CUBO, A Mathematical Journal Vol. 24, no. 01, pp. 01–20, April 2022 DOI: 10.4067/S0719-06462022000100001 Quasi bi-slant submersions in contact geometry Rajendra Prasad 1 Mehmet Akif Akyol 2 Sushil Kumar 3 Punit Kumar Singh 1 1Department of Mathematics and Astronomy, University of Lucknow, Lucknow, India. rp.manpur@rediffmail.com singhpunit1993@gmail.com 2Department of Mathematics, Faculty of Sciences and Arts, Bingöl University, 12000, Bingöl, Turkey. mehmetakifakyol@bingol.edu.tr 3Department of Mathematics, Shri Jai Narain Post Graduate College, Lucknow, India. sushilmath20@gmail.com ABSTRACT The aim of the paper is to introduce the concept of quasi bi- slant submersions from almost contact metric manifolds onto Riemannian manifolds as a generalization of semi-slant and hemi-slant submersions. We mainly focus on quasi bi-slant submersions from cosymplectic manifolds. We give some non-trivial examples and study the geometry of leaves of distributions which are involved in the definition of the sub- mersion. Moreover, we find some conditions for such sub- mersions to be integrable and totally geodesic. RESUMEN El objetivo de este art́ıculo es introducir el concepto de sub- mersiones cuasi bi-inclinadas desde variedades casi contacto métricas hacia variedades Riemannianas, como una genera- lización de submersiones semi-inclinadas y hemi-inclinadas. Principalmente nos enfocamos en submersiones cuasi bi- inclinadas desde variedades cosimplécticas. Damos algunos ejemplos no triviales y estudiamos la geometŕıa de hojas de distribuciones que están involucradas en la definición de la submersión. Más aún, encontramos algunas condiciones para que estas submersiones sean integrables y totalmente geodésicas. Keywords and Phrases: Riemannian submersion, semi-invariant submersion, bi-slant submersion, quasi bi-slant submersion, horizontal distribution. 2020 AMS Mathematics Subject Classification: 53C15, 53C43, 53B20. Accepted: 04 September, 2021 Received: 02 January, 2021 c©2022 R. Prasad et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/S0719-06462022000100001 https://orcid.org/0000-0002-7502-0239 https://orcid.org/0000-0003-2334-6955 https://orcid.org/0000-0003-2118-4374 https://orcid.org/0000-0002-8700-5976 mailto:rp.manpur@rediffmail.com mailto:singhpunit1993@gmail.com mailto:mehmetakifakyol@bingol.edu.tr mailto:sushilmath20@gmail.com 2 R. Prasad, M. A. Akyol, S. Kumar & P. K. Singh CUBO 24, 1 (2022) 1 Introductions In differential geometry, there are so many important applications of immersions and submersions both in mathematics and in physics. The properties of slant submersions became an interesting subject in differential geometry, both in complex geometry and in contact geometry. In 1966 and 1967, the theory of Riemannian submersions was initiated by O’Neill [17] and Gray [11] respectively. Nowadays, Riemannian submersions are of great interest not only in mathematics, but also in theoretical pyhsics, owing to their applications in the Yang-Mills theory, Kaluza-Klein theory, supergravity and superstring theories (see [7, 8, 10, 13, 14] ). In 1976, the almost complex type of Riemannian submersions was studied by Watson [29]. He also introduced almost Hermitian submersions between almost Hermitian manifolds requiring that such Riemannian submersions are almost complex maps. In 1985, D. Chinea [9] extended the notion of almost Hermitian submersion to several kinds of sub-classes of almost contact manifolds. In [4] and [5], there are so many im- portant and interesting results about Riemannian and almost Hermitian submersions. In 2010, B. Şahin introduced anti invariant submersions from almost Hermitian manifolds onto Riemannian manifolds [25]. Inspired by B. Şahin’s article, many geometers introduced several new types of Riemannian submersions in different ambient spaces such as semi-invariant submersion [21, 23], generic submersion [27], slant submersion [12, 22], hemi-slant submersion [28], semi-slant submer- sion [18], bi-slant submersion [26], quasi hemi-slant submersion [16], quasi bi-slant submersion [19, 20], conformal anti-invariant submersion [1], conformal slant submersion [2] and conformal semi-slant submersion [3, 15]. Also, these kinds of submersions were considered in different kinds of structures such as cosymplectic, Sasakian, Kenmotsu, nearly Kaehler, almost product, para- contact, etc. Recent developments in the theory of submersions can be found in the book [24]. Inspired from the good and interesting results of above studies, we introduce the notion of quasi bi-slant submersions from cosymplectic manifolds onto Riemannian manifolds. The paper is organized as follows: In the second section, we gather some basic definitions related to quasi bi-slant Riemannian submersion. In the third section, we obtain some results on quasi bi- slant Riemannian submersions from a cosymplectic manifold onto a Riemannian manifold. We also study the geometry of the leaves of the distributions involved in the considered submersions and discuss their totally geodesicity. We obtain conditions for the fibres or the horizontal distribution to be totally geodesic. In the last section, we provide some examples for such submersions. CUBO 24, 1 (2022) Quasi bi-slant submersions in contact geometry 3 2 Preliminaries An n−dimensional smooth manifold M is said to have an almost contact structure, if there exist on M, a tensor field φ of type (1,1), a vector field ξ and 1−form η such that: φ2 = −I + η ⊗ ξ, φξ = 0, η ◦ φ = 0, (2.1) η(ξ) = 1. (2.2) There exists a Riemannian metric g on an almost contact manifold M satisfying the next conditions: g(φU,φV ) = g(U,V ) − η(U)η(V ), (2.3) g(U,ξ) = η(U), (2.4) where U,V are vector fields on M. An almost contact structure (φ,ξ,η) is said to be normal if the almost complex structure J on the product manifold M × R is given by J ( U, α d dt ) = ( φU − αξ, η(U) d dt ) (2.5) and α is the differentiable function on M × R has no torsion, i.e., J is integrable. The condition for normality in terms of φ, ξ, and η is [φ,φ] + 2dη ⊗ ξ = 0 on M, where [φ,φ] is the Nijenhuis tensor of φ. Finally, the fundamental 2−form Φ is defined by Φ(U,V ) = g(U,φV ). An almost contact metric manifold with almost contact structure (φ,ξ,η,g) is said to be cosym- plectic if (∇Uφ)V = 0, (2.6) for any U,V on M. It is both normal and closed and the structure equation of a cosymplectic manifold is given by ∇Uξ = 0, (2.7) for any U on M, where ∇ denotes the Riemannian connection of the metric g on M. Example 2.1 ([6]). R2n+1 with Cartesian coordinates (xi,yi,z)(i = 1, . . . ,n) and its usual contact form η = dz. The characteristic vector field ξ is given by ∂ ∂z and its Riemannian metric g and tensor field φ are given by g = n ∑ i=1 ((dxi) 2 + (dyi) 2) + (dz)2, φ =     0 δij 0 −δij 0 0 0 0 0     , i = 1, . . . ,n. 4 R. Prasad, M. A. Akyol, S. Kumar & P. K. Singh CUBO 24, 1 (2022) This gives a cosymplectic manifold on R2n+1. The vector fields ei = ∂ ∂yi , en+i = ∂ ∂xi , ξ form a φ-basis for the cosymplectic structure. Before giving our definition, we recall the following definition: Definition 2.2 ([28]). Let M be an almost Hermitian manifold with Hermitian metric gM and almost complex structure J, and let N be a Riemannian manifold with Riemannian metric gN. A Riemannian submersion f : (M,gM,J) → (N,gN) is called a hemi-slant submersion if the vertical distribution kerf∗ of f admits two orthogonal complementary distributions D θ and D⊥ such that Dθ is slant with angle θ and D⊥ is anti-invariant, i.e, we have kerf∗ = D θ ⊕ D⊥. In this case, the angle θ is called the hemi-slant angle of the submersion. Definition 2.3. Let (M,φ,ξ,η,gM ) be an almost contact metric manifold and (N,gN) a Rieman- nian manifold. A Riemannian submersion f : (M,φ,ξ,η,gM ) → (N,gN), is called a quasi bi-slant submersion if there exist four mutually orthogonal distributions D,D1,D2 and < ξ > such that (i) kerf∗ = D ⊕ D1 ⊕ D2⊕ < ξ >, (ii) φ(D) = D i.e., D is invariant, (iii) φ(D1) ⊥ D2 and φ(D2) ⊥ D1, (iv) for any non-zero vector field U ∈ (D1)p, p ∈ M, the angle θ1 between φU and (D1)p is constant and independent of the choice of the point p and U in (D1)p, (v) for any non-zero vector field U ∈ (D2)q, q ∈ M, the angle θ2 between φU and (D2)q is constant and independent of the choice of point q and U in (D2)q, These angles θ1 and θ2 are called the slant angles of the submersion. We easily observe that (a) If dimD 6= 0, dim D1 = 0 and dimD2 = 0, then f is an invariant submersion. (b) If dimD 6= 0, dimD1 6= 0, 0 < θ1 < π2 and dim D2 = 0, then f is proper semi-slant submersion. (c) If dimD = 0, dimD1 6= 0, 0 < θ1 < π2 and dim D2 = 0, then f is slant submersion with slant angle θ1. CUBO 24, 1 (2022) Quasi bi-slant submersions in contact geometry 5 (d) If dimD = 0, dimD1 = 0 and dimD2 6= 0, 0 < θ2 < π2 , then f is slant submersion with slant angle θ2. (e) If dimD = 0, dim D1 6= 0, θ1 = π2 and dim D2 = 0, then f is an anti-invariant submersion. (f) If dimD 6= 0, dim D1 6= 0, θ1 = π2 and dim D2 = 0, then f is an semi-invariant submersion. (g) If dimD = 0, dimD1 6= 0, 0 < θ1 < π2 and dimD2 6= 0, θ2 = π 2 , then f is a hemi-slant submersion. (h) If dimD = 0, dimD1 6= 0, 0 < θ1 < π2 and dimD2 6= 0, 0 < θ2 < π 2 , then f is a bi-slant submersion. (i) If dimD 6= 0, dimD1 6= 0, 0 < θ1 < π2 and dimD2 6= 0, θ2 = π 2 , then we may call f is an quasi-hemi-slant submersion. (j) If dimD 6= 0, dimD1 6= 0, 0 < θ1 < π2 and dimD2 6= 0, 0 < θ2 < π 2 , then f is proper quasi bi-slant submersion. Define O’Neill’s tensors T and A by AEF = H∇HEVF + V∇HEHF, (2.8) TEF = H∇VEVF + V∇VEHF, (2.9) for any vector fields E,F on M, where ∇ is the Levi-Civita connection of gM. It is easy to see that TE and AE are skew-symmetric operators on the tangent bundle of M reversing the vertical and the horizontal distributions. From equations (2.8) and (2.9) we have ∇UV = TUV + V∇UV, (2.10) ∇UX = TUX + H∇UX, (2.11) ∇XU = AXU + V∇XU, (2.12) ∇XY = H∇XY + AXY, (2.13) for U,V ∈ Γ(kerf∗) and X,Y ∈ Γ(kerf∗)⊥, where H∇UY = AY U, if Y is basic. It is not difficult to observe that T acts on the fibers as the second fundamental form, while A acts on the horizontal distribution and measures the obstruction to the integrability of this distribution. It is seen that for q ∈ M, U ∈ Vq and X ∈ Hq the linear operators AX, TU : TqM → TqM 6 R. Prasad, M. A. Akyol, S. Kumar & P. K. Singh CUBO 24, 1 (2022) are skew-symmetric, that is gM(AXE,F) = −gM(E,AXF) and gM (TUE,F) = −gM(E,TUF) (2.14) for each E,F ∈ TqM. Since TU is skew-symmetric, we observe that f has totally geodesic fibres if and only if T ≡ 0. Let (M,φ,ξ,η,gM ) be a cosymplectic manifold, (N,gN) be a Riemannian manifold and f : M → N a smooth map. Then the second fundamental form of f is given by (∇f∗)(Y,Z) = ∇fY f∗Z − f∗(∇Y Z), for Y,Z ∈ Γ(TpM), (2.15) where we denote conveniently by ∇ the Levi-Civita connections of the metrics gM and gN and ∇f is the pullback connection. We recall that a differentiable map f between two Riemannian manifolds is totally geodesic if (∇f∗)(Y,Z) = 0, for all Y,Z ∈ Γ(TM). A totally geodesic map is that it maps every geodesic in the total space into a geodesic in the base space in proportion to arc lengths. 3 Quasi bi-slant submersions Let f be quasi bi-slant submersion from an almost contact metric manifold (M,φ,ξ,η,gM ) onto a Riemannian manifold (N,gN). Then, we have TM = kerf∗ ⊕ (kerf∗)⊥. (3.1) Now, for any vector field U ∈ Γ(kerf∗), we put U = PU + QU + RU + η(U)ξ, (3.2) where P , Q and R are projection morphisms of kerf∗ onto D, D1 and D2, respectively. For any U ∈ Γ(kerf∗), we set φU = ψU + ωU, (3.3) where ψU ∈ Γ(kerf∗) and ωU ∈ Γ(kerf∗)⊥. Now, let U1, U2 and U3 be vector fields in D, D1 and D2 respectively. Since D is invariant, i.e. φD = D, we get ωU1 = 0. For any U2 ∈ Γ(D1) we get ωU2 ∈ Γ(ωD1) and for any U3 ∈ Γ(D2) we get ωU3 ∈ Γ(ωD2), hence ωU2 ⊕ ωU3 ∈ Γ(ωD1 ⊕ ωD2) ⊆ Γ(kerf∗)⊥. From equations (3.2) and (3.3), we have φU = φ(PU) + φ(QU) + φ(RU), = ψ(PU) + ω(PU) + ψ(QU) + ω(QU) + ψ(RU) + ω(RU). CUBO 24, 1 (2022) Quasi bi-slant submersions in contact geometry 7 Since φD = D, we get ωPU = 0. Hence above equation reduces to φU = ψPU + ψQU + ωQU + ψRU + ωRU. (3.4) Thus we have the following decomposition according to equation (3.4) φ(kerf∗) = (ψD) ⊕ (ψD1 ⊕ ψD2) ⊕ (ωD1 ⊕ ωD2), (3.5) where ⊕ denotes orthogonal direct sum. Further, let U ∈ Γ(D1) and V ∈ Γ(D2). Then gM (U,V ) = 0. From Definition 2.3 (iii), we have gM (φU,V ) = gM(U,φV ) = 0. Now, consider gM (ψU,V ) = gM (φU − ωU,V ) = gM(φU,V ) = 0. Similarly, we have gM(U,ψV ) = 0. Let W ∈ Γ(D) and U ∈ Γ(D1). Then we have gM (ψU,W) = gM (φU − ωU,W) = gM (φU,W) = −g(U,φW) = 0, as D is invariant, i.e., φW ∈ Γ(D). Similarly, for W ∈ Γ(D) and V ∈ Γ(D2), we obtain gM(ψV,W) = 0, From above equations, we have gM (ψU,ψV ) = 0, and gM(ωU,ωV ) = 0, for all U ∈ Γ(D1) and V ∈ Γ(D2). So, we can write ψD1 ∩ ψD2 = {0}, ωD1 ∩ ωD2 = {0}. If θ2 = π 2 , then ψR = 0 and D2 is anti-invariant, i.e., φ(D2) ⊆ (kerf∗)⊥. 8 R. Prasad, M. A. Akyol, S. Kumar & P. K. Singh CUBO 24, 1 (2022) We also have φ(kerf∗) = ψD ⊕ ψD1 ⊕ ωD1 ⊕ ωD2. (3.6) Since ωD1 ⊆ (kerf∗)⊥, ωD2 ⊆ (kerf∗)⊥. So we can write (kerf∗) ⊥ = ωD1 ⊕ ωD2 ⊕ V, where V is invariant and orthogonal complement of (ωD1 ⊕ ωD2) in (kerf∗)⊥. Also for any non-zero vector field W ∈ Γ(kerf)⊥, we have φW = BW + CW, (3.7) where BW ∈ Γ(kerf) and CW ∈ Γ(V). Lemma 3.1. Let f be a quasi bi-slant submersion from an almost contact metric manifold (M,φ,ξ,η,gM) onto a Riemannian manifold (N,gN). Then, we have ψ2U + BωU = −U + η(U)ξ, ωψU + CωU = 0, ωBW + C2W = −W, ψBW + BCW = 0, for all U ∈ Γ(kerf∗) and W ∈ Γ(kerf∗)⊥. Lemma 3.2. Let f be a quasi bi-slant submersion from an almost contact metric manifold (M,φ,ξ,η,gM) onto a Riemannian manifold (N,gN). Then, we have (i) ψ2U = −(cos2 θ1)U, (ii) gM(ψU,ψV ) = cos 2 θ1gM (U,V ), (iii) gM(ωU,ωV ) = sin 2 θ1gM (U,V ), for all U,V ∈ Γ(D1). Lemma 3.3. Let f be a quasi bi-slant submersion from an contact metric manifold (M,φ,ξ,η,gM ) onto a Riemannian manifold (N,gN). Then, we have (i) ψ2W = −(cos2 θ2)W, (ii) gM(ψW,ψZ) = cos 2 θ2gM(W,Z), (iii) gM(ωW,ωZ) = sin 2 θ2gM(W,Z), for all W,Z ∈ Γ(D2). CUBO 24, 1 (2022) Quasi bi-slant submersions in contact geometry 9 Lemma 3.4. Let f be a quasi bi-slant submersion from a cosymplectic manifold (M,φ,ξ,η,gM ) onto a Riemannian manifold (N,gN). Then, we have V∇UψV + TUωV = ψV∇UV + BTUV, (3.8) TUψV + H∇UωV = ωV∇UV + CTUV, (3.9) V∇XBY + AXCY = ψAXY + BH∇XY, (3.10) AXBY + H∇XCY = ωAXY + CH∇XY, (3.11) V∇UBX + TUCX = ψTUX + BH∇UX, (3.12) TUBX + H∇UCX = ωTUX + CH∇UX, (3.13) V∇Y ψU + AY ωU = BAY U + ψV∇Y U, (3.14) AY ψU + H∇Y ωU = CAY U + ωV∇Y U, (3.15) for any U,V ∈ Γ(kerf∗) and X,Y ∈ Γ(kerf∗)⊥. Now, we define (∇Uψ)V = V∇UψV − ψV∇UV, (3.16) (∇Uω)V = H∇UωV − ωV∇UV, (3.17) (∇XC)Y = H∇XCY − CH∇XY, (3.18) (∇XB)Y = V∇XBY − BH∇XY, (3.19) for any U,V ∈ Γ(kerf∗) and X,Y ∈ Γ(kerf∗)⊥. Lemma 3.5. Let f be a quasi bi-slant submersion from a cosymplectic manifold (M,φ,ξ,η,gM ) onto a Riemannian manifold (N,gN). Then, we have (∇Uψ)V = BTUV − TUωV, (∇Uω)V = CTUV − TUψV, (∇XC)Y = ωAXY − AXBY, (∇XB)Y = ψAXY − AXCY, for any vectors U,V ∈ Γ(kerf∗) and X,Y ∈ Γ(kerf∗)⊥. The proofs of above Lemmas follow from straightforward computations, so we omit them. If the tensors ψ and ω are parallel with respect to the linear connection ∇ on M respectively, then BTUV = TUωV, and CTUV = TUψV, for any U,V ∈ Γ(TM). 10 R. Prasad, M. A. Akyol, S. Kumar & P. K. Singh CUBO 24, 1 (2022) Lemma 3.6. Let f be a quasi bi-slant submersion from a cosymplectic manifold (M,φ,ξ,η,gM ) onto a Riemannian manifold (N,gN). Then, we have (i) gM(∇XY,ξ) = 0 for all X,Y ∈ Γ(D ⊕ D1 ⊕ D2), (ii) gM([X,Y ],ξ) = 0 for all X,Y ∈ Γ(D ⊕ D1 ⊕ D2). Proof. Let X,Y ∈ Γ(D ⊕ D1 ⊕ D2), consider ∇X{gM(Y,ξ)} = (∇XgM)(Y,ξ) + gM (∇XY,ξ) + gM(Y,∇Xξ). Since X and Y are orthogonal to ξ ie. gM(∇XY,ξ) = −gM(Y,∇Xξ), using equation (2.7) and the property that metric tensor is ∇−parallel, we have both results of this lemma. Theorem 3.7. Let f be a proper quasi bi-slant submersion from a cosymplectic manifold (M,φ,ξ,η,gM) onto a Riemannian manifold (N,gN). Then, the invariant distribution D is in- tegrable if and only if gM(TV ψU − TUψV,ωQW + ωRW) = gM (V∇UψV − V∇V ψU,ψQW + ψRW), (3.20) for U,V ∈ Γ(D) and W ∈ Γ(D1 ⊕ D2). Proof. For U,V ∈ Γ(D), and W ∈ Γ(D1 ⊕ D2), using equations (2.1)–(2.4), (2.6), (2.7), (2.10), (3.2), (3.3) and Lemma 3.6 we have gM([U,V ],W) = gM (∇UφV,φW) + η(W)η(∇UV ) − gM(∇V φU,φW) − η(W)η(∇V U), = gM (∇UψV,φW) − gM(∇V ψU,φW), = gM (TUψV − TV ψU,ωQW + ωRW) − gM(V∇UψV − V∇V ψU,ψQW + ψRW), which completes the proof. Theorem 3.8. Let f be a proper quasi bi-slant submersion from a cosymplectic manifold (M,φ,ξ,η,gM) onto a Riemannian manifold (N,gN). Then, the slant distribution D1 is integrable if and only if gM(TW ωψZ − TZωψW,X) = gM (TW ωZ − TZωW,φPX + ψRX) +gM(H∇W ωZ − H∇ZωW,ωRX), (3.21) for all W,Z ∈ Γ(D1) and X ∈ Γ(D ⊕ D2). CUBO 24, 1 (2022) Quasi bi-slant submersions in contact geometry 11 Proof. For all W,Z ∈ Γ(D1) and X ∈ Γ(D ⊕ D2), we have gM([W,Z],X) = gM(∇W Z,X) − gM (∇ZW,X). Using equations (2.1)–(2.4), (2.6), (2.7), (2.11), (3.2), (3.3) and Lemma 3.2 we have gM([W,Z],X) = gM (∇W φZ,φX) − gM(∇ZφW,φX), = gM (∇W ψZ,φX) + gM(∇W ωZ,φX) − gM (∇ZψW,φX) − gM (∇W ωZ,φX), = cos2 θ1gM(∇W Z,X) − cos2 θ1gM(∇ZW,X) − gM(TW ωψZ − TZωψW,X) +gM(H∇W ωZ + TW ωZ,φPX + ψRX + ωRX) −gM(H∇ZωW + TZωW,φPX + ψRX + ωRX). Now, we obtain sin2 θ1gM([W,Z],X) = gM(TW ωZ − TZωW,φPX + ψRX) + gM (H∇W ωZ − H∇ZωW,ωRX) −gM(TW ωψZ − TZωψW,X), which completes the proof. Theorem 3.9. Let f be a proper quasi bi-slant submersion from a cosymplectic manifold (M,φ,ξ,η,gM) onto a Riemannian manifold (N,gN). Then, the slant distribution D2 is integrable if and only if gM(TUωψV − TV ωψU,Y ) = gM(H∇UωV − H∇V ωU,ωQY ) +gM(TUωV − TV ωU,φPY + ψQY ), (3.22) for all U,V ∈ Γ(D2) and Y ∈ Γ(D ⊕ D1). Proof. For all U,V ∈ Γ(D2) and Y ∈ Γ(D ⊕ D1), using equations (2.1)–(2.4), (2.6), (2.7), (3.3) and Lemma 3.6 we have gM ([U,V ],Y ) = gM(∇UψV,φY ) + gM(∇UωV,φY ) − gM(∇V ψU,φY ) − gM(∇V ωU,φY ). From equations (2.9), (3.2) and Lemma 3.3 we have gM ([U,V ],Y ) = cos 2 θ2gM ([U,V ],Y ) + gM(H∇UωV − H∇V ωU,ωQY ) +gM(TUωV − TV ωU,φPY + ψQY ) − gM (TUωψV − TV ωψU,Y ). Now, we have sin2 θ2gM([U,V ],Y ) = gM(TUωV − TV ωU,φPY + ψQY ) − gM(TUωψV − TV ωψU,Y ) +gM(H∇UωV − H∇V ωU,ωQY ), which the proof follows from the above equations. 12 R. Prasad, M. A. Akyol, S. Kumar & P. K. Singh CUBO 24, 1 (2022) Theorem 3.10. Let f be a proper quasi bi-slant submersion from a cosymplectic manifold (M,φ,ξ,η,gM) onto a Riemannian manifold (N,gN). Then the horizontal distribution (kerf∗) ⊥ defines a totally geodesic foliation on M if and only if gM(AUV,PW + cos2 θ1QW + cos2 θ2RW) = gM (H∇UV,ωψPW + ωψQW + ωψRW) −gM(AUBV + H∇UCV,ωW), (3.23) for all U,V ∈ Γ(kerf∗)⊥ and W ∈ Γ(kerf∗). Proof. For U,V ∈ Γ(kerf∗)⊥ and W ∈ Γ(kerf∗), we have gM(∇UV,W) = gM(∇UV,PW + QW + RW + η(W)ξ). Using equations (2.1)–(2.4), (2.6), (2.7), (2.12), (2.13), (3.2), (3.3), (3.7) and Lemmas 3.2 and 3.3 we have gM (∇UV,W) = gM (φ∇UV,φPW) + gM(φ∇UV,φQW) + gM(φ∇UV,φRW), = gM (AUV,PW + cos2 θ1QW + cos2 θ2RW) −gM(H∇UV,ωψPW + ωψQW + ωψRW) +gM(AUBV + H∇UCV,ωPW + ωQW + ωRW). Taking into account ωPW + ωQW + ωRW = ωW and ωPW = 0 in the above, one obtains gM(∇UV,W) = gM(AUV,PW + cos2 θ1QW + cos2 θ2RW) −gM(H∇UV,ωψPW + ωψQW + ωψRW) +gM(AUBV + H∇UCV,ωW). Theorem 3.11. Let f be a proper quasi bi-slant submersion from a cosymplectic manifold (M,φ,ξ,η,gM) onto a Riemannian manifold (N,gN). Then the vertical distribution (kerf∗) defines a totally geodesic foliation on M if and only if gM(TXPY + cos2 θ1TXQY + cos2 θ2TXRY,U) = gM(H∇XωψPY + H∇XωψQY + H∇XωψRY,U) + gM(TXωY,BU) + gM(H∇XωY,CU), (3.24) for all X,Y ∈ Γ(kerf∗) and U ∈ Γ(kerf∗)⊥. Proof. For all X,Y ∈ Γ(kerf∗) and U ∈ Γ(kerf∗)⊥, by using equations (2.1)–(2.4), (2.6) and (2.7) we have gM(∇XY,U) = gM (∇XφPY,φU) + gM(∇XφQY,φU) + gM(∇XφRY,φU). CUBO 24, 1 (2022) Quasi bi-slant submersions in contact geometry 13 Taking into account of (2.10), (2.11), (3.2), (3.3), (3.7) and Lemmas 3.2 and 3.3 we have gM (∇XY,U) = gM(TXPY,U) + cos2 θ1gM(TXQY,U) + cos2 θ2gM (TXRY,U) −gM(H∇XωψPY + H∇XωψQY + H∇XωψRY,U) +gM(∇XωPY + ∇XωQY + ∇XωRY,φU). Since ωPY + ωQY + ωRY = ωY and ωPY = 0, we derive gM (∇XY,U) = gM(TXPY + cos2 θ1TXQY + cos2 θ2TXRY,U) −gM(H∇XωψPY + H∇XωψQY + H∇XωψRY,U) +gM(TXωY,BU) + gM(H∇XωY,CU), which completes the proof. From Theorems 3.10 and 3.11 we also have the following decomposition results. Theorem 3.12. Let f be a proper quasi bi-slant submersion from a cosymplectic manifold (M,φ,ξ,η,gM) onto a Riemannian manifold (N,gN). Then, the total space is locally a product manifold of the form Mker f∗ × M(ker f∗)⊥, where Mker f∗ and M(ker f∗)⊥ are leaves of kerf∗ and (kerf∗) ⊥ respectively if and only if gM(AUV,PY + cos2 θ1QY + cos2 θ2RY ) = gM (H∇UV,ωψPY + ωψQY + ωψRY ) +gM(AUBV + H∇UCV,ωY ), and gM(TXY + cos2 θ1TXQY + cos2 θ2TXRY,U) = gM (H∇XωψPY + H∇XωψQY + H∇XωψRY,U) + gM(TXωY,BU) + gM(H∇XωY,CU), for all X,Y ∈ Γ(kerf∗) and U,V ∈ Γ(kerf∗)⊥. Theorem 3.13. Let f be a proper quasi bi-slant submersion from a cosymplectic manifold (M,φ,ξ,η,gM) onto a Riemannian manifold (N,gN). Then the distribution D defines a totally geodesic foliation if and only if gM(TUφPV,ωQW + ωRW) = −gM(V∇UφPV,ψQW + ψRW), (3.25) and gM (TUφPV,CY ) = −gM(V∇UφPV,BY ), (3.26) for all U,V ∈ Γ(D),W ∈ Γ(D1 ⊕ D2) and Y ∈ Γ(kerf∗)⊥. 14 R. Prasad, M. A. Akyol, S. Kumar & P. K. Singh CUBO 24, 1 (2022) Proof. For all U,V ∈ Γ(D), W ∈ Γ(D1 ⊕ D2) and Y ∈ Γ(kerf∗)⊥, using equations (2.1)–(2.4), (2.6), (2.7), (3.2), (3.3) and Lemma 3.6 we have gM(∇UV,W) = gM(∇UφV,φW), = gM(∇UφPV,φQW + φRW), = gM(TUφPV,ωQW + ωRW) + gM(V∇UφPV,ψQW + ψRW). Now, again using equations (2.10), (3.2), (3.3) and (3.7) we have gM(∇UV,Y ) = gM(∇UφV,φY ), = gM(∇UφPV,BY + CY ), = gM(V∇UφPV,BY ) + gM(TUφPV,CY ), which completes the proof. Theorem 3.14. Let f be a proper quasi bi-slant submersion from a cosymplectic manifold (M,φ,ξ,η,gM) onto a Riemannian manifold (N,gN). Then the distribution D1 defines a totally geodesic foliation if and only if gM(TW ωψZ,U) = gM (TW ωQZ,φPU + ψRU) + gM(H∇W ωQZ,ωRU), (3.27) and gM(H∇W ωψZ,Y ) = gM(H∇W ωZ,CY ) + gM (TW ωZ,BY ), (3.28) for all W,Z ∈ Γ(D1),U ∈ Γ(D ⊕ D2) and Y ∈ Γ(kerf∗)⊥. Proof. For all W,Z ∈ Γ(D1), U ∈ Γ(D ⊕ D2) and Y ∈ Γ(kerf∗)⊥, using equations (2.1)–(2.4), (2.6), (2.7), (2.11), (3.2), (3.3) and Lemma 3.2, we have gM (∇W Z,U) = gM(∇W φZ,φU) = gM(∇W ψZ,φU) + gM (∇W ωZ,φU), = cos2 θ1gM(∇W Z,U) − gM (TW ωψZ,U) +gM(TW ωQZ,φPU + ψRU) + gM(H∇W ωQZ,ωRU). Now, we obtain sin2 θ1gM(∇W Z,U) = −gM(TW ωψZ,U) + gM(TW ωQZ,φPU + ψRU) + gM(H∇W ωQZ,ωRZ) Next, from equations (2.1)–(2.4), (2.6), (2.7), (2.12), (3.3), (3.7) and Lemma 3.2, we have gM (∇W Z,Y ) = gM(∇W φZ,φY ), = gM(∇W ψZ,φY ) + gM(∇W ωZ,φY ), = cos2 θ1gM (∇W Z,Y ) − gM(H∇W ωψZ,Y ) +gM(H∇W ωZ,CY ) + gM(TW ωZ,BY ). CUBO 24, 1 (2022) Quasi bi-slant submersions in contact geometry 15 Now, we arrive sin2 θ1gM(∇W Z,Y ) = −gM(H∇W ωψZ,Y ) + gM(H∇W ωZ,CY ) + gM(TW ωZ,BY ), which completes the proof. Theorem 3.15. Let f be a proper quasi bi-slant submersion from a cosymplectic manifold (M,φ,ξ,η,gM) onto a Riemannian manifold (N,gN). Then the distribution D2 defines a totally geodesic foliation if and only if gM (TUωψV,W) = gM (TUωQV,φPW + φRW) + gM(H∇UωQV,ωRW), (3.29) and gM(H∇UωψV,Y ) = gM (H∇UωV,CY ) + gM(TUωV,BY ), (3.30) for all U,V ∈ Γ(D2),W ∈ Γ(D ⊕ D1) and Y ∈ Γ(kerf∗)⊥. Proof. For all U,V ∈ Γ(D2),W ∈ Γ(D ⊕ D1) and Y ∈ Γ(kerf∗)⊥, by using equations (2.1)–(2.4), (2.6), (2.7), (2.10), (3.3) and from Lemma 3.2 and Lemma 3.6, we have gM(∇UV,W) = gM (∇UψV,φW) + gM(∇UωV,φW), = cos2 θ2gM(∇UV,W) − gM(TUωψV,W) +gM(TUωQV,φPW + ψRW) + gM(H∇UωQV,ωRW). Now, we get sin2 θ2gM(∇UV,W) = −gM(TUωψV,W) + gM (TUωQV,φPW + ψRW) + gM (H∇UωQV,ωRW). Next, from equations (2.1)–(2.4), (2.6), (2.7), (2.12), (3.2) (3.3), (3.7) and Lemma 3.2, we have gM(∇UV,Y ) = gM (∇UψV,φY ) + gM(∇UωV,φY ), = cos2 θ2gM(∇UV,Y ) − gM(H∇UωψV,Y ) +gM(H∇UωV,CY ) + gM(TUωV,BY ). Now, we obtain sin2 θ1gM(∇UV,Y ) = −gM(H∇UωψV,Y ) + gM(H∇UωV,CY ) + gM (TUωV,BY ), which completes the proof. We recall that a differentiable map f between two Riemannian manifolds is totally geodesic if (∇f∗)(Y,Z) = 0, for all Y,Z ∈ Γ(TM). A totally geodesic map is that it maps every geodesic in the total space into a geodesic in the base space in proportion to arc lengths. 16 R. Prasad, M. A. Akyol, S. Kumar & P. K. Singh CUBO 24, 1 (2022) Theorem 3.16. Let f be a proper quasi bi-slant submersion from a cosymplectic manifold (M,φ,ξ,η,gM) onto a Riemannian manifold (N,gN). Then the map f is totally geodesic if and only if gM(H∇UωψQV + H∇UωψRV − cos2 θ1TUQV − cos2 θ2TURV,W) = gM (V∇UφPV + TUωQV + TUωRV,BW) + gM (TUφPV + H∇UωQV + H∇UωRV,CW), and gM(H∇W ωψQU + H∇W ωψRU − cos2 θ1AW QU − cos2 θ2AW RU,Z) = gM (V∇W φPU + AW ωQU + AW ωRU,BZ) + gM (AW φPU + H∇W ωQU + H∇W ωRU,CZ), for all U,V ∈ Γ(kerf∗) and W,Z ∈ Γ(kerf∗)⊥. Proof. For all U,V ∈ Γ(kerf∗) and W,Z ∈ Γ(kerf∗)⊥, making use of (2.1)–(2.4), (2.6), (2.7), (2.10), (2.11), (3.2), (3.3), (3.7) and from Lemma 3.2 and 3.3, we derive gM(∇UV,W) = gM (∇UφV,φW) = gM (∇UφPV,φW) + gM(∇UφQV,φW) + gM(∇UφRV,φW), = gM (∇UφPV,φW) + gM(∇UψQV,φW) + gM (∇UψRV,φW) +gM(∇UωQV,φW) + gM(∇UωRV,φW), = gM (V∇UφPV + TUωQV + TUωRV,W) +gM(TUφPV + H∇UωQV + H∇UωRV,CW) +gM(cos 2 θ1TUQV + cos2 θ2TURV − H∇UωψQV − H∇UωψRV,W). Next, taking account of (2.1)–(2.4), (2.6), (2.7), (2.10), (2.12), (2.13), (3.2), (3.3), (3.7) and Lemmas 3.2 and 3.3, we have gM(∇W U,Z) = gM(φ∇W U,φZ) = gM(∇W φU,φZ), = gM(∇W φPU,φZ) + gM(∇W φQU,φZ) + gM(∇W φRU,φZ), = gM(∇W φPU,φZ) + gM(∇W ψQU,φZ) + gM(∇W ψRU,φZ) +gM(∇W ωQU,φZ) + gM(∇W ωRU,φZ), = gM(V∇W φPU + AW ωQU + AW ωRU,BZ) +gM(AW φPU + H∇W ωQU + H∇W ωRU,CZ) +gM(cos 2 θ1AW QU + cos2 θ2AW RU − H∇W ωψQU − H∇W ωψRU,Z), which completes the proof. CUBO 24, 1 (2022) Quasi bi-slant submersions in contact geometry 17 4 Examples In this section, we are going to give some non-trivial examples. We will use the notation mentioned in Example 2.1. Example 4.1. Define a map π : R15 → R6 π(x1,x2, . . . ,x7,y1,y2, . . . ,y7,z) = (x2 cosθ1 − y3 sinθ1,y2,x4 sinθ2 − y5 cosθ2,x5,x7,y7), which is a quasi bi-slant submersion such that X1 = ∂ ∂x1 , X2 = ∂ ∂y1 , X3 = ∂ ∂x2 sinθ1 + ∂ ∂y3 cosθ1, X4 = ∂ ∂x3 , X5 = ∂ ∂x4 cosθ2 + ∂ ∂y5 sinθ2, X6 = ∂ ∂y4 , X7 = ∂ ∂x6 , X8 = ∂ ∂y6 , X9 = ξ = ∂ ∂z , (kerπ∗) = (D ⊕ D1 ⊕ D2 ⊕ 〈ξ〉) , where D = 〈 X1 = ∂ ∂x1 ,X2 = ∂ ∂y1 ,X7 = ∂ ∂x6 ,X8 = ∂ ∂y6 〉 , D1 = 〈 X3 = ∂ ∂x2 sinθ1 + ∂ ∂y3 cosθ1,X4 = ∂ ∂x3 〉 , D2 = 〈 X5 = ∂ ∂x4 cosθ2 + ∂ ∂y5 sinθ2,X6 = ∂ ∂y4 〉 , 〈ξ〉 = 〈 X9 = ∂ ∂z 〉 , and (kerπ∗) ⊥ = 〈 ∂ ∂x2 cosθ1 − ∂ ∂y3 sinθ1, ∂ ∂y2 , ∂ ∂x4 sinθ2 − ∂ ∂y5 cosθ2, ∂ ∂x5 , ∂ ∂x7 , ∂ ∂y7 〉 , with bi-slant angles θ1 and θ2. Thus the above example verifies the Lemmas 3.1, 3.2, 3.3 and 3.6. Example 4.2. Define a map π : R13 → R6 π (x1,x2, . . . ,x6,y1,y2, . . . ,y6,z) = ( x1 − x2√ 2 ,y1, √ 3x4 − x5 2 ,y5,x6,y6 ) , which is a quasi bi-slant submersion such that X1 = 1√ 2 ( ∂ ∂x1 + ∂ ∂x2 ) , X2 = ∂ ∂y2 , X3 = ∂ ∂x3 , X4 = ∂ ∂y3 , X5 = 1 2 ( ∂ ∂x4 + √ 3 ∂ ∂x5 ) , X6 = ∂ ∂y4 , 18 R. Prasad, M. A. Akyol, S. Kumar & P. K. Singh CUBO 24, 1 (2022) X7 = ξ = ∂ ∂z , (kerπ∗) = (D ⊕ D1 ⊕ D2 ⊕ 〈ξ〉) , where D = 〈 X3 = ∂ ∂x3 ,X4 = ∂ ∂y3 〉 , D1 = 〈 X1 = 1√ 2 ( ∂ ∂x1 + ∂ ∂x2 ) ,X2 = ∂ ∂y2 〉 , D2 = 〈 X5 = 1 2 ( ∂ ∂x4 + √ 3 ∂ ∂x5 ) ,X6 = ∂ ∂y4 〉 , 〈ξ〉 = 〈 X7 = ∂ ∂z 〉 , and (kerπ∗) ⊥ = 〈 ∂ ∂y1 , 1√ 2 ( ∂ ∂x1 + ∂ ∂x2 ) , 1 2 (√ 3 ∂ ∂x4 − ∂ ∂x5 ) , ∂ ∂y5 , ∂ ∂x6 , ∂ ∂y6 〉 , with bi-slant angles θ1 = π 4 and θ2 = π 3 . Therefore, the above example verifies the Lemmas 3.1, 3.2, 3.3 and 3.6. CUBO 24, 1 (2022) Quasi bi-slant submersions in contact geometry 19 References [1] M. A. Akyol, “Conformal anti-invariant submersions from cosymplectic manifolds”, Hacet. J. Math. Stat., vol. 46, no. 2, pp. 177–192, 2017. [2] M. A. Akyol and B. Şahin, “Conformal slant submersions”, Hacet. J. Math. Stat., vol 48, no.1, pp. 28–44, 2019. [3] M. A. Akyol, “Conformal semi-slant submersions”, Int. J. Geom. Methods Mod. Phys., vol. 14, no. 7, 1750114, 25 pages, 2017. [4] P. Baird and J. C. Wood, Harmonic morphism between Riemannian manifolds, Oxford science publications, Oxford, 2003. [5] D. E. Blair, Riemannian geometry of contact and symplectic manifolds. Progress in Mathe- matics. 203, Birkhäuser Boston, Basel, Berlin, 2002. [6] M. Cengizhan and I. K. Erken, “Anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian submersions”, Filomat, vol. 29, no. 7, pp. 1429–1444, 2015. [7] J.-P. Bourguignon and H. B. Lawson, “Stability and isolation phenomena for Yang-Mills fields”, Comm. Math. Phys., vol. 79, no. 2, pp.189–230, 1981. [8] J.-P. Bourguignon, “A mathematician’s visit to Kaluza-Klein theory”, Rend. Sem. Mat. Univ. Politec. Torino, Special Issue, pp. 143–163, 1989. [9] D. Chinea, “Almost contact metric submersions”, Rend. Circ. Mat. Palermo, vol. 34, no. 1, pp. 89–104, 1985. [10] M. Falcitelli, A. M. Pastore and S. Ianus, Riemannian submersions and related topics, World Scientific, River Edge, NJ, 2004. [11] A. Gray, “Pseudo-Riemannian almost product manifolds and submersions”, J. Math. Mech., vol. 16, pp. 715–738, 1967. [12] Y. Gündüzalp and M. A. Akyol, “Conformal slant submersions from cosymplectic manifolds”, Turkish J. Math., vol. 42, no. 5, pp. 2672–2689, 2018. [13] S. Ianuş and M. Visinescu, “Kaluza-Klein theory with scalar fields and generalized Hopf manifolds”, Classical Quantum Gravity, vol. 4, no. 5, pp. 1317–1325, 1987. [14] S. Ianuş, A. M. Ionescu, R. Mocanu and G. E. Vı̂lcu, “Riemannian submersions from almost contact metric manifolds”, Abh. Math. Semin. Univ. Hambg., vol. 81, no. 1, pp. 101–114, 2011. 20 R. Prasad, M. A. Akyol, S. Kumar & P. K. Singh CUBO 24, 1 (2022) [15] S. Kumar, R. Prasad and P. K. Singh, “Conformal semi-slant submersions from Lorentzian para Sasakian manifolds”, Commun. Korean Math. Soc., vol. 34, no. 2, pp. 637–655, 2019. [16] S. Longwap, F. Massamba and N. E. Homti, “On quasi-hemi-slant Riemannian submersion”, Journal of Advances in Mathematics and Computer Science, vol. 34, no. 1, pp. 1–14, 2019. [17] B. O’Neill, “The fundamental equations of a submersion”, Michigan Math. J., vol. 33, no. 13, pp. 459–469, 1966. [18] K.-S. Park and R. Prasad, “Semi-slant submersions”, Bull. Korean Math. Soc, vol. 50, no. 3, pp. 951–962, 2013. [19] R. Prasad, S. S. Shukla and S. Kumar, “On quasi-bi-slant submersions”, Mediterr. J. Math., vol. 16, no. 6, paper no. 155, 18 pages, 2019. [20] R. Prasad, P. K. Singh and S. Kumar, “On quasi-bi-slant submersions from Sasakian manifolds onto Riemannian manifolds”, Afr. Mat., vol. 32, no. 3-4, pp. 403–417, 2020. [21] B. S.ahin, “Semi-invariant submersions from almost Hermitian manifolds”, Canad. Math. Bull., vol. 56, no. 1, pp. 173–182, 2013. [22] B. S.ahin, “Slant submersions from almost Hermitian manifolds”, Bull. Math. Soc. Sci. Math. Roumanie, vol. 54(102), no. 1, pp. 93–105, 2011. [23] B. Şahin, “Riemannian submersion from almost Hermitian manifolds”, Taiwanese J. Math., vol. 17, no. 2, pp. 629–659, 2013. [24] B. S.ahin, Riemannian submersions, Riemannian maps in Hermitian geometry and their ap- plications, Elsevier, Academic Press, London, 2017. [25] B. S.ahin, “Anti-invariant Riemannian submersions from almost Hermitian manifolds”, Cent. Eur. J. Math., vol. 8, no. 3, pp. 437–447, 2010. [26] C. Sayar, M. A. Akyol and R. Prasad, “Bi-slant submersions in complex geometry”, Int. J. Geom. Methods Mod. Phys., vol. 17, no. 4, 17 pages, 2020. [27] C. Sayar, H. M. Taştan, F. Özdemir and M. M. Tripathi, “Generic submersions from Kaehler manifolds”, Bull. Malays. Math. Sci. Soc., vol. 43, no. 1, pp. 809–831, 2020. [28] H. M. Taştan, B. Şahin and Ş. Yanan, “Hemi-slant submersions”, Mediterr. J. Math., vol 13, no. 4, pp. 2171–2184, 2016. [29] B. Watson, “Almost Hermitian submersions”, J. Differential Geometry, vol. 11, no. 1, pp. 147–165, 1976. Introductions Preliminaries Quasi bi-slant submersions Examples