Int. J. Anal. Appl. (2022), 20:59 Geometry of Warped Product CR and Semi-Slant Submanifolds in Quasi-Para-Sasakian Manifolds Shamsur Rahman1, Abdul Haseeb2,∗, Nargis Jamal3 1Department of Mathematics, Maulana Azad National Urdu University, Polytechnic, Satellite Campus Darbhanga, Bihar 846001, India 2Department of Mathematics, College of Science, Jazan University, Jazan-45142, Kingdom of Saudi Arabia 3Department of Mathematics, College of Science (Girls Campus Mehliya), Jazan University, Jazan-45142, Kingdom of Saudi Arabia ∗Corresponding author: malikhaseeb80@gmail.com, haseeb@jazanu.edu.sa Abstract. In the present paper we study the existence or non-existence of warped product semi-slant submanifolds in quasi-para-Sasakian manifolds and prove that there are no proper warped product semi- slant submanifolds in a quasi-para-Sasakian manifold such that totally geodesic and totally umbilical submanifolds of warped product are proper semi-slant and invariant (or anti-invariant), respectively. 1. Introduction The concept of warped product manifolds was introduced by Bishop and O’Neill for constructing manifolds of non-positive curvature, as one of the most effective generalization of Riemannian product manifold [15]. About two decades ago, Chen extended the work of Bishop and O’Neill and studied the warped product CR-submanifold of Kaehler manifolds [3,4], this study was also extended by many geometers in different settings [2,13,14]. The existence or non-existence of warped product manifolds plays an important role in differential geometry as well as in physics. In [6], Blair introduced the notion of quasi-Sasakian manifolds that unifies Sasakian and cosymplectic manifolds. Tanno [19] also contributed some remarkable results on quasi-Sasakian structure. Recently, quasi-Sasakian structure have been studied in [1, 17, 18]). The geometry of almost paracontact manifold was studied by Received: Sep. 16, 2022. 2010 Mathematics Subject Classification. 53C40, 53C42, 53B26. Key words and phrases. warped product; semi-slant submanifolds; quasi-para-Sasakian manifolds. https://doi.org/10.28924/2291-8639-20-2022-59 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-59 2 Int. J. Anal. Appl. (2022), 20:59 Kaneyuki and Williams in [16] as a natural generalization of natural odd-dimensional analogue to almost para-Hermitian structures. The study of almost paracontact metric manifolds was carried out in one of Zamkovoy’s papers [20]. In [21], Olszak studied normal almost contact metric manifolds of dimension 3. In 2009, Welyczko [10] investigated curvature and torsion of Frenet-Legendre curves in 3-dimensional normal almost paracontact metric manifolds. Recently, 3-dimensional normal almost paracontact metric manifolds were studied in [5,7,8]. 2. Preliminaries Let M̄ be a (2n + 1)-dimensional almost paracontact manifold with structure tensor (f ,ξ,υ,<,>), where f , ξ and υ be a tensor field of type (1, 1), a vector field, and a 1-form, respectively on M̄ satisfying f ξ = 0, f 2 = I −υ ⊗ξ, υ ◦ f = 0, (2.1) υ(ξ) = 1, υ(X) =< X ,ξ >, < f ·, f · >= − <,> +υ ⊗υ, (2.2) where I is the identity on the tangent bundle TM̄ of M̄. We say that M̄ is a paracontact metric manifold if there exists a one-form υ such that < X , fY >= dυ(X ,Y) = 1 2 (Xυ(Y) −Yυ(X) −υ([X ,Y]), for all X ,Y ∈ X(M̄), where X(M̄) denotes the Lie algebra of vector fields on M̄, and < fX ,Y > + < X , fY >= 0 (2.3) for all vector fields X and Y on M̄. Further, an almost paracontact metric manifold is called a quasi-para-Sasakian manifold if (∇̄Xf )Y = υ(Y)FX− < FX ,Y > ξ, (2.4) and ∇̄Xξ = −fFX , fFX = FfX , < FX ,Y >= − < X ,FY >, (2.5) where ∇̄ denotes the Levi-Civita connection with respect to the metric tensor <,> and F is a tensor field of type (1, 1). By applying f to (2.5) and using (2.1), we obtain FX = υ(FX)ξ− f (∇̄Xξ). (2.6) Also by replacing X by ξ in (2.5) it follows that ∇̄ξξ = 0. (2.7) Using (2.4), (2.6) and (2.7) we infer Fξ = υ(Fξ)ξ, (2.8) Int. J. Anal. Appl. (2022), 20:59 3 and (∇̄ξf )X = 0 (2.9) for any X ∈ Γ(TM̄). If M is a contact CR-submanifold of M̄ and the projections on D and D⊥ are denoted by P and Q, respectively; then for all vector field X tangent to M, we infer X = PX + QX + υ(X)ξ. (2.10) Now we put Bλ + Cλ = f λ, (2.11) where Bλ and Cλ are tangential and normal part of f λ on M. Next we define the tensor field of type (1, 1) on M by fX = f PX , (2.12) and the Γ(TM⊥)-valued 2-form ω by ωX = f QX . (2.13) Since D is invariant by f , then it follows from (2.11) and (2.12) that B is Γ(D⊥)-valued and t is Γ(D)-valued, respectively. By using (2.1), (2.10), (2.12) and (2.13), we obtain ωX + tX = fX , (2.14) and t3 + t = 0; C3 + C = 0. (2.15) Then by (2.15) we conclude that t and C are f -structure in sense of Yano [11] on TM and TM⊥, respectively. Now suppose <,> be the induced metric and ξ be tangent to M. Further, we suppose ∇ and ∇⊥ be the induced connections on the tangent bundle TM and the normal bundle T⊥M of M, respectively. Then the Gauss and Weingarten formulas are given respectively by ∇̄XY = σ(X ,Y) + ∇XY, (2.16) ∇̄Xλ = −ΛλX + ∇⊥Xλ (2.17) for all vector fields X ,Y tangent to M and any vector field λ normal to M, where σ and Λλ are the second fundamental form and the shape operator for the immersion of M into M̄. The second fundamental form σ and shape operator Λλ are related by < σ(X ,Y),λ >=< ΛλX ,Y > (2.18) for all vector fields X ,Y tangent to M and vector field λ normal to M. 4 Int. J. Anal. Appl. (2022), 20:59 Furthermore, for any Z ∈ Γ(TM̄), we put FZ = αZ + βZ, (2.19) where αZ and βZ are the tangent part and the normal part of FZ, respectively. From (2.3) we have < tX ,Y > + < X ,tY >= 0. (2.20) In account of (2.6), (2.11), (2.12) and (2.16) we obtain αX = υ(X)υ(FX)ξ− t(∇Xξ) −Bσ(X ,ξ), (2.21) and βX = −ω(∇Xξ) −Cσ(X ,ξ). (2.22) Proposition 2.1. If M is a contact CR-submanifold of a quasi-para-Sasakian manifold M̄, then Γ(TM) is invariant with respect to the action of f if and only if we have ω(∇Xξ) = 0, (2.23) and Cσ(X ,ξ) = 0. (2.24) Proof. From (2.22) it follows that F is a tensor field of type (1, 1) on M if and only if ω(∇Xξ) + Cσ(X ,ξ) = 0. (2.25) Then (2.23) and (2.24) follows from (2.25) (since < ωY,Cλ >= 0 for any Y ∈ Γ(TM)). Corollary 2.1. If M is a contact CR-submanifold of a quasi-para-Sasakian manifold M̄ such that Γ(TM) is invariant with respect to the action of F, then both the distributions D and D⊥ are invariant with respect to the action of F. Proof. Let X ∈ Γ(D), then by using the third relation of (2.5) and (2.8) we obtain < FX ,ξ >= − < X ,Fξ >= υ(Fξ) < X ,ξ >= 0. On the other hand, by using (2.2), the second relation of (2.5) and the invariace of D with respect to the action of f we infer < FX ,Z >=< FfX ′,Z >= − < FX ′, fZ >= 0, where X ′ ∈ Γ(D) and Z ∈ Γ(D⊥). Hence D is invariant by F. In a similar way it follows that D⊥ is invariant by the action of F. The Riemannian connections ∇ and ∇⊥ allow us to define the usual covariant derivatives as (∇Xt)Y = ∇XtY − t∇XY, (2.26) Int. J. Anal. Appl. (2022), 20:59 5 and (∇Xω)Y = ∇⊥XωY −ω∇XY. (2.27) Now, the canonical structures t and ω on a submanifold M are said to be parallel if ∇t = 0 and ∇ω = 0, respectively. On a CR-submanifold of a quasi-para-Sasakian manifold, it follows from (2.5) and (2.16) that ∇Xξ = −fFX , (2.28) and σ(X ,ξ) = 0 (2.29) for each X ∈ TM . Furthermore, from (2.29) we obtain Λωξ = 0; υ(Λω)X = 0. (2.30) Lemma 2.1. For a contact CR-submanifold M of a quasi-para-Sasakian manifold M̄, we infer (∇Xt)Y = ΛωYX + Bσ(X ,Y) + υ(Y)αX− < FX ,Y > ξ, (2.31) (∇Xω)Y = Cσ(X ,Y) −σ(X ,tY) + υ(Y)βX . (2.32) Proof. By using (2.4), (2.16)-(2.19), (2.26) and (2.27), we obtain (αX + βX)υ(Y)− < FX ,Y > ξ = (∇Xt)Y + (∇Xω)Y − ΛωYX −Bσ(X ,Y) −Cσ(X ,Y) + σ(X ,tY) for any X ,Y ∈ Γ(TM). By equating the tangential and the normal parts in above relation, (2.31) and (2.32), respectively follows. The covariant derivatives of B and C are given respectively by (∇XB)λ = ∇XBλ−B(∇⊥Xλ), (2.33) and (∇⊥XC)λ = ∇ ⊥ XCλ−C(∇ ⊥ Xλ) (2.34) for any X ∈ Γ(TM) and ≥∈ Γ(TM⊥). Lemma 2.2. For a contact CR-submanifold M of a quasi-para-Sasakian manifold M̄, we infer (∇XB)λ = ΛCλX − t(ΛλX)− < FX ,λ > ξ, (2.35) and (∇⊥XC)λ = −σ(X ,Bλ) −ω(ΛλX) (2.36) for any X ∈ Γ(TM) and λ ∈ Γ(TM⊥). 6 Int. J. Anal. Appl. (2022), 20:59 Lemma 2.3. For a contact CR-submanifold M of a quasi-para-Sasakian manifold M̄, we infer ΛfXY = ΛfYX , (2.37) and < σ(U,V), fZ >=< ∇UZ, fV > (2.38) for all U ∈ Γ(TM),V ∈ Γ(D) and X ,Y,Z ∈ Γ(D⊥). Proof. By using (2.2), (2.4) and (2.16)-(2.18), we have < ΛfXY,U >=< σ(Y,U), fX >=< ∇̄UY, fX > − < ∇UY, fX > =< ∇UY, fX >= − < f (∇UY),X >= − < −(∇̄Uf )Y + ∇̄UfY,X > + < υ(Y)FU− < FU,Y > ξ,X > − < ∇̄UfY,X > − < −ΛfYU + ∇⊥U fY,X >=< ΛfYU,X >=< ΛfYX ,U > . Since υ(Y) = υ(X) = 0, therefore we find (2.37). Next, by using (2.2), (2.4) and (2.16), we obtain < σ(U,V), fZ >=< ∇̄UV, fZ > − < V,∇̄UfZ > − < V, (∇̄Uf )Z + f (∇̄UZ) > − < V,υ(Z)FU− < FU,Z > ξ > − < V, f (∇̄UZ) >=< fV,∇̄UZ >=< fV,∇UZ > which leads to (2.38). A submanifold M of an almost para contact metric manifold M̄ is said to be invariant if F is identically zero, that is, fX ∈ TM and anti-invariant if t is identically zero, that is, fX ∈ T⊥M, for any X ∈ TM. For each non-zero vector X tangent to M at any point x such that X is not proportional to ξ, we denote by θ(X), the angle between fX and TxM for all x ∈M. Definition 2.1. A submanifold N is said to be slant if the angle θ(X) is constant for all X ∈ TXN−{ξ} and x ∈ N. The angle θ is called a slant angle or Wirtinger angle. Obviously, if θ = 0, then N is invariant; and if θ = π/2, then M is an anti-invariant submanifold. If the slant angle of N is different from 0 and π/2 then it is called proper slant. A characterization of slant submanifolds is given by the following theorem: Theorem 2.1. [9] Let N be slant submanifold of a quasi-para-Sasakian manifold M̄ such that ξ is tangent to N. Then N is slant submanifold if and only if there exists a constant λ ∈ [0, 1] such that t2X = µ(X −υ(X))ξ. (2.39) Furthermore, if θ is the slant angle of N, then µ = cos2θ. Int. J. Anal. Appl. (2022), 20:59 7 Corollary 2.2. Let N be a slant submanifold with slant angle θ of a quasi-para-Sasakian manifold M̄ such that ξ is tangent to N. Then we have < tZ,tW >= cos2θ{− < Z,W > +υ(Z)υ(W)}, (2.40) < ωZ,ωW >= sin2θ{− < Z,W > +υ(Z)υ(W)} (2.41) for any Z,W tangent to N. 3. Warped product semi-slant submanifolds a quasi-para-Sasakian manifold For two Riemannian manifolds (N1,<,>1) and (N2,<,>2) and a positive differentiable function δ on N1, the warped product of N1 and N2 is the Riemannian manifold N1×δN2 = (N1×N2,<,>), where <,>=<,>1 +δ 2 <,>2 . (3.1) More explicitly, if the vector fields X and Y are tangent to N1×δN2 at (x,y), then < X ,Y >=<,>1 (π1 ∗X ,π1 ∗Y) + δ2(x) <,>2 (π2 ∗X ,π2 ∗Y), (3.2) where πi (i = 1, 2) are the canonical projections of N1×δN2 onto N1 and N2, respectively, and ∗ stands for derivative map. If M̃ = N1×δN2 is a warped product manifold, this means that N1 and N2 are totally geodesic and totally umbilical submanifolds of M̃, respectively. For warped product manifolds, we have the following proposition [12,15]: Proposition 3.1. On a warped product manifold M̃ = N1×δN2, we have (1) ∇XY ∈ Γ(TN1) is the lift of ∇XY on N1, (2) ∇UX = ∇XU = X(lnδ)U, (3) ∇UV = ∇ ′ UV− < U,V > ∇lnδ for any X,Y ∈ Γ(TN1) and U,V ∈ Γ(TN2), where ∇ and ∇ ′ denote the Levi-Civita connections on M and N2, respectively. Let us suppose that M̄ be a quasi-para-Sasakian manifold and N1×δN2 be a warped product semi- slant submanifold of a quasi-para-Sasakian manifold M̄. Such submanifolds are always tangent to the structure vector field ξ. If the manifolds Nθ and NT (resp., N⊥) are slant and invariant (resp., anti- invariant) submanifolds of a quasi-para-Sasakian manifold M̄, then their warped product semi-slant submanifolds may be given by one of the following forms: (i) Nθ×δNT , (ii) Nθ×δN⊥, (iii) NT×δNθ, (iv) N⊥×δNθ. Here, we are concerned with cases (i) and (ii). Theorem 3.1. If M̄ is a quasi-para-Sasakian manifold, then there do not exist proper warped product semi-slant submanifolds Nθ×δNT such that Nθ is a proper slant submanifold, NT is an invariant submanifold of M̄ and ξ is tangent to N . 8 Int. J. Anal. Appl. (2022), 20:59 Proof. Let Nθ×δNT be a proper warped product semi-slant submanifold of a quasi-para-Sasakian manifold M̄. For any X ,Y ∈ Γ(TNθ) and U,V ∈ Γ(TNT ), we have (∇̄Xf )U = ∇̄XfU − f (∇̄XU). (3.3) Thus, from (2.4), (2.11), (2.14) and (2.16) we obtain υ(U)FX− < FX ,U > ξ = σ(X ,tU) −Bσ(X ,U) −Cσ(X ,U). This means that Bσ(X ,U) = 0, (3.4) and Cσ(X ,U) −σ(X ,tU) = 0. (3.5) On the other hand, by interchanging roles of U and X in (3.3), we conclude tX log(δ)U = ΛωXU + X log(δ)tU + Bσ(U,X), (3.6) and ∇⊥UωX + σ(U,tX) −Cσ(U,X) = 0. (3.7) From (3.6), we arrive at tX log(δ) < U,U > = < ΛωXU,U > + < Bσ(U,X),U > (3.8) = < σ(U,U),ωX > + < Bσ(U,X),U > = < σ(U,U),ωX > − < σ(X ,U), fU > = < σ(U,U),ωX > . On the other hand, since the ambient space M̄ is a quasi-para-Sasakian manifold, then by using (3.5) and (3.7) we get Ch(Z,ξ) = 0 (3.9) for any Z ∈ Γ(TN). By using (3.5) and (3.7), we get ωX = Cσ(X ,ξ) = 0. Thus we have tX log(δ) < U,U >= 0, this implies that tX log(δ) = 0, that is, the warping function δ is constant on Nθ. � Theorem 3.2. If M̄ is a quasi-para-Sasakian manifold, then there do not exist proper warped product semi-slant submanifolds Nθ×δN⊥ such that Nθ is a proper slant submanifold, N⊥ is an invariant submanifold of M̄ and ξ is tangent to N . Proof. Let Nθ×δN⊥ be a proper warped product semi-slant submanifold of a quasi-para-Sasakian manifold M̄ such that ξ is tangent to N. For any X ,Y ∈ Γ(TNθ) and U,V ∈ Γ(TN⊥), we have (∇̄Xf )U = ∇̄XfU − f (∇̄XU). Int. J. Anal. Appl. (2022), 20:59 9 Using (2.4), (2.14), (2.16), (2.17) and Proposition 3.1, the above equation takes the form υ(U)FX −g(FX ,U)ξ = −ΛωUX + ∇⊥XωU −X(logδ)ωU (3.10) −f σ(X ,U). This means that ΛωUX + Bσ(X ,U) = 0, (3.11) and ∇⊥XωU −X(logδ)ωU −Cσ(X ,U) = 0. (3.12) By interchanging roles of X and U in (3.10), we arrive at υ(U)FX− < FX ,U > ξ = tX log(δ)U + σ(U,tX) − ΛωXU (3.13) +∇⊥UωX −X log(δ)ωU −Bσ(U,X) −Cσ(U,X). Equating the tangential and normal components in (3.13), we find tX log(δ)U = ΛωXU + Bσ(U,X), (3.14) and σ(U,tX) + ∇⊥UωX −X log(δ)ωU −Cσ(U,X) = 0, (3.15) respectively. From (3.14), we find < ΛωXU,tY > + < Bσ(U,X),tY >= 0. (3.16) Since the ambient space M̄ is a quasi-para-Sasakian manifold, ξ is tangent to N and using (2.2), we obtain < Bσ(X ,U),tY > = < f σ(X ,U), fY > = − < σ(X ,U),Y > +υ(Y)υ(σ(X ,U)) = 0. This implies that < Bσ(X ,U),tY >=< σ(U,tY),ωX >= 0. (3.17) Thus we have < σ(U,tY), fX >= 0 (3.18) for any X ,Y ∈ Γ(TNθ). Moreover, making use of (3.11) and (3.18), we get < σ(X ,tY), fU >= 0. (3.19) 10 Int. J. Anal. Appl. (2022), 20:59 By using the Gauss-Weingarten formulas and considering that Nθ is totally geodesic in N, we arrive at < σ(X ,tY), fU > = < ∇̄tYX , fU) = − < f (∇̄tYX),U > (3.20) = − < ∇̄tYfX − (∇̄tYf )X ,U > = − < ∇̄tYtX ,U > − < ∇̄tYωX ,U > + < υ(X)FtY,U > − < FtY,X >< ξ,U > = < ΛωXtY,U > −υ(U) < FtY,X > = < σ(tY,U),ωX > −υ(U) < FtY,X > = υ(U) < tY,FX > . Thus from (3.19) and (3.20), we conclude υ(U) < tY,FX >=< σ(X ,tY, fU >= 0. (3.21) Here, if υ(U) = 0, then by using (2.32) and (3.12), we leads to X log(δ)ωU = υ(∇XU) = − < −fFX ,U >= 0. This is impossible. 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Math. 47 (1986), 41–50. https://doi.org/10.4064/ap-47-1-41-50. https://doi.org/10.1017/s0017089500010156 https://doi.org/10.1007/s00025-009-0364-2 https://doi.org/10.3906/mat-0901-6 https://doi.org/10.2298/fil2101125k https://doi.org/10.2298/fil2101125k https://doi.org/10.1007/s10474-007-6013-x https://doi.org/10.1090/s0002-9947-1969-0251664-4 https://doi.org/10.1090/s0002-9947-1969-0251664-4 https://doi.org/10.1017/S0027763000021565 https://doi.org/10.4067/S0719-06462022000100105 https://doi.org/10.1007/s10455-008-9147-3 https://doi.org/10.1007/s10455-008-9147-3 https://doi.org/10.4064/ap-47-1-41-50 1. Introduction 2. Preliminaries 3. Warped product semi-slant submanifolds a quasi-para-Sasakian manifold References