Int. J. Anal. Appl. (2023), 21:32 A Note on LP-Kenmotsu Manifolds Admitting Conformal Ricci-Yamabe Solitons Mobin Ahmad1,∗, Gazala1, Maha Atif Al-Shabrawi2 1Department of Mathematics and statistics, Integral University, Kursi Road, Lucknow-226026, India 2Department of Mathematical Sciences, Umm Ul Qura University, Makkah, Saudi Arabia ∗Corresponding author: mobinahmad68@gmail.com Abstract. In the current note, we study Lorentzian para-Kenmotsu (in brief, LP-Kenmotsu) manifolds admitting conformal Ricci-Yamabe solitons (CRYS) and gradient conformal Ricci-Yamabe soliton (gra- dient CRYS). At last by constructing a 5-dimensional non-trivial example we illustrate our result. 1. Introduction As a generalization of the classical Ricci flow [8], the concept of conformal Ricci flow was introduced by Fischer [5], which is defined on an n-dimensional Riemannian manifold M by the equations ∂g ∂t = −2(S + g n ) −pg, r(g) = −1, where p defines a time dependent non-dynamical scalar field (also called the conformal pressure), g is the Riemannian metric; r and S represent the scalar curvature and the Ricci tensor of M, respectively. The term −pg plays a role of constraint force to maintain r in the above equation. In [1], the authors Basu and Bhattacharyya proposed the concept of conformal Ricci soliton on M and is defined by £Kg + 2S + (2Λ − (p + 2 n ))g = 0, where £K represents the Lie derivative operator along the smooth vector field K on M and Λ ∈ R (the set of real numbers). Received: Feb. 3, 2023. 2010 Mathematics Subject Classification. 53C20, 53C21, 53C25, 53E20. Key words and phrases. Lorentzian para-Kenmotsu manifolds; conformal Ricci-Yamabe solitons; Einstein manifolds; ν-Einstein manifolds. https://doi.org/10.28924/2291-8639-21-2023-32 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-32 2 Int. J. Anal. Appl. (2023), 21:32 Very recently, a scalar combination of Ricci and Yamabe flows was proposed by the authors Güler and Crasmareanu [7], this advanced class of geometric flows called Ricci-Yamabe (RY) flow of type (σ,ρ) and is defined by ∂ ∂t g(t) + 2σS(g(t)) + ρr(t)g(t) = 0, g(0) = g0 for some scalars σ and ρ. A solution to the RY flow is called Ricci-Yamabe soliton (RYS) if it depends only on one parameter group of diffeomorphism and scaling. A Riemannian (or semi-Riemannian) manifold M is said to have a RYS if [9,10] £Kg + 2σS + (2Λ −ρr)g = 0. (1.1) A Riemannian (or semi-Riemannian) manifold M is said to have a conformal Ricci-Yamabe soliton (CRYS) if [20] £Kg + 2σS + (2Λ −ρr − (p + 2 n ))g = 0, (1.2) where σ,ρ, Λ ∈R. If K is the gradient of a smooth function v on M, then (1.2) is called the gradient conformal Ricci-Yamabe soliton (gradient CRYS) and hence (1.2) turns to ∇2v + σS + (Λ − ρr 2 − 1 2 (p + 2 n ))g = 0, (1.3) where ∇2v is the Hessian of v and is defined by Hessv = ∇∇v. A CRYS is said to be shrinking, steady or expanding if Λ < 0, = 0 or > 0, respectively. A CRYS is said to be a • Conformal Ricci soliton if σ = 1,ρ = 0, • Conformal Yamabe soliton if σ = 0,ρ = 1, • Conformal Einstein soliton if σ = 1,ρ = −1. As a continuation of this study, we tried to study CRYS and gradient CRYS in the frame-work of LP-Kenmotsu manifolds of dimension n. We recommend the papers [2–4,6,13–17] and the references therein for more details about the related studies. 2. Preliminaries An n-dimensional differentiable manifold M with structure (ϕ,ζ,ν,g) is said to be a Lorentzian almost paracontact metric manifold, if it admits a (1, 1)-tensor field ϕ, a contravariant vector field ζ, a 1-form ν and a Lorentzian metric g satisfying ν(ζ) + 1 = 0, (2.1) ϕ2E = E + ν(E)ζ, (2.2) ϕζ = 0, ν(ϕE) = 0, g(ϕE,ϕF ) = g(E,F ) + ν(E)ν(F ), Int. J. Anal. Appl. (2023), 21:32 3 g(E,ζ) = ν(E), (2.3) ϕ(E,F ) = ϕ(F,E) = g(E,ϕF ) for any vector fields E,F ∈ χ(M), where χ(M) is the Lie algebra of vector fields on M. If ζ is a killing vector field, the (para) contact structure is called a K-(para) contact. In such a case, we have ∇Eζ = ϕE. Recently, the authors Haseeb and Prasad defined and studied the following notion: Definition 2.1. A Lorentzian almost paracontact manifold M is called Lorentzian para-Kenmostu manifold if [11] (∇Eϕ)F = −g(ϕE,F )ζ −ν(F )ϕE for any E,F on M. In an LP-Kenmostu manifold, we have ∇Eζ = −E −ν(E)ζ, (2.4) (∇Eν)F = −g(E,F ) −ν(E)ν(F ), (2.5) where ∇ denotes the Levi-Civita connection respecting to the Lorentzian metric g. Furthermore, in an LP-Kenmotsu manifold, the following relations hold [11]: g(R(E,F )G,ζ) = ν(R(E,F )G) = g(F,G)ν(E) −g(E,G)ν(F ), R(ζ,E)F = −R(E,ζ)F = g(E,F )ζ −ν(F )E, R(E,F )ζ = ν(F )E −ν(E)F, R(ζ,E)ζ = E + ν(E)ζ, (2.6) S(E,ζ) = (n− 1)ν(E), S(ζ,ζ) = −(n− 1), (2.7) Qζ = (n− 1)ζ, for any E,F,G ∈ χ(M), where R,S and Q represent the curvature tensor, the Ricci tensor and the Q Ricci operator, respectively. Definition 2.2. [19] An LP-Kenmotsu manifold M is said to be ν-Einstein manifold if its S( 6= 0) is of the form S(E,F ) = ag(E,F ) + bν(E)ν(F ), where a and b are smooth functions on M. In particular, if b = 0, then M is termed as an Einstein manifold. 4 Int. J. Anal. Appl. (2023), 21:32 Remark 2.1. [12] In an LP-Kenmotsu manifold of n-dimension, S is of the form S(E,F ) = ( r n− 1 − 1)g(E,F ) + ( r n− 1 −n)ν(E)ν(F ), (2.8) where r is the scalar curvature of the manifold. Lemma 2.1. In an n-dimensional LP-Kenmotsu manifold, we have ζ(r) = 2(r −n(n− 1)), (2.9) (∇EQ)ζ = QE − (n− 1)E, (2.10) (∇ζQ)E = 2QE − 2(n− 1)E, (2.11) for any E on M. Proof. Equation (2.8) yields QE = ( r n− 1 − 1)E + ( r n− 1 −n)ν(E)ζ. (2.12) Taking the covariant derivative of (2.12) with respect to F and making use of (2.4) and (2.5), we lead to (∇FQ)E = F (r) n− 1 (E + ν(E)ζ) − ( r n− 1 −n)(g(E,F )ζ + ν(E)F + 2ν(E)ν(F )ζ). By contracting F in the foregoing equation and using trace {F → (∇FQ)E} = 12E(r), we find n− 3 2(n− 1) E(r) = { ζ(r) n− 1 − (r −n(n− 1)) } ν(E), which by replacing E by ζ and using (2.1) gives (2.9). We refer the readers to see [13] for the proof of (2.10) and (2.11). � Remark 2.2. From the equation (2.9), it is noticed that if an n-dimensional LP-Kenmotsu manifold possesses the constant scalar curvature, then r = n(n − 1) and hence (2.8) reduces to S(E,F ) = (n− 1)g(E,F ). Thus the manifold under consideration is an Einstein manifold. 3. CRYS on LP-Kenmotsu Manifolds Let the metric of an n-dimensional LP-Kenmotsu manifold be a conformal Ricci-Yamabe soliton, thus (1.2) holds. By differentiating (1.2) covariantly with resprct to G, we have (∇G£Kg)(E,F ) = −2σ(∇GS)(E,F ) + ρ(Gr)g(E,F ). (3.1) Since ∇g = 0, then the following formula [18] (£K∇Eg −∇E£Kg −∇[K,E]g)(F,G) = −g((£K∇)(E,F ),G) −g((£K∇)(E,G),F ) turns to (∇E£Kg)(F,G) = g((£K∇)(E,F ),G) + g((£K∇)(E,G),F ). Int. J. Anal. Appl. (2023), 21:32 5 Since the operator £K∇ is symmetric, therefore we have 2g((£K∇)(E,F ),G) = (∇E£Kg)(F,G) + (∇F£Kg)(E,G) − (∇G£Kg)(E,F ), which by using (3.1) takes the form 2g((£K∇)(E,F ),G) = −2σ[(∇ES)(F,G) + (∇FS)(G,E) − (∇GS)(E,F )] +ρ[(Er)g(F,G) + (Fr)g(G,E) − (Gr)g(E,F )]. (3.2) Putting F = ζ in (3.2) and using (2.3), we find 2g((£K∇)(E,ζ),G) = −2σ[(∇ES)(ζ,G) + (∇ζS)(G,E) − (∇GS)(E,ζ)] +ρ[(Er)ν(G) + 2(r −n(n− 1))g(E,G) − (Gr)ν(E)]. (3.3) By virtue of (2.10) and (2.11), (3.3) leads to 2g((£K∇)(E,ζ),G) = −4σ[S(E,G) − (n− 1)g(E,G)] +ρ[(Er)ν(G) + 2(r −n(n− 1))g(E,G) − (Gr)ν(E)]. By eliminating G from the foregoing equation, we have 2(£K∇)(F,ζ) = ρg(Dr,F )ζ −ρ(Dr)ν(F ) − 4σQF (3.4) +[4σ(n− 1) + 2ρ(r −n(n− 1))]F. If we take r as constant, then from (2.9) it follows that r = n(n− 1), and hence (3.4) reduces to (£K∇)(F,ζ) = −2σQF + 2σ(n− 1)F. (3.5) Taking covariant derivative of (3.5) with respect to E, we have (∇E£K∇)(F,ζ) = (£K∇)(F,E) − 2σν(E)[QF − (n− 1)F ] (3.6) − 2σ(∇EQ)F. Again from [18], we have (£KR)(E,F )G = (∇E£K∇)(F,G) − (∇F£K∇)(E,G), which by putting G = ζ and using (3.6) takes the form (£KR)(E,F )ζ = 2σν(F )(QE − (n− 1)E) − 2σν(E)(QF − (n− 1)F ) (3.7) −2σ((∇EQ)F − (∇FQ)E). Putting F = ζ in (3.7) then using (2.1), (2.2), (2.10) and (2.11), we arrive at (£KR)(E,ζ)ζ = 0. (3.8) The Lie derivative of (2.6) along K leads to (£KR)(E,ζ)ζ −g(E,£Kζ)ζ + 2ν(£Kζ)E = −(£Kν)(E)ζ. (3.9) 6 Int. J. Anal. Appl. (2023), 21:32 From (3.8) and (3.9), we have (£Kν)(E)ζ = −2ν(£Kζ)E + g(E,£Kζ)ζ. (3.10) Taking the Lie derivative of g(E,ζ) = ν(E), we find (£Kν)(E) = g(E,£Kζ) + (£Kg)(E,ζ). (3.11) By putting F = ζ in (1.2) and using (2.7), we have (£Kg)(E,ζ) = −{2σ(n− 1) + 2Λ −ρn(n− 1) − (p + 2 n )}ν(E), (3.12) where r = n(n− 1) being used. Taking the Lie derivative of g(ζ,ζ) = −1 along K we lead to (£Kg)(ζ,ζ) = −2ν(£Kζ). (3.13) From (3.12) and (3.13), we find ν(£Kζ) = −{σ(n− 1) + Λ − ρn(n− 1) 2 − 1 2 (p + 2 n )}. (3.14) Now combining the equations (3.10), (3.11), (3.12) and (3.14), we find Λ = ρn(n− 1) 2 −σ(n− 1) + 1 2 (p + 2 n ). (3.15) Thus we have Theorem 3.1. Let (M,g) be an n-dimensional LP-Kenmotsu manifold admitting CRYS with constant scalar curvature tensor, then Λ = ρn(n−1) 2 −σ(n− 1) + 1 2 (p + 2 n ). Corollary 3.1. Let the metric of n-dimensional LP-Kenmotsu manifold is CRYS. Then we have Values of σ,ρ Soliton type Soliton constant CRYS to be expanding, shrinking or steady σ = 1, ρ = 0 conformal Ricci soliton Λ = 1 2 (p+ 2 n )−(n− 1) CRYS is shrinking, steady and expanding if p > 2(n2−n−1) n , p = 2(n 2−n−1) n and p < 2(n 2−n−1) n , resp. σ = 0, ρ = 1 conformal Yam- abe soliton Λ = 1 2 (p + 2 n ) + n(n−1) 2 CRYS is shrinking, steady and expanding if p < −(n3−n2 +2) n , p = −(n 3−n2 +2) n and p > −(n 3−n2 +2) n , resp. σ = 1, ρ = −1 conformal Ein- stein soliton Λ = 1 2 (p + 2 n ) − (n−1)(n+2) 2 CRYS is shrinking, steady and expand- ing if p < (n+1)(n 2−2) n , p = (n+1)(n2−2) n and p > (n+1)(n2−2) n , resp. Int. J. Anal. Appl. (2023), 21:32 7 4. Gradient CRYS on LP-Kenmotsu Manifolds Let M be an n-dimensional LP-Kenmotsu manifold with g as a gradient CRYS. Then equation (1.3) can be written as ∇EDv + σQE + (Λ − ρr 2 − 1 2 (p + 2 n ))E = 0, (4.1) for all vector fields E on M, where D denotes the gradient operator of g. Taking the covariant derivative of (4.1) with respect to F , we have ∇F∇EDv = −σ{(∇FQ)E + Q(∇FE)} + ρ F (r) 2 E (4.2) −(Λ − ρr 2 − 1 2 (p + 2 n ))∇FE. Interchanging E and F in (4.2), we lead to ∇E∇FDv = −σ{(∇EQ)F + Q(∇EF )} + ρ E(r) 2 F (4.3) −(Λ − ρr 2 − 1 2 (p + 2 n ))∇EF. By making use of (4.1)-(4.3), we find R(E,F )Dv = σ{(∇FQ)E − (∇EQ)F} + ρ 2 {E(r)F −F (r)E}. (4.4) Now from (2.8), we find QE = ( r n− 1 − 1)E + ( r n− 1 −n)ν(E)ζ, which on taking covariant derivative with repect to F leads to (∇FQ)E = F (r) n− 1 (E + ν(E)ζ) − ( r n− 1 −n)(g(E,F )ζ (4.5) +2ν(E)ν(F )ζ + ν(E)F ). By using (4.5) in (4.4), we have R(E,F )Dv = (n− 1)ρ− 2σ 2(n− 1) {E(r)F −F (r)E} + σ n− 1 {F (r)ν(E)ζ −E(r)ν(F )ζ} −σ( r n− 1 −n)(ν(E)F −ν(F )E). (4.6) Contracting forgoing equation along E gives S(F,Dv) = − {(n− 1)2ρ− 2σ(n− 2) 2(n− 1) } F (r) (4.7) + σ(n− 3)(r −n(n− 1)) n− 1 ν(F ). From the equation (2.8), we have S(F,Dv) = ( r n− 1 − 1)F (v) + ( r n− 1 −n)ν(F )ζ(v). (4.8) 8 Int. J. Anal. Appl. (2023), 21:32 Now by equating (4.7) and (4.8), then putting F = ζ and using (2.1), (2.9), we find ζ(v) = r −n(n− 1) n− 1 {σ − (n− 1)ρ}. (4.9) Taking the inner product of (4.6) with ζ, we get F (v)ν(E) −E(v)ν(F ) = ρ 2 {E(r)ν(F ) −F (r)ν(E)}, which by replacing E by ζ then using (2.9) and (4.9), we infer F (v) = − σ(r −n(n− 1)) n− 1 ν(F ) − ρ 2 F (r). (4.10) If we take r as constant, then from Remark 2.5, we get r = n(n−1). Thus (4.10) leads to F (v) = 0. This implies that v is constant. Thus the soliton under the consideration is trivial. Hence we state: Theorem 4.1. If the metric of an n-dimensional LP-Kenmotsu manifold of constant scalar curvature tensor admitting a special type of vector field is gradient CRYS, then the soliton is trivial. For v constant, (1.3) turns to σQE = −(Λ − ρr 2 − 1 2 (p + 2 n ))E, which leads to S(E,F ) = − 1 σ (Λ − ρn(n− 1) 2 − 1 2 (p + 2 n ))g(E,F ), σ 6= 0. (4.11) By putting E = F = ζ in (4.11) and using (2.7), we obtain Λ = ρn(n− 1) 2 −σ(n− 1) + 1 2 (p + 2 n ). (4.12) Corollary 4.1. If an n-dimensional LP-Kenmotsu manifold admits a gradient CRYS with the constant scalar curvature, then the manifold under the consideration is an Einstein manifold and Λ = ρn(n−1) 2 − σ(n− 1) + 1 2 (p + 2 n ). 5. Example We consider the 5-dimensional manifold M5 = { (x1,x2,x3,x4,x5) ∈R5 : x5 > 0 } , where (x1,x2,x3,x4,x5) are the standard coordinates in R5. Let %1, %2, %3, %4 and %5 be the vector fields on M5 given by %1 = e x5 ∂ ∂x1 , %2 = e x5 ∂ ∂x2 , %3 = e x5 ∂ ∂x3 , %4 = e x5 ∂ ∂x4 , %5 = ∂ ∂x5 = ζ, which are linearly independent at each point of M5. Let g be the Lorentzian metric defined by g(%i,%i ) = 1, for 1 ≤ i ≤ 4 and g(%5,%5) = −1, g(%i,%j) = 0, for i 6= j, 1 ≤ i, j ≤ 5. Int. J. Anal. Appl. (2023), 21:32 9 Let ν be the 1-form defined by ν(E) = g(E,%5) = g(E,ζ) for all E ∈ χ(M5), and let ϕ be the (1, 1)-tensor field defined by ϕ%1 = −%2, ϕ%2 = −%1, ϕ%3 = −%4, ϕ%4 = −%3, ϕ%5 = 0. By applying linearity of ϕ and g, we have ν(ζ) = g(ζ,ζ) = −1, ϕ2E = E + ν(E)ζ and g(ϕE,ϕF ) = g(E,F ) + ν(E)ν(F ) for all E,F ∈ χ(M5). Thus for %5 = ζ, the structure (ϕ,ζ,ν,g) defines a Lorentzian almost paracon- tact metric structure on M5. Then we have [%i,%j] = −%i, for 1 ≤ i ≤ 4, j = 5, [%i,%j] = 0, otherwise. By using Koszul’s formula, we can easily find we obtain ∇%i%j =   −%5, 1 ≤ i = j ≤ 4, −%i, 1 ≤ i ≤ 4, j = 5, 0, otherwise. Also one can easily verify that ∇Eζ = −E −η(E)ζ and (∇Eϕ)F = −g(ϕE,F )ζ −ν(F )ϕE. Therefore, the manifold is an LP-Kenmotsu manifold. From the above results, we can easily obtain the non-vanishing components of R as follows: R(%1,%2)%1 = −%2, R(%1,%2)%2 = %1, R(%1,%3)%1 = −%3, R(%1,%3)%3 = %1, R(%1,%4)%1 = −v4, R(%1,%4)%4 = %1, R(%1,%5)%1 = −%5, R(%1,%5)%5 = −%1, R(%2,%3)%2 = −%3, R(%2,%3)%3 = %2, R(%2,%4)%2 = −%4, R(%2,%4)%4 = %2, R(%2,%5)%2 = −%5, R(%2,%5)%5 = −%2, R(%3,%4)%3 = −%4, R(%3,%4)%4 = %3, R(%3,%5)%3 = −%5, R(%3,%5)%5 = −%3, R(%4,%5)%4 = −%5, R(%4,%5)%5 = −%4. Also, we calculate the Ricci tensors as follows: S(%1,%1) = S(%2,%2) = S(%3,%3) = S(%4,%4) = 4, S(%5,%5) = −4. Therefore, we have r = S(%1,%1) + S(%2,%2) + S(%3,%3) + S(%4,%4) −S(%5,%5) = 20. Now by taking Dv = (%1v)%1 + (%2v)%2 + (%3v)%3 + (%4v)%4 + (%5v)%5, we have ∇%1Dv = (%1(%1v) − (%5v))%1 + (%1(%2v))%2 + (%1(%3v))%3 + (%1(%4v))%4 +(%1(%5v) − (%1v))%5, 10 Int. J. Anal. Appl. (2023), 21:32 ∇%2Dv = (%2(%1v))%1 + (%2(%2v) − (%5v))%2 + (%2(%3v))%3 + (%2(%4v))%4 +(%2(%5v) − (%2v))%5, ∇%3Dv = (%3(%1v))%1 + (%3(%2v))%2 + (%3(%3v) − (%5v))%3 + (%3(%4v))%4 +(%3(%5v) − (%3v))%5, ∇%4Dv = (%4(%1v))%1 + (%4(%2v))%2 + (%4(%3v))%3 + (%4(%4v) − (%5v))%4 +(%4(%5v) − (%4v))%5, ∇%5Dv = (%5(%1v))%1 + (%5(%2v))%2 + (%5(%3v))%3 + (%5(%4v))%4 + (%5(%5v))%5. Thus by virtue of (4.1), we obtain  %1(%1v) −%5v = −(Λ + 4σ − 10ρ− 12 (p + 2 5 )), %2(%2v) −%5v = −(Λ + 4σ − 10ρ− 12 (p + 2 5 )), %3(%3v) −%5v = −(Λ + 4σ − 10ρ− 12 (p + 2 5 )), %4(%4v) −%5v = −(Λ + 4σ − 10ρ− 12 (p + 2 5 )), %5(%5v) = −(Λ + 4σ − 10ρ− 12 (p + 2 5 )), %1(%2v) = %1(%3v) = %1(%4v) = 0, %2(%1v) = %2(%3v) = %2(%4v) = 0, %3(%1v) = %3(%2v) = %3(%4v) = 0, %4(%1v) = %4(%2v) = %4(%3v) = 0, %1(%5v) − (%1v) = %2(%5v) − (%2v) = 0, %3(%5v) − (%3v) = %4(%5v) − (%4v) = 0. (5.1) Thus the equations in (5.1) are respectively amounting to e2x5 ∂2v ∂x21 − ∂v ∂x5 = −(Λ + 4σ − 10ρ− 1 2 (p + 2 5 )), e2x5 ∂2v ∂x22 − ∂v ∂x5 = −(Λ + 4σ − 10ρ− 1 2 (p + 2 5 )), e2x5 ∂2v ∂x23 − ∂v ∂x5 = −(Λ + 4σ − 10ρ− 1 2 (p + 2 5 )), e2x5 ∂2v ∂x24 − ∂v ∂x5 = −(Λ + 4σ − 10ρ− 1 2 (p + 2 5 )), ∂2v ∂x25 = −(Λ + 4σ − 10ρ− 1 2 (p + 2 5 )), ∂2v ∂x1∂x2 = ∂2v ∂x1∂x3 = ∂2v ∂x1∂x4 = ∂2v ∂x2∂x3 = ∂2v ∂x2∂x4 = ∂2v ∂x3∂x4 = 0, Int. J. Anal. Appl. (2023), 21:32 11 ex5 ∂2v ∂x5∂x1 + ∂v ∂x1 = ex5 ∂2v ∂x5∂x2 + ∂v ∂x2 = ex5 ∂2v ∂x5∂x3 + ∂v ∂x3 = ex5 ∂2v ∂x5∂x4 + ∂v ∂x4 = 0. From the above equations it is observed that v is constant for Λ = −4σ + 10ρ + 1 2 (p + 2 5 ). Hence equation (4.1) is satisfied. Thus, g is a gradient RYS with the soliton vector field K = Dv, where v is constant and Λ = −4σ + 10ρ + 1 2 (p + 2 5 ). Thus, Theorem 4.1 is verified. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. 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