corollary 3.5 However, in the sequel, we show that the notion of anti-invariant ξ⊥-Riemannian submersion puts some restrictions on the source manifold. Theorem 4.5. Let (M, gM, φ, ξ, η) be a hyperbolic β-Kenmotsu manifold and (N, gN) be a Rie- mannian manifold . Then there does not exist an anti-invariant ξ⊥-Riemannian submersion from M to N with (kerF∗) ⊥ = φ(kerF∗) ⊥⊕ < ξ > such that M is a locally proper twisted product manifold of the form MkerF∗ ×f M(kerF∗)⊥. Proof. Suppose that F : (M, gM, φ, ξ, η) −→ (N, gN) is an anti-invaraiant ξ⊥-Riemannian sub- mersion with (kerF∗) ⊥ = φ(kerF∗) ⊥⊕ < ξ > and M is a locally twisted product of the form MkerF∗ ×f M(kerF∗)⊥ .Then MkerF∗ is a totally geodesic foliation and M(kerF⊥∗ ) is a totally um- bilical foliation. We denote the second fundamental form of M(kerF⊥ ∗ ) by h. Then we have gM(∇XY, V) = gM(h(X, Y), V) X, Y ∈ ((kerF∗)⊥, V ∈ (kerF∗). (4.1) Since M(⊥kerF∗ ) is a totally umbilical foliation, we have gM(∇XY, V) = gM(H, V)gM(X, Y), where H is the mean curvature vector field of M(kerF∗)⊥. On the other hand, from (3.5), we derive gM(∇XY, V) = −gM(φY, ∇XφV) − βη(Y)g(X, V) − βη(X)η(Y)η(V). (4.2) CUBO 20, 1 (2018) Anti-invariant ξ⊥-Riemannian Submersions . . . 93 Using (2.13), we obtain gM(∇XY, V) = gM(φY, AXφV) − βη(Y)g(X, V) − βη(X)η(Y)η(V) (4.3) = gM(Y, AXφV) − βg(X, V) − βη(X)η(V)ξ) Therefore, from (4.1), (4.3) and (2.2), we have AXφV = gM(H, V)φX + η(AXφV)ξ. Since AXφV ∈ (kerF∗), η(AXφV) = gM(AXφV, ξ) = 0. Thus, we have AXφV = gM(H, V)φX. Hence, we derive gM(AXφV, φX) − βη(X)η(V)g(Y, φX) = −gM(H, V) { ‖X‖2 − η2(X) } gM(∇XφV, φX) = −gM(H, V) { ‖X‖2 − η2(X) } + βη(X)η(V)g(Y, φX) gM(∇XY, V) + βη(Y)g(X, V) − βη(X)η(Y)η(V) = −gM(H, V) { ‖X‖2 − η2(X) } + βη(X)η(V)g(Y, φX). Thus using (2.9), we have AXX = 0, which implies βη(X)gM(X, V) = −gM(H, V) { ‖X‖2 − η2(X) } + βη(X)η(Y)[η(V) − gM(Y, φX)] for every X ∈ ((kerF⊥ ∗ ), V ∈ (kerF∗). Choosing X which is orthogonal to ξ gM(H, V) ‖X‖2 = 0. Since gM is the Riemannian metric and H ∈ (kerF∗), we conclude that H = 0, which shows kerF⊥∗ is totally geodesic, so M is usual product of Riemannian manifolds. References [1] Chinea, C. Almost contact metric submersions, Rend. Circ. Mat. Palermo, 43(1), 89-104, 1985. [2] Eells, J., Sampson, J. H. Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86, 109-160. 1964. [3] Falcitelli, M., Ianus, S., Pastore, A. M. Riemannian submersions and Related topics, (World Scientific, River Edge, NJ, 2004. [4] Gray, A. Pseudo-Riemannian almost product manifolds and submersion, J. Math. Mech., 16, 715-737, 1967. 94 Mohd Danish Siddiqi and Mehmet Akif Akyol CUBO 20, 1 (2018) [5] Ianus, S., Pastore, A. M., Harmonic maps on contact metric manifolds, Ann. Math. Blaise Pascal, 2(2), 43-53, 1995. [6] Lee, J. W., Anti-invariant ξ⊥− Riemannian submersions from almost contact manifolds, Hacettepe J. Math. Stat. 42(2), 231-241, 2013. [7] O’Neill , B. The fundamental equations of a submersions, Mich. Math. J., 13, 458-469, 1996. [8] Sahin, B. Anti-invariant Riemannian submersions from almost hermition manifolds, Cent. Eur. J. Math., 8(3), 437-447, 2010. [9] Siddiqi, M. D., Ahmed, M and Ojha, J.P., CR-submanifolds of nearly-trans hyperbolic sasakian manifolds admitting semi-symmetric non-metric connection, Afr. Diaspora J. Math. (N.S.), Vol 17(1), 93-105, 2014. [10] Upadhyay, M. D, Dube., K. K., Almost contact hyperbolic (f, g, η, ξ) structure, Acta. Math. Acad. Scient. Hung., Tomus, 28, 1-4, 1976. [11] Watson, B. Almost Hermitian submersions, J. Differential Geometry, 11(1), 147-165, 1976. Introduction 0.2cmPreliminaries 0.2cm Anti-invariant - Riemannian Submersions 0.2cmDecomposition theorems