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Riemannian Maps Whose Base Manifolds Admit a Ricci Soliton and Their Harmonicity 3 Whose Base Manifold Admits a Ricci Soliton

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Riemannian Maps Whose Base Manifolds Admit a Ricci Soliton and Their Harmonicity 3 Whose Base Manifold Admits a Ricci Soliton Riemannian Maps Whose Base Manifolds Admit A Ricci Soliton And Their Harmonicity Akhilesh Yadav and Kiran Meena Abstract: In this paper, we study Riemannian maps whose base manifolds admit a Ricci soliton and give a non-trivial example of such Riemannian maps. First, we find Riemannian curvature tensor of base manifolds for Riemannian map F . Further, we obtain Ricci tensors and calculate scalar curvature of base manifolds. Moreover, we obtain necessary conditions for rangeF∗ to be Ricci soliton, almost Ricci soliton and ⊥ Einstein. We also obtain necessary conditions for (rangeF∗) to be Ricci soliton ⊥ and Einstein. Also, we calculate scalar curvatures of rangeF∗ and (rangeF∗) by using Ricci soliton. Finally, we study harmonicity and biharmonicity of such Riemannian maps and obtain a necessary and sufficient condition for Riemannian map between Riemannian manifolds whose base manifold admits a Ricci soliton to be harmonic. We also obtain necessary and sufficient conditions for Riemannian map from Riemannian manifold to space form which admits Ricci soliton to be harmonic and biharmonic. Finally, some applications are presented for further studies on base manifolds of Riemannian maps. M. S. C. 2010: 53B20; 53C25; 53C43. Keywords: Riemannian manifold, Ricci soliton, Einstein manifold, Riemannian map, Harmonic map, Biharmonic map. 1 Introduction In 1992, Fischer introduced Riemannian map between Riemannian manifolds in [4] as a generalization of the notion of an isometric immersion and Riemannian submersion. The geometry of Riemannian submersions have been discussed in [3]. We note that a remarkable property of Riemannian maps is that a Riemannian 2 map satisfies the generalized eikonal equation F∗ = rankF , which is a bridge between geometric optics and physical optics [4].k Thek eikonal equation of geomet- rical optics solved by using Cauchy’s method of characteristics. In [4] Fischer also proposed an approach to build a quantum model and he pointed out the succes arXiv:2107.01049v1 [math.DG] 1 Jul 2021 of such a programme of building a quantum model of nature using Riemannian maps would provide an interesting relationship between Riemannian maps, har- monic maps and Lagrangian field theory on the mathematical side, and Maxwell’s equation, Shr¨odinger’s equation and their proposed generalization on the physical side. In [11], B. S¸ahin develop certain geometric structures along a Riemannian map to investigate the geometry of such maps. He constructed Gauss-Weingarten formulas and obtained Gauss, Codazzi and Ricci equations for Riemannian maps by using second fundamental form and linear connection. On the other hand, in 1988, the notion of Ricci soliton was introduced by Hamil- ton in [5]. A Ricci soliton is a natural generalization of an Einstein metric. A Rie- mannian manifold (N,gN ) is called a Ricci soliton if there exists a smooth vector 2 Akhilesh Yadav and Kiran Meena field ξ (called potential vector field) on N such that 1 (1.1) (L g )(X , Y )+ Ric(X , Y )+ λg (X , Y )=0, 2 ξ N 1 1 1 1 N 1 1 where Lξ is the Lie derivative of the metric tensor of gN with respect to ξ, Ric is the Ricci tensor of (N,gN ), λ is a constant and X1, Y1 are arbitrary vector fields on N. We shall denote a Ricci soliton by (N,gN ,ξ,λ). The Ricci soliton (N,gN ,ξ,λ) is said to be shrinking, steady or expanding accordingly as λ< 0, λ =0 or λ > 0, respectively. It is obvious that a trivial Ricci soliton is an Einstein manifold with ξ zero or killing, that is, Lie derivative of metric tensor gN with respect to ξ is vanish. Hamilton showed that the self-similar solutions of Ricci flow are Ricci solitons. The Ricci soliton (N,gN ,ξ,λ) is said to be a gradient Ricci soliton, if the potential vector field ξ is the gradient of some smooth function f on N, which is denoted by (N,gN ,f,λ). Moreover, in [2], a non killing tangent vector field ξ on a Riemannian manifold (N,gN ) is called conformal, if it satisfies LξgN =2fgN , where Lξ is the Lie derivative of the metric tensor of gN with respect to ξ and f is called the potential function of ξ. In [9], Pigola introduced a natural extension of the concept of gradient Ricci soliton by taking λ as a variable function instead of constant and then the Ricci soliton (M,g1,ξ,λ) is called an almost Ricci soliton. Hence, the almost Ricci soliton becomes a Ricci soliton, if the function λ is a constant. The almost Ricci soliton is called shrinking, steady or expanding accordingly as λ < 0, λ = 0 or λ > 0, respectively. In [8], Perelman used the Ricci soliton in order to solve the Poincar´e conjecture and then the geometry of Ricci solitons has been the focus of attention of many mathematicians. Moreover, Ricci solitons have been studied on contact, paracontact and Sasakian manifolds. In [7], Meri¸cand Kili¸cstudy Riemannian sub- mersions whose total manifolds admit a Ricci soliton. In [18], Siddiqi and Akyol study η-Ricci-Yamabe solitons on Riemannian submersions from Riemannian mani- folds. In [6], Meri¸cstudy the Riemannian submersions admitting an almost Yamabe soliton. A harmonic map between Riemannian manifolds has played an important role in linking the geometry to global analysis on Riemannian manifolds as well as its importance in physics is also well established. Therefore it is in interesting question to find harmonic maps to Ricci soliton. In [19], we study Riemannian maps whose total manifolds admit a Ricci soliton. In this paper, we study Riemannian maps whose base manifolds admit a Ricci soli- ton. In section 2, we give some basic informations about Riemannian map which is needed for this paper. In section 3, a Riemannian map F between Riemannian manifold is considered and we find Riemannian curvature tensor of base manifolds for Riemannian map. Moreover, we calculate Ricci tensors and scalar curvature of base manifolds for Riemannian map. In section 4, we obtain necessary conditions for rangeF∗ to be Ricci soliton, almost Ricci soliton and Einstein. We also obtain ⊥ necessary conditions for (rangeF∗) to be Ricci soliton and Einstein. Moreover, we ⊥ calculate scalar curvature of rangeF∗ and (rangeF∗) for totally geodesic Rieman- nian map F by using Ricci soliton. The section 5 is devoted to harmonicity and biharmonicity. We obtain a necessary and sufficient condition for Riemannian map between Riemannian manifolds whose base manifold admits a Ricci soliton to be harmonic. We also obtain necessary and sufficient conditions for Riemannian map from Riemannian manifold to space form which admits Ricci soliton to be harmonic and biharmonic. In section 6, we give a non-trivial example of Riemannian map Riemannian maps whose base manifolds admit a Ricci soliton and their harmonicity 3 whose base manifold admits a Ricci soliton. In the last section, we propose some applications for further studies on base manifolds of Riemannian maps. 2 Preliminaries In this section, we recall the notion of Riemannian maps between Riemannian man- ifolds and give a brief review of basic facts of Riemannian maps. m n Let F : (M ,gM ) (N ,gN ) be a smooth map between Riemannian manifolds such that 0 < rankF →min m,n , where dim(M)= m and dim(N)= n. Then we ≤ { } denote the kernel space of F∗ by νp = kerF∗p at p M and consider the orthogonal ⊥ ∈ complementary space p = (kerF∗p) to kerF∗p in TpM. Then the tangent space H ⊥ TpM of M at p has the decomposition TpM = (kerF∗p) (kerF∗p) = νp p. ⊕ ⊕ H We denote the range of F∗ by rangeF∗ at p M and consider the orthogonal ⊥ ∈ complementary space (rangeF∗p) to rangeF∗p in the tangent space TF (p)N of ⊥ N at F (p) N. Since rankF min m,n , we have (rangeF∗) = 0 . Thus ∈ ≤ { } 6 { } the tangent space TF (p)N of N at F (p) N has the decomposition TF (p)N = ⊥ ∈ (rangeF∗p) (rangeF∗p) . ⊕ m n Now, a smooth map F : (M ,gM ) (N ,gN ) is called Riemannian map →h ⊥ at p M if the horizontal restriction F∗p : (kerF∗p) (rangeF∗p) is a lin- ∈ →⊥ ear isometry between the inner product spaces ((kerF∗p) ,g ⊥ ) and M(p)|(kerF∗p) (rangeF∗p,gN(p1) (rangeF∗p)), where F (p) = p1. In other words, F∗ satisfies the equation | (2.1) gN (F∗X, F∗Y )= gM (X, Y ), ⊥ for all X, Y vector field tangent to Γ(kerF∗p) . It follows that isometric immersions and Riemannian submersions are particular ⊥ Riemannian maps with kerF∗ = 0 and (rangeF∗) = 0 . { } { } Let F : (M,gM ) (N,gN ) is a smooth map between Riemannian manifolds. → −1 Then the differential F∗ of F can be viewed as a section of bundle Hom(TM,F TN) −1 −1 M, where F TN is the pullback bundle whose fibers at p M is (F TN)p = → −1 ∈ TF (p)N, p M. The bundle Hom(TM,F TN) has a connection induced from ∈ N ∇ the Levi-Civita connection M and the pullback connection F . Then the second fundamental form of F is given∇ by ∇ N F M (2.2) ( F∗)(X, Y )= F∗Y F∗( Y ), ∇ ∇X − ∇X N F N for all X, Y Γ(TM), where F∗Y F = F∗Y . It is known that the second ∈ ∇X ◦ ∇F∗X fundamental form is symmetric. In [12] B. S¸ahin proved that ( F∗)(X, Y ) has no ⊥ ∇ component in rangeF∗, for all X, Y Γ(kerF∗) . More precisely, we have ∈ ⊥ (2.3) ( F∗)(X, Y ) Γ(rangeF∗) . ∇ ∈ ⊥ F ⊥ For any vector field X on M and any section V of (rangeF∗) , we have X V , N ⊥ F ⊥ ∇ which is the orthogonal projection of X V on (rangeF∗) , where is linear ⊥ ∇F ⊥ ∇ connection on (rangeF∗) such that gN = 0.
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