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

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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 Poincaré 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 submersions whose total manifolds admit a Ricci soliton. In [18], Siddiqi and Akyol study η-Ricci-Yamabe solitons on Riemannian submersions from Riemannian manifolds. 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 soliton. 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 manifolds 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. ∇ 4 Akhilesh Yadav and Kiran Meena

Now, for a Riemannian map F we deﬁne V as ([14], p. 188) S N F ⊥ (2.4) V = V F∗X + V, ∇F∗X −S ∇X N where is Levi-Civita connection on N, V F∗X is the tangential component (a ∇ N S N vector ﬁeld along F ) of F∗X V. Thus at p M, we have F∗X V (p) TF (p)N, ∇ F ⊥ ∈ ⊥ ∇ ∈ V F∗X F∗p(TpM) and V (p) (F∗p(TpM)) . It is easy to see that V F∗X S ∈ ∇X ∈ S is bilinear in V , and F∗X at p depends only on Vp and F∗pXp. Hence from (2.2) and (2.4), we obtain

(2.5) gN ( V F∗X, F∗Y )= gN (V, ( F∗)(X, Y )), S ∇ ⊥ ⊥ for X, Y Γ(kerF∗) and V Γ(rangeF∗) . Using (2.5), we obtain ∈ ∈ ∗ (2.6) gN (( F∗)(X, F∗( V F∗Y )), W )= gN ( W F∗X, V F∗Y ), ∇ S S S ∗ where V is self adjoint operator and F∗ is the adjoint map of F∗. For details, we refer toS ([14], p. 186).

Definition 2.1 [15] A Riemannian map F between Riemannian manifolds (M,gM ) and (N,gN ) is said to be an umbilical Riemannian map at p M, if ∈ (2.7) V F∗,p(Xp)= fF∗pXp, S ⊥ for F∗X Γ(rangeF∗) and V Γ(rangeF∗) , where f is a differential function on M. If F is∈ umbilical for every p∈ M then we say that F is an umbilical Riemannian map. ∈ The Riemannian curvature tensor RN of N is a (1,3) tensor field defined by N R (X1, Y1)Z1 = X1 Y1 Z1 Y1 X1 Z1 [X1,Y1]Z1, for any X1, Y1,Z1 Γ(TN). ∇ ∇ −∇ ∇ −∇ ⊥ ∈ Now for F∗X, F∗Y, F∗Z Γ(rangeF∗) and V Γ(rangeF∗) , we have [11] ∈ ∈ N gN (R (F∗X, F∗Y )V, F∗Z)= gN (( ˜ Y )V F∗X, F∗Z) (2.8) ∇ S gN (( ˜ X )V F∗Y, F∗Z), − ∇ S where ( ˜ X )V F∗Y is defined by ∇ S N ˜ M ∗ F (2.9) ( X )V F∗Y = F∗( X F∗( V F∗Y )) ∇F ⊥V F∗Y V P X F∗Y, ∇ S ∇ S − S X − S ∇ N where P denotes the projection morphism on rangeF∗ and R is Riemannian curvature tensor of N (which is metric connection on N). ∇ If N is of constant sectional curvature c, denoted by N(c) (known as space form), whose curvature tensor field RN is given by [20]

N (2.10) R (X , Y )Z = c gN (Y ,Z )X gN (X ,Z )Y , 1 1 1 { 1 1 1 − 1 1 1} for X , Y ,Z Γ(TN). Now we denote Ricci tensor and scalar curvature by Ric 1 1 1 ∈ and s, respectively and defined as Ric(X1, Y1) = trace(Z1 R(Z1,X1)Y1) and s = traceRic(X , Y ), for X , Y Γ(TN). 7→ 1 1 1 1 ∈ The gradient of a smooth function f is defined as a vector field, denoted by gradf and given by

(2.11) gN (gradf, X1)= X1(f), for X Γ(TN). 1 ∈ Riemannian maps whose base manifolds admit a Ricci soliton and their harmonicity 5

3 The equations of Riemannian curvature, Ricci tensor and scalar curvature for base manifolds of Riemannian maps

In this section, we develop some geometric structures for base manifolds of Rieman- nian maps to investigate geometry of such maps. We will ﬁnd Riemannian curvature tensor and calculate Ricci tensors and scalar curvature of base manifold.

Proposition 3.1. Let F : (M,gM ) (N,gN ) be a Riemannian map between Riemannian manifolds. Then →

N F ⊥ F ⊥ N R (F∗X, V )W = F ⊥ F∗X + W + W F∗X ∇V W X V V −S F ⊥ F ⊥ ∇ ∇ ∇F ⊥S W F X + ∗ W (3.1) V X W V ∗ F∗(SV F∗X) −∇F ⊥∇ − S NS ∇F ⊥ F ⊥ W F X + ∗ N W, ∇ V W V ∗ F∗(∇ F∗X) −∇ X − S ∇ ∇ V ⊥ part of Γ(rangeF∗) in (3.1) is

F ⊥ F ⊥ F ⊥ F ⊥ F ⊥ F ⊥ R (F∗X, V )W = X V W V X W F ⊥ W ∇X V (3.2) ∇ F∇⊥ − ∇ ∇F ⊥ − ∇ + ∗ W + ∗ N W, F∗(SV F∗X) F∗(∇ F∗X) ∇ ∇ V ⊥ N for F∗X Γ(rangeF∗) and V, W Γ(rangeF∗) , where R denotes Riemannian ∈ N ∈ ∗ curvature tensor of which is metric connection on N and F∗ is the adjoint ∇ map of F∗.

Proof. Let F : (M,gM ) (N,gN ) be a Riemannian map between Riemannian → ⊥ manifolds. Now for F∗X Γ(rangeF∗) and V, W Γ(rangeF∗) , we have ∈ ∈ N N N N N N (3.3) R (F∗X, V )W = W W W. ∇F∗X ∇V − ∇V ∇F∗X − ∇[F∗X,V ] Now, using (2.4), we get

N N N F ⊥ F ⊥ F ⊥ ⊥ (3.4) F∗X V W = F∗X V W = ∇F W F∗X + X V W, ∇ ∇ ∇ ∇ −S V ∇ ∇

N N N F ⊥ F ⊥ (3.5) W = W F∗X + W, ∇V ∇F∗X −∇V S ∇V ∇X and

N F ⊥ F ⊥ N ∗ F ⊥ N (3.6) [F∗X,V ]W = W V F∗X F∗(SV F∗X)W + ∇ V W ∇ F∗X W, ∇ S S − ∇ ∇ X − ∇ V ⊥ ⊥ N where V Γ(rangeF∗) ,X Γ(kerF∗) , F∗X Γ(rangeF∗) and is Levi- ∈ F ⊥ ∈ ∈ ⊥ ∇N Civita connection on N, is connection on (rangeF∗) . Since gN ( V F∗X,U)= ∇⊥ N ∇ 0 for all U Γ(rangeF∗) , therefore V F∗X Γ(rangeF∗). Then using (2.4) in (3.6), we get∈ ∇ ∈

N F ⊥ F ⊥ ∗ ⊥ F∗X,V W = W V F∗X F∗ S F∗X W + F W [ ] ( V ) ∇X V (3.7) ∇ S S N − ∇ F ⊥ ∇ + F X ∗ N W. W V ∗ F∗(∇ F∗X) S ∇ − ∇ V Now using (3.4), (3.5) and (3.7) in (3.3), we get (3.1), which completes the proof. Now, above proposition implies following results: 6 Akhilesh Yadav and Kiran Meena

Results: Taking inner product of (3.1) with F∗Y Γ(rangeF∗), we get ∈ N N gN (R (F∗X, V )W, F∗Y )= gN ( F ⊥ F∗X, F∗Y )+ gN ( W F∗X, F∗Y ) ∇V W V − S ∇ SN gN ( W V F∗X, F∗Y ) gN ( W ( F∗X), F∗Y ). − S S − S ∇V Since V is self-adjoint then from above equation, we get S N gN (R (F∗X, V )W, F∗Y )= gN ( F ⊥ F∗X, F∗Y ) ∇V W − S N +gN ( W F∗X, F∗Y ) (3.8) ∇V S gN ( V F∗X, W F∗Y ) − S N S gN ( W ( F∗X), F∗Y ). − S ∇V ⊥ Now, taking inner product of (3.1) with U Γ(rangeF∗) , we get ∈ N F ⊥ F ⊥ F ⊥ F ⊥ gN (R (F∗X, V )W, U)= gN X V W V X W F∇⊥ ∇ − ∇ F ⊥∇ + ∗ W F ⊥ W (3.9) F∗(SV F∗X) ∇ V ∇ − ∇ X F ⊥ + ∗ N W, U , F∗(∇ F∗X) ∇ V ⊥ where F∗X, F∗Y Γ(rangeF∗) and U,V,W Γ(rangeF∗) . ∈ ∈ m n Theorem 3.1. Let F : (M ,gM ) (N ,gN ) be a Riemannian map between → Riemannian manifolds. Then Ricci tensors on (N,gN ) are:

n1 rangeF∗ Ric(F∗X, F∗Y )= Ric (F∗X, F∗Y ) gN ( F ⊥ F∗X, F∗Y ) n ∇e ek − kP=1 S k (3.10) N gN ( ek ek F∗X, F∗Y )+ gN ( ek F∗X, ek F∗Y ) − ∇N S S S +gN ( F∗X, e F∗Y ) , ∇ek S k o

⊥ m (rangeF∗) Ric(V, W )= Ric (V, W ) gN ( F ⊥ F∗Xj, F∗Xj) n ∇V W − j=Pr+1 S (3.11) N +gN ( V F∗Xj , W F∗Xj ) V (gN ( W F∗Xj , F∗Xj )) S S N − ∇ S +2gN ( W F∗Xj , F∗Xj) , S ∇V o and m ˜ Ric(F∗X, V )= ngN (( X )V F∗Xj , F∗Xj ) j=Pr+1 ∇ S

gN (( ˜ X )V F∗X, F∗Xj ) − ∇ j S o (3.12) n1 F ⊥ F ⊥ F ⊥ F ⊥ F ⊥ 2 gN V V F ⊥ V X ek ek X ∇ ek − kP=1 ∇ ∇ − ∇ ∇ − ∇ X F ⊥ F ⊥ + ∗ V + ∗ N V,e , F∗(Se F∗X) F∗(∇ F∗X) k ∇ k ∇ ek ⊥ ⊥ for X, Y Γ(kerF∗) , V, W Γ(rangeF∗) and F∗X, F∗Y Γ(rangeF∗), where ∈ ∈ ∈ ⊥ F∗Xj r+1≤j≤m and ek 1≤k≤n1 are orthonormal bases of rangeF∗ and (rangeF∗) , { } ∗ { } respectively and F∗ is the adjoint map of F∗.

Proof. We know that m N Ric(F∗X, F∗Y )= gN (R (F∗Xj , F∗X)F∗Y, F∗Xj) j=Pr+1 n1 N + gN (R (ek, F∗X)F∗Y,ek), kP=1 Riemannian maps whose base manifolds admit a Ricci soliton and their harmonicity 7

⊥ for X, Y Γ(kerF∗) , where F∗Xj r+1≤j≤m and ek 1≤k≤n1 are orthonormal ∈ {⊥ } { } bases of rangeF∗ and (rangeF∗) , respectively. Then using (3.8) in above equation, we get (3.10). Also, we know that

m n1 Ric(V, W )= g (RN (F X , V )W, F X )+ g (RN (e , V )W, e ). X N ∗ j ∗ j X N k k j=r+1 k=1

⊥ for V, W Γ(rangeF∗) , where F∗Xj r+1≤j≤m and ek 1≤k≤n1 are orthonormal ∈ ⊥{ } { } bases of rangeF∗ and (rangeF∗) , respectively. Then using (3.8) in above equation, we get

⊥ m (rangeF∗) Ric(V, W )= Ric (V, W )+ gN ( F ⊥ F∗Xj , F∗Xj ) n ∇V W j=Pr+1 − S (3.13) N +gN ( V W F∗Xj , F∗Xj ) gN ( V F∗Xj , W F∗Xj ) ∇ S N − S S gN ( W F∗Xj , F∗Xj ) . − S ∇V o Since N is metric connection on N therefore using metric compatibility condition in (3.13),∇ we get (3.11). Similarly by using (2.8) and (3.9), we get (3.12), which completes the proof. m n Theorem 3.2. Let F : (M ,gM ) (N ,gN ) be a Riemannian map between Riemannian manifolds. Then →

⊥ m n1 rangeF∗ (rangeF∗) s = s + s 2 g ( F ⊥ F X , F X ) N ∇e ek ∗ j ∗ j − j=Pr+1kP=1 S k m n1 m n1 N + gN ( ek ek F∗Xj, F∗Xj ) 2 gN ( ek F∗Xj, ek F∗Xj ) j=Pr+1kP=1 ∇ S − j=Pr+1kP=1 S S m n1 m n1 N N 3 gN ( ek F∗Xj , ek F∗Xj )+ ek (gN ( ek F∗Xj , F∗Xj )), − j=Pr+1kP=1 ∇ S j=Pr+1kP=1∇ S

⊥ rangeF∗ (rangeF∗) where s,s and s denote the scalar curvatures of N, rangeF∗ and ⊥ (rangeF∗) , respectively and F∗Xj r+1≤j≤m and ek 1≤k≤n1 are orthonormal {⊥ } { } bases of rangeF∗ and (rangeF∗) .

Proof. Since scalar curvature of N deﬁned by

s = traceRic(F∗X, F∗Y )+ traceRic(V, W ), ⊥ for F∗X, F∗Y Γ(rangeF∗) and V, W Γ(rangeF∗) . Then ∈ ∈ m n1 (3.14) s = Ric(F X , F X )+ Ric(e ,e ), X ∗ l ∗ l X t t l=r+1 t=1

⊥ where F∗Xl r+1≤l≤m and et 1≤t≤n1 are orthonormal bases of rangeF∗ and (rangeF∗) , respectively.{ } Now, using (3.10){ } and (3.11) in (3.14), we get

m n1 rangeF∗ s = Ric (F∗Xl, F∗Xl) gN ( F ⊥ F∗Xl, F∗Xl) n ∇e ek l=Pr+1kP=1 − S k N N +gN ( e F∗Xl, F∗Xl) gN ( e F∗Xl, e F∗Xl) gN ( F∗Xl, e F∗Xl) ∇ek S k − S k S k − ∇ek S k o m n1 ⊥ ∗ (rangeF ) ⊥ + Ric (et,et) gN ( ∇F e F∗Xj , F∗Xj ) gN ( et F∗Xj, et F∗Xj ) n et t j=Pr+1tP=1 − S − S S N N + (gN ( e F∗Xj , F∗Xj)) 2gN ( e F∗Xj , F∗Xj ) , ∇et S t − S t ∇et o 8 Akhilesh Yadav and Kiran Meena which implies

m n Corollary 3.1. Let F : (M ,gM ) (N ,gN ) be a totally geodesic Riemannian map between Riemannian manifolds.→ Then

⊥ s = srangeF∗ + s(rangeF∗) .

⊥ Proof. Since F is totally geodesic then V F∗X = 0, for all X Γ(kerF∗) and ⊥ S ∈ V Γ(rangeF∗) . Then above corollary follows by Theorem (3.2). ∈ m n Corollary 3.2. Let F : (M ,gM ) (N ,gN ) be an umbilical Riemannian map between Riemannian manifolds. Then→

⊥ n1 rangeF∗ (rangeF∗) 2 s = s + s + 2f 2f +2 ek(f)(m r). − − kP=1 −

Proof. Since F is an umbilical map then using (2.7) in Theorem (3.2), we get

⊥ m n1 rangeF∗ (rangeF∗) s = s + s 2 gN (fF∗Xj , F∗Xj) − j=Pr+1kP=1 m n1 m n1 N + gN ( ek fF∗Xj, F∗Xj ) 2 gN (fF∗Xj , fF∗Xj ) j=Pr+1kP=1 ∇ − j=Pr+1kP=1 m n1 m n1 N N 3 gN ( ek F∗Xj, fF∗Xj )+ ek (gN (fF∗Xj , F∗Xj)), − j=Pr+1kP=1 ∇ j=Pr+1kP=1∇ which implies the proof.

4 Riemannian maps whose base manifolds admitting a Ricci soliton

In this section, we study an application of Ricci tensor in detail which we are find in section 3. We consider a Riemannian map F : (M,gM ) (N,gN ) from → Riemannian manifold to Ricci soliton and give some characterizations for rangeF∗ ⊥ and (rangeF∗) . Proposition 4.1. [13] Let F be a Riemannian map from a Riemannian manifold (M,gM ) to a Riemannian manifold (N,gN ). Then F is totally geodesic if and only if (i) AX Y =0, (ii) the fibers of F define totally geodesic foliation on M, (iii) SV F∗X =0, ⊥ ⊥ for X, Y Γ(kerF∗) and V Γ(rangeF∗) . ∈ ∈ Riemannian maps whose base manifolds admit a Ricci soliton and their harmonicity 9

Theorem 4.1. Let F : (M,gM ) (N,gN ) be a totally geodesic Riemannian map → between Riemannian manifolds and (N,gN ,ξ,λ) be a Ricci soliton with potential vector field ξ Γ(TN). Then the following statements are true: ∈ ⊥ (i) If the vector field ξ = F∗Z(say) Γ(rangeF∗) for all Z Γ(kerF∗) , then ∈ ∈ rangeF∗ is a Ricci soliton. ⊥ (ii) If the vector field ξ = V (say) Γ(rangeF∗) , then rangeF∗ is an Einstein. ∈

Proof. Since (N,gN ,ξ,λ) is a Ricci soliton then, we have 1 (4.1) (L g )(F∗X, F∗Y )+ Ric(F∗X, F∗Y )+ λg (F∗X, F∗Y )=0, 2 ξ N N for F∗X, F∗Y Γ(rangeF∗). Then from (4.1), we get ∈ 1 N N gN ( ξ, F∗Y )+ gN ( ξ, F∗X) (4.2) 2 { ∇F∗X ∇F∗Y } +Ric(F∗X, F∗Y )+ λgN (F∗X, F∗Y )=0.

Since F is totally geodesic then using (iii) of Proposition 4.1 and (3.10) in (4.2), we get

1 N N 2 gN ( F∗X ξ, F∗Y )+ gN ( F∗Y ξ, F∗X) (4.3) { rangeF∇ ∗ ∇ } +Ric (F∗X, F∗Y )+ λgN (F∗X, F∗Y )=0.

Now, if the vector ﬁeld ξ = F∗Z(say) Γ(rangeF∗), then from (4.3), we get ∈ 1 N N 2 gN ( F∗X F∗Z, F∗Y )+ gN ( F∗Y F∗Z, F∗X) { rangeF∇ ∗ ∇ } +Ric (F∗X, F∗Y )+ λgN (F∗X, F∗Y )=0, which implies rangeF∗ is a Ricci soliton, which implies (i). ⊥ Also, if the vector ﬁeld ξ = V (say) Γ(rangeF∗) , then from (4.3), we get ∈ 1 N N 2 gN ( F∗X V, F∗Y )+ gN ( F∗Y V, F∗X) { rangeF∇ ∗ ∇ } +Ric (F∗X, F∗Y )+ λgN (F∗X, F∗Y )=0.

Using (2.4) in above equation, we get

1 2 gN ( V F∗X, F∗Y )+ gN ( V F∗Y, F∗X) − { rangeFS ∗ S } +Ric (F∗X, F∗Y )+ λgN (F∗X, F∗Y )=0.

Since V is self-adjoint then from above equation, we get S rangeF∗ gN ( V F∗X, F∗Y )+ Ric (F∗X, F∗Y )+ λgN (F∗X, F∗Y )=0. − S Since F is totally geodesic therefore again using (iii) of Proposition 4.1 in above equation, we get

rangeF∗ (4.4) Ric (F∗X, F∗Y )+ λgN (F∗X, F∗Y )=0, which means rangeF∗ is an Einstein, which implies (ii), which completes the proof.

m n Theorem 4.2. Let F : (M ,gM ) (N ,gN ) be a totally geodesic Riemannian → map between Riemannian manifolds and (N,gN ,ξ,λ) be a Ricci soliton with the ⊥ potential vector ﬁeld ξ Γ(rangeF∗) then the scalar curvature of rangeF∗ is ∈ λ(m r), where dim(rangeF∗)= m r. − − − 10 Akhilesh Yadav and Kiran Meena

Proof. By taking trace of (4.4), we get proof of this theorem.

Theorem 4.3. Let F : (M,gM ) (N,gN ) be a totally geodesic Riemannian map → between Riemannian manifolds and (N,gN ,ξ,λ) be a Ricci soliton with potential vector field ξ Γ(TN). Then the following statements are true: ∈ ⊥ ⊥ (i) If the vector field ξ = V (say) Γ(rangeF∗) , then (rangeF∗) is a Ricci soliton. ∈ ⊥ (ii) If the vector field ξ = F∗X(say) Γ(rangeF∗), then (rangeF∗) is an Einstein. ∈

Proof. Since (N,gN ,ξ,λ) be a Ricci soliton then, we have

1 (L g )(U, W )+ Ric(U, W )+ λg (U, W )=0 2 ξ N N

⊥ for U, W Γ(rangeF∗) . Then from above equation, we get ∈

1 N N gN ( ξ, W )+ gN ( ξ,U + Ric(U, W )+ λgN (U, W )=0. 2{ ∇U ∇W } Since F is totally geodesic then using (iii) of Proposition 4.1 and (3.11) in above equation, we get

⊥ 1 N N (rangeF∗) (4.5) gN ( ξ, W )+ gN ( ξ,U + Ric (U, W )+ λgN (U, W )=0. 2{ ∇U ∇W }

⊥ Now, if the vector ﬁeld ξ = V (say) Γ(rangeF∗) , then from (4.5), we get ∈

⊥ 1 N N (rangeF∗) gN ( V, W )+ gN ( V,U) + Ric (U, W )+ λgN (U, W )=0. 2{ ∇U ∇W }

F ⊥ ⊥ Since is connection on (rangeF∗) therefore above equation can be written as ∇

⊥ 1 F ⊥ F ⊥ (rangeF∗) gN ( V, W )+ gN ( V,U) + Ric (U, W )+ λgN (U, W )=0, 2{ ∇U ∇W }

⊥ which implies (rangeF∗) is a Ricci soliton, which implies (i). Also, if the vector ﬁeld ξ = F∗X(say) Γ(rangeF∗), then from (4.5), we get ∈

⊥ 1 N N (rangeF∗) gN ( F∗X, W )+ gN ( F∗X,U + Ric (U, W )+ λgN (U, W )=0. 2{ ∇U ∇W }

Since N is metric connection therefore using metric compatibility condition in above∇ equation, we get

⊥ (rangeF∗) (4.6) Ric (U, W )+ λgN (U, W )=0,

⊥ which means (rangeF∗) is an Einstein, which implies (ii), which completes the proof.

m n Theorem 4.4. Let F : (M ,gM ) (N ,gN ) be a totally geodesic Riemannian → map between Riemannian manifolds and (N,gN ,ξ,λ) be a Ricci soliton. If the potential vector ﬁeld ξ = F∗X(say) Γ(rangeF∗) then the scalar curvature of ⊥ ∈ ⊥ (rangeF∗) is λn , where dim(rangeF∗) = n . − 1 1 Proof. By taking trace of (4.6), we get proof of this theorem. Riemannian maps whose base manifolds admit a Ricci soliton and their harmonicity 11

Theorem 4.5. Let F : (M,gM ) (N,gN ) be an umbilical Riemannian map be- → tween Riemannian manifolds and (N,gN ,ξ,λ) be a Ricci soliton with potential vector field ξ Γ(TN). Then the following statements are true: ∈ ⊥ (i) If the vector field ξ = V (say) Γ(rangeF∗) , then rangeF∗ is an Einstein. ∈ (ii) If the vector field ξ = F∗Z(say) Γ(rangeF∗), then rangeF∗ is an almost Ricci soliton. ∈

⊥ Proof. Since (N,gN ,ξ,λ) is a Ricci soliton. If ξ = V (say) Γ(rangeF∗) then using (2.4) and (3.10) in (4.2), we get ∈ 1 F ⊥ F ⊥ 2 gN ( V F∗X + X V, F∗Y )+ gN ( V F∗Y + Y V, F∗X) { −S ∇ n1 −S ∇ } rangeF∗ +Ric (F∗X, F∗Y ) gN ( F ⊥ F∗X, F∗Y ) n ∇e ek (4.7) − kP=1 S k N gN ( ek ek F∗X, F∗Y )+ gN ( ek F∗X, ek F∗Y ) − ∇N S S S +gN ( F∗X, e F∗Y ) + λgN (F∗X, F∗Y )=0. ∇ek S k o

Since V is self-adjoint then from (4.7), we get S rangeF∗ gN ( V F∗X, F∗Y )+ Ric (F∗X, F∗Y ) − n1 S N gN ( F ⊥ F∗X, F∗Y ) gN ( e F∗X, F∗Y )+ gN ( e F∗X, e F∗Y ) n ∇e ek ek k k k −kP=1 S k − ∇ S S S N +gN ( F∗X, e F∗Y ) + λgN (F∗X, F∗Y )=0. ∇ek S k o Since F is an umbilical Riemannian map then using (2.7) in above equation, we get

rangeF∗ 2fgN (F∗X, F∗Y )+ Ric (F∗X, F∗Y ) − n1 2 + ek(f)gN (F∗X, F∗Y ) f gN (F∗X, F∗Y )+ λgN (F∗X, F∗Y )=0. kP=1 − Thus from above equation, we get

rangeF∗ Ric (F∗X, F∗Y ) µgN (F∗X, F∗Y )=0, − n1 2 where, µ =2f + f ek(f) λ is a diﬀerentiable function. Hence rangeF∗ is an − kP=1 − Einstein, which implies (i).

Also, if ξ = F∗Z(say) Γ(rangeF∗) then using (3.10) in (4.1), we get ∈ 1 rangeF∗ 2 (LF∗Z gN )(F∗X, F∗Y )+ Ric (F∗X, F∗Y ) n1 N gN ( F ⊥ F∗X, F∗Y ) gN ( e F∗X, F∗Y ) n ∇e ek ek k −kP=1 S k − ∇ S N +gN ( e F∗X, e F∗Y )+ gN ( F∗X, e F∗Y ) + λgN (F∗X, F∗Y )=0. S k S k ∇ek S k o Since F is an umbilical Riemannian map then using (2.7) in above equation, we get

1 rangeF∗ 2 (LF∗Z gN )(F∗X, F∗Y )+ Ric (F∗X, F∗Y ) fgN (F∗X, F∗Y ) n1 − (4.8) 2 + ek(f)gN (F∗X, F∗Y ) f gN (F∗X, F∗Y )+ λgN (F∗X, F∗Y )=0. kP=1 − Thus from (4.8), we get

1 rangeF∗ 2 (LF∗Z gN )(F∗X, F∗Y )+ Ric (F∗X, F∗Y ) n1 2 f + f ek(f) λgN (F∗X, F∗Y )=0, − − kP=1 − which means rangeF∗ is an almost Ricci soliton, which implies (ii), which completes the proof. 12 Akhilesh Yadav and Kiran Meena

Theorem 4.6. Let F : (M,gM ) (N,gN ) be an umbilical Riemannian map from a → Riemannian manifold to an Einstein manifold and (N,gN , F∗Z, λ) be a Ricci soliton with potential vector ﬁeld F∗Z Γ(rangeF∗). Then vector ﬁeld F∗Z is conformal ∈ on rangeF∗.

rangeF∗ Proof. Since N is Einstein, then rangeF∗ is also Einstein, so Ric (F∗X, F∗Y ) = λgN (F∗X, F∗Y ). Then from (4.8), we get − 1 (LF∗Z gN )(F∗X, F∗Y ) µgN (F∗X, F∗Y )=0, 2 −

n1 2 where, µ = f + f ek(f) is a diﬀerentiable function. Hence vector ﬁeld F∗Z is − kP=1 conformal, which completes the proof.

Theorem 4.7. Let F : (M,gM ) (N,gN ) be a Riemannian map between Rie- → mannian manifolds and (N,gN , F∗U, λ) be a Ricci soliton with potential vector ﬁeld F∗U, for U kerF∗. Then (N,gN ) is an Einstein manifold. ∈

Proof. We know that for all U Γ(kerF∗), F∗U = 0 and since (N,gN , F∗U, λ) is a Ricci soliton then from (1.1), we∈ get

Ric(X1, Y1)+ λgN (X1, Y1)=0, for any X1, Y1 Γ(TN), which means N is an Einstein manifold, which completes the proof. ∈

5 Riemannian maps to Ricci solitons and their harmonicity and biharmonicity

This section deals with the harmonicity and biharmonicity of Riemannian map from a Riemannian manifold to a Ricci soliton. It is known that Riemannian maps need not be harmonic maps and also harmonic maps need not to be Riemannian maps. Now, we give necessary and sufficient conditions for Riemannian maps to Ricci solitons to be harmonic. We first recall m n that a map F : (M ,gM ) (N ,gN ) between Riemannian manifolds is harmonic if and only if the tension field→ of F vanishes at each point p M, i.e., ∈ m τ(F )= trace( F∗)= ( F∗)(ei,ei)=0, ∇ X ∇ i=1 where ei 1≤i≤m is local orthonormal frame around a point p M and F∗ is the second{ fundamental} form of F . ∈ ∇

m n Lemma 5.1. [16] Let F : (M ,gM ) (N ,gN ) be a Riemannian map between → Riemannian manifolds. Then the tension ﬁeld of F is given by τ(F )= rF∗(H)+ − (m r)H , where r = dim(kerF∗), (m r) = rankF , H and H are the mean − 2 − 2 curvature vector ﬁelds of the distribution kerF∗ and rangeF∗, respectively. Riemannian maps whose base manifolds admit a Ricci soliton and their harmonicity 13

Moreover, the mean curvature vector ﬁeld of rangeF∗ is deﬁned by [16]

m 1 F (5.1) H = F∗(Xj ), 2 m r X ∇Xj − j=r+1

⊥ where Xj r ≤j≤m is an orthonormal basis of (kerF∗) . { } +1 m n Lemma 5.2. [15] Let F : (M ,gM ) (N ,gN ) be a Riemannian map between Riemannian manifolds. Then F is an umbilical→ map if and only if

(5.2) ( F∗)(X, Y )= gM (X, Y )H , ∇ 2 ⊥ ⊥ for X, Y Γ(kerF∗) and H is, nowhere zero, vector ﬁeld on (rangeF∗) . ∈ 2 Theorem 5.1. Let (N,gN ,V,λ) be a Ricci soliton with potential vector ﬁeld V ⊥ m n ∈ Γ(rangeF∗) and F : (M ,gM ) (N ,gN ) be an umbilical Riemannian map → between Riemannian manifolds such that the scalar curvature of rangeF∗ is λ(m − n1 − ∇N f 2 r) =0 and kerF∗ is minimal. Then F is harmonic if and only if = ek, f n1 6 kP=1 ⊥ ∞ where ek ≤k≤n is an orthonormal basis of (rangeF∗) and f C (M). { }1 1 ∈ Proof. By using (2.6) in (4.7), we get

1 F ⊥ F ⊥ 2 gN ( V F∗X + X V, F∗Y )+ gN ( V F∗Y + Y V, F∗X) { −S ∇ n1 −S ∇ } rangeF∗ +Ric (F∗X, F∗Y ) gN ( F ⊥ F∗X, F∗Y ) n ∇e ek (5.3) − kP=1 S k N ∗ gN ( ek ek F∗X, F∗Y )+ gN (( F∗)(X, F∗ ek F∗Y ),ek) − ∇N S ∇ S +gN ( F∗X, e F∗Y ) + λgN (F∗X, F∗Y )=0, ∇ek S k o ⊥ where ek 1≤k≤n1 is an orthonormal basis of (rangeF∗) . Since V is self-adjoint then from{ } (5.3), we get S

rangeF∗ gN ( V F∗X, F∗Y )+ Ric (F∗X, F∗Y )+ λgN (F∗X, F∗Y ) − n1 S N gN ( F ⊥ F∗X, F∗Y ) gN ( e F∗X, F∗Y ) (5.4) n ∇e ek ek k −kP=1 S k − ∇ S ∗ N +gN (( F∗)(X, F∗ e F∗Y ),ek)+ gN ( e F∗X, F∗Y ) =0. ∇ S k S k ∇ek o Since F is an umbilical Riemannian map then using (2.7) and (5.2) in (5.4), we get

rangeF∗ 2fgN (F∗X, F∗Y )+ Ric (F∗X, F∗Y )+ λgN (F∗X, F∗Y ) − n n (5.5) 1 1 + ek(f)gN (F∗X, F∗Y ) f gN (F∗X, F∗Y )gN (H2,ek)=0. kP=1 − kP=1

Taking trace of (5.5), we get

n1 n1 rangeF∗ 2f(m r)+ s + λ(m r)+ ek(f)(m r) f(m r) gN (H2,ek)=0. − − − kP=1 − − − kP=1

Putting srangeF∗ = λ(m r) in above equation, we get − −

n1 n1 2f + ek(f) f gN (H ,ek)=0, − X − X 2 k=1 k=1 14 Akhilesh Yadav and Kiran Meena which implies n n n 2f 1 1 1 − gN (ek,ek)+ ek(f) f gN (H2,ek)=0. n1 X X − X k=1 k=1 k=1 Using (2.11) in above equation, we get

n1 n1 n1 2f N − gN (ek,ek)+ gN ( f,ek) f gN (H2,ek)=0, n1 X X ∇ − X k=1 k=1 k=1 which implies n1 2f N − ek + f fH2 =0. n1 X ∇ − k=1 Hence n 2 1 N f (5.6) H2 = ek ∇ . n1 X − f k=1

Since kerF∗ is minimal and using (5.6) in Lemma 5.1, we get

n1 2 ∇N f τ(F ) = (m r) ek , which completes the proof. n n1 f o − kP=1 −

Theorem 5.2. Let (N,gN , F∗X, λ) be a Ricci soliton with potential vector ﬁeld m F∗X Γ(rangeF∗) and F : (M ,gM ) (N(c),gN ) be a Riemannian map from a ∈ → Riemannian manifold to a space form. Then F is harmonic if and only if kerF∗ is minimal.

Proof. Since (N,gN ) be a Ricci soliton then, we have 1 (L ∗ g )(F∗Y, V )+ Ric(F∗Y, V )+ λg (F∗Y, V )=0, 2 F X N N ⊥ for V Γ(rangeF∗) and F∗X, F∗Y Γ(rangeF∗). Then from above equation, we get ∈ ∈ 1 N N gN ( F∗X, V )+ gN ( F∗X, F∗Y ) + Ric(F∗X, V )=0, 2{ ∇F∗Y ∇V } or N 1 F N (5.7) gN ( F∗X F, V )+ gN ( F∗X, F∗Y ) + Ric(F∗X, V )=0. 2{ ∇Y ◦ ∇V } By deﬁnition of Ricci tensor and using (2.2) in (5.7), we get

1 M N 2 gN (F∗( Y X) + ( F∗)(Y,X), V )+ gN ( V F∗X, F∗Y ) { m ∇ ∇ n1 ∇ } N N + gN (R (F∗Xj , F∗X)V, F∗Xj)+ gN (R (ek, F∗X)V,ek)=0, j=Pr+1 kP=1

where F∗Xj r+1≤j≤m and ek 1≤k≤n1 are orthonormal bases of rangeF∗ and { ⊥ } { } (rangeF∗) , respectively. Using (2.10) in above equation, we get

1 N 2 gN (( F∗)(Y,X), V )+ gN ( V F∗X, F∗Y ) { m ∇ ∇ } + cngN (gN (F∗X, V )F∗Xj gN (F∗Xj , V )F∗X, F∗Xj )o (5.8) j=Pr+1 − n1 + cngN (gN (F∗X, V )ek gN (ek, V )F∗Y,ek)o =0. kP=1 − Riemannian maps whose base manifolds admit a Ricci soliton and their harmonicity 15

Taking trace of (5.8), we get

m n1 1 N gN (( F∗)(Xl,Xl),ea)+ gN ( F∗Xj , F∗Xj) =0. (5.9) 2 n ea o l=Pr+1aP=1 ∇ ∇

Since N is metric connection on N and using (2.2) in (5.9), we get ∇

m n1 N g ( F F (X ),e )=0. N Xl ∗ l a l=Pr+1aP=1 ∇

Now using (5.1) in above equation, we obtain

n1 gN (H2,ea)=0. aP=1

Hence H2 = 0. Then by Lemma 5.1, F is harmonic if and only if H = 0, which completes the proof. Biharmonic maps have been extensively studied in last decade and many au- thors have obtained classiﬁcation results. Biharmonicity of Riemannian maps was studied by B. S¸ahin in [17]. For the biharmonicity of Riemannian maps, we have the following result:

m Theorem 5.3. [17] Let F : (M ,gM ) (N(c),gN ) be a Riemannian map from a Riemannian manifold to a space form.→ Then F is biharmonic if and only if

rtrace (∇F∗ )(.,H)F∗(.) rtraceF∗( (.) (.)H) S ∗− ∇ ∇ F ⊥ (5.10) (m r)traceF∗( (.) F∗( H2 F∗(.))) (m r)trace ∇ H F∗(.) − − ∇ S − − S (.) 2 rc(m r 1)F∗(H)=0, − − − and

F ⊥ rtrace (.) ( F∗)(., H)+ rtrace( F∗)(., (.)H) ∇ ∇ ∇ ∇ ⊥ (5.11) ∗ R +(m r)trace( F∗)(., F∗( H2 F∗(.))) (m r)∆ H2 (m − r)2cH =0∇ , S − − − − 2 ⊥ where dim(kerF∗)= r and dim(kerF∗) = m r. −

Theorem 5.4. Let (N,gN , F∗X, λ) be a Ricci soliton with potential vector ﬁeld F∗X Γ(rangeF∗) and F : (M,gM ) (N(c),gN ) be a Riemannian map from a ∈ → Riemannian manifold to a space form. Then F is biharmonic if and only if kerF∗ is minimal.

Proof. We see in Theorem 5.2, H2 = 0 then from (5.10) and (5.11), F is biharmonic if and only if H = 0, which completes the proof.

6 Example

Example 6.1 Let M = (x , x , x ) R3 : x = 0, x = 0, x = 0 be a 3- { 1 2 3 ∈ 1 6 2 6 3 6 } dimensional Riemannian manifold with Riemannian metric gM on M given by gM = 2x 2 2x2 2 2 2 e 3 dx + e 3 dx + dx . Let N = (y ,y ) R be a Riemannian manifold 1 2 3 { 1 2 ∈ } 16 Akhilesh Yadav and Kiran Meena

2y2 2 2 with Riemannian metric gN on N given by gN = e dy1 + dy2 . Consider a map F : (M,gM ) (N,gN ) deﬁned by → x1 + x2 + x3 F (x1, x2, x3)= , 0 . √3 By direct computations

kerF∗ = Span U = e + e ,U = e + e n 1 − 1 2 2 − 1 3o and ⊥ (kerF∗) = SpannX = e1 + e2 + e3o,

′ ′ −x3 ∂ −x3 ∂ ∂ −y2 ∂ ∂ where e1 = e ,e2 = e ,e3 = , e = e ,e = are bases n ∂x1 ∂x2 ∂x3 o n 1 ∂y1 2 ∂y2 o on TpM and TF (p)N respectively, for all p M. By direct computations, we can see ′ ∈ ⊥ that F∗(X)= √3e1 and gM (X,X)= gN (F∗X, F∗X) for X Γ(kerF∗) . Thus F is ′ ∈ ⊥ ′ Riemannian map with rangeF∗ = SpannF∗X = e1o and (rangeF∗) = Spanne2o. Now, we will show that base manifold N admits Ricci soliton, i.e., 1 (6.1) (L g )(X , Y )+ Ric(X , Y )+ λg (X , Y )=0, 2 Z1 N 1 1 1 1 N 1 1 for any X , Y ,Z Γ(TN). 1 1 1 ∈ Now,

1 1 N N (6.2) (LZ gN )(X1, Y1)= gN ( Z1, Y1)+ gN ( Z1,X1) . 2 1 2n ∇X1 ∇Y1 o ⊥ Since dimension of rangeF∗ and (rangeF∗) is one therefore we can decompose ′ ′ ′ ′ ′ ′ X1, Y1 and Z1 such that X1 = a1e1 + a2e2, Y1 = a3e1 + a4e2 and Z1 = a5e1 + a6e2, ′ ′ ⊥ where e1 and e2 denote for component of rangeF∗ and (rangeF∗) , respectively and ai ≤i≤ R are some scalars. Then from (6.2), we get { }1 6 ∈ ′ ′ ′ ′ 1 1 N (L g )(X , Y )= g ( ′ ′ a e + a e ,a e + a e ) 2 Z1 N 1 1 2 N a e a e 5 1 6 2 3 1 4 2 n ∇ 1 1+ 2 2 N ′ ′ ′ ′ +g ( ′ ′ a e + a e ,a e + a e ) . N a e a e 5 1 6 2 1 1 2 2 ∇ 3 1+ 4 2 o Since N is metric connection then from above equation, we get ∇ ′ ′ ′ ′ 1 1 N N (L g )(X , Y )= 2a a a g ( ′ e ,e )+2a a a g ( ′ e ,e ) 2 Z1 N 1 1 2 1 3 6 N e 2 1 2 4 5 N e 1 2 n ∇ 1 ∇ 2 N ′ ′ N ′ ′ +a a a g ( ′ e ,e )+ a a a g ( ′ e ,e ) (6.3) 2 3 6 N e 2 1 1 4 5 N e 1 2 ∇ 2 ∇ 1 N ′ ′ N ′ ′ +a a a g ( ′ e ,e )+ a a a g ( ′ e ,e ) . 1 4 6 N e 2 1 2 3 5 N e 1 2 ∇ 2 ∇ 1 o F ⊥ ⊥ Since is connection on (rangeF∗) and using metric compatibility condition of N∇in (6.3), we get ∇ 1 1 N ′ ′ N ′ ′ (L g )(X , Y )= ( 2a a a + a a a )g ( ′ e ,e ) 2a a a g ( ′ e ,e ) 2 Z1 N 1 1 2 1 3 6 1 4 5 N e 1 2 2 4 5 N e 2 1 n − ∇ 1 − ∇ 2 F ⊥ ′ ′ N ′ ′ +(a a a + a a a )g ( ′ e ,e )+ a a a g ( ′ e ,e ) , 2 3 6 1 4 6 N e 2 1 2 3 5 N e 1 2 ∇ 2 ∇ 1 o which implies

1 1 N ′ ′ (L g )(X , Y )= (a a a 2a a a + a a a )g ( ′ e ,e ), 2 Z1 N 1 1 2 1 4 5 1 3 6 2 3 5 N e 1 2 − ∇ 1 Riemannian maps whose base manifolds admit a Ricci soliton and their harmonicity 17 or we can write

′ 1 1 N (LZ gN )(X , Y )= (a a a 2a a a + a a a )gN ( F∗X,e ). 2 1 1 1 2 1 4 5 − 1 3 6 2 3 5 ∇F∗X 2 Thus N (6.4) 1 1 F ′ (LZ gN )(X , Y )= (a a a 2a a a + a a a )gN ( F∗X F,e ). 2 1 1 1 2 1 4 5 − 1 3 6 2 3 5 ∇X ◦ 2 Using (2.2) in (6.4), we get

1 1 ′ (LZ gN )(X , Y )= (a a a 2a a a + a a a )gN (( F∗)(X,X)+ F∗( X X),e ). 2 1 1 1 2 1 4 5 − 1 3 6 2 3 5 ∇ ∇ 2 ′ Using (2.3), we can write ( F∗)(X,X)= a e . Then from above equation, we get ∇ 7 2 1 1 (6.5) (LZ gN )(X , Y )= (a a a 2a a a + a a a )a . 2 1 1 1 2 1 4 5 − 1 3 6 2 3 5 7 Also,

′ ′ ′ ′ (6.6) gN (X1, Y1)= gN (a1e1 + a2e2,a3e1 + a4e2) = (a1a3 + a2a4),

and ′ ′ ′ ′ Ric(X1, Y1)= Ric(a1e1 + a2e2,a3e1 + a4e2), which implies

′ ′ ′ ′ ′ ′ (6.7) Ric(X1, Y1)= a1a3Ric(e1,e1) + (a1a4 + a2a3)Ric(e1,e2)+ a2a4Ric(e2,e2). By (3.10) and (3.11), we get

′ ′ ′ ′ ′ ′ rangeF∗ Ric(e ,e )= Ric (e ,e ) gN ( F ⊥ ′ e ,e ) 1 1 1 1 ∇ ′ e2 1 1 − S e ′ ′ 2 ′ ′ N (6.8) +gN ( ′ ′ e ,e ) gN ( ′ e , ′ e ) e e2 1 1 e2 1 e2 1 ∇ 2 S′ ′ − S S N g ( ′ e , ′ e ), N e 1 e 1 − ∇ 2 S 2 and

′ ′ ⊥ ′ ′ ′ ′ (rangeF∗) Ric(e ,e )= Ric (e ,e ) gN ( F ⊥ ′ e ,e ) 2 2 2 2 ∇ ′ e2 1 1 − S e ′ ′ 2′ ′ N (6.9) + ′ (gN ( ′ e ,e )) gN ( ′ e , ′ e ) e e2 1 1 e2 1 e2 1 ∇ 2 S′ ′ − S S N 2g ( ′ e , ′ e ). N e 1 e 1 − ∇ 2 S 2 By (3.2) and (3.12), we get

′ ′ F ⊥ ′ ′ ′ (6.10) Ric(e ,e )= 2gN (R (F∗X,e )e ,e ). 1 2 − 2 2 2 Using (6.8), (6.9) and (6.10) in (6.7), we get

′ ′ ′ ′ rangeF∗ N Ric(X1, Y1) = (a1a3)Ric (e ,e ) + (a1a3)gN ( ′ ′ e ,e ) 1 1 e e2 1 1 ′ ′ ∇ 2 S (a1a3 + a2a4)gN ( F ⊥ ′ e ,e ) ∇ ′ e2 1 1 − S e 2′ ′ (a1a3 + a2a4)gN ( ′ e , ′ e ) e2 1 e2 1 − S N ′S ′ (6.11) (a1a3 +2a2a4)gN ( ′ e , ′ e ) e 1 e2 1 − ⊥∇ 2′ ′S (rangeF∗) +(a2a4)Ric (e2,e2) N ′ ′ +(a a ) ′ (g ( ′ e ,e )) 2 4 e N e 1 1 ∇ 2 S 2 F ⊥ ′ ′ ′ 2(a a + a a )gN (R (F∗X,e )e ,e ). − 1 4 2 3 2 2 2 18 Akhilesh Yadav and Kiran Meena

′ ′ ⊥ rangeF∗ Since dimension of rangeF∗ and (rangeF∗) is one therefore Ric (e1,e1) = ⊥ ′ ′ ′ ′ (rangeF∗) 0 and Ric (e ,e ) = 0. Also, since F ⊥ ′ e , ′ e Γ(rangeF∗) and 2 2 ∇ ′ e2 1 e2 1 e S 2 S ∈ F ⊥ ′ ′ ⊥ ′ ′ ′ R (F∗X,e )e Γ(rangeF∗) therefore we can write F ⊥ ′ e = a8e , ′ e = 2 2 ∇ ′ e2 1 1 e2 1 ∈ S e S ′ ′ ′ ′ 2 F ⊥ R a9e1 and R (F∗X,e2)e2 = a10e2 for some scalars a8,a9,a10 . Then by putting these values in (6.11), we get ∈

N ′ ′ ′ ′ Ric(X , Y ) = (a a )g ( ′ a e ,e ) (a a + a a )g (a e ,e ) 1 1 1 3 N e 9 1 1 1 3 2 4 N 8 1 1 ∇ 2 − ′ ′ N ′ ′ (a a + a a )g (a e ,a e ) (a a +2a a )g ( ′ e ,a e ) 1 3 2 4 N 9 1 9 1 1 3 2 4 N e 1 9 1 − − ∇ 2 N ′ ′ ′ ′ +(a a ) ′ (g (a e ,e )) 2(a a + a a )g (a e ,e ), 2 4 e N 9 1 1 1 4 2 3 N 10 2 2 ∇ 2 − which implies

(6.12) Ric(X , Y )= (a a + a a )a (a a + a a )a2 2(a a + a a )a . 1 1 − 1 3 2 4 8 − 1 3 2 4 9 − 1 4 2 3 10 Now, using (6.5), (6.6) and (6.12) in (6.1), we obtain that metric gN admits Ricci soliton for

1 1 1 λ = a1a3a6a7 a1a4a5a7 a2a3a5a7 + a1a3a8 (a1a3+a2a4) n − 2 − 2 2 2 +a2a4a8 + a1a3a9 + a2a4a9 +2a1a4a10 +2a2a3a10o, where a a = a a . Since all ai R therefore for some choices of ai’s Ricci soliton 1 3 6 − 2 4 ∈ (N,gN ) will be shrinking, expanding or steady according to λ< 0, λ> 0 or λ = 0.

7 Applications for further studies

In this section, we present some more applications of curvature tensors, Ricci tensors and scalar curvature of base manifolds of Riemannian maps, which are we find in section 3. Similar problems can be formulated to further studies by using solitons in the following situations: Problem 7.1 Like [10] by using Ricci tensors study of Einstein metrices on the base space of Riemannian map and find conditions for the base space of a Riemannian map to be an Einstein manifold. Problem 7.2 By using Ricci tensors study of invariant, anti-invariant and semi- invariant Riemannian maps (cf. [12], [13]) to obtain some characterizations. Problem 7.3 Like [7] study of Riemannian maps whose base manifolds admit gradient Ricci soliton, conformal gradient Ricci soliton, almost Ricci soliton and η-Ricci soliton to obtain some characterizations. Problem 7.4 Like (cf. [6], [18]) study of Riemannian maps whose base manifolds admit an almost Yamabe soliton, η-Ricci-Yamabe soliton, η-Yamabe soliton and quasi-Yamabe soliton to obtain some characterizations. Problem 7.5 Characterizations of Riemannian maps whose base manifolds admit ρ-Einstein soliton. Acknowledgement: Kiran Meena gratefully acknowledges the financial support provided by the Council of Scientific and Industrial Research (C.S.I.R.), New Delhi, India [ File No.: 09/013(0887)/2019-EMR-I ]. Data availability statement: Not applicable to this article. Riemannian maps whose base manifolds admit a Ricci soliton and their harmonicity 19

References

[1] A. L. Besse. Einstein Manifolds. Springer-Verlag, Berlin, Heidelberg, New York, 1987.

[2] S. Deshmukh and F. R. Al-Solamy. Conformal vector ﬁelds on a Riemannian manifold. Balkan Journal of Geometry and Its Application, Balkan Society of Geometers, Geometry Balkan Press, Vol 19(2), 86-93, 2014.

[3] M. Falcitelli, S. Ianus and A. M. Pastore. Riemannian Submersions and Related Topics. World Scientiﬁc, River Edge, NJ, 2004.

[4] A. E. Fischer. Riemannian maps between Riemannian manifolds. Mathematical aspects of classical ﬁeld theory (Seattle, WA, 1991), contemp. Math. Amer. Math. soc., Providence, RI, Vol 132, 331-366, 1992.

[5] R. S. Hamilton. The Ricci ﬂow on surfaces, mathematics and general relativity. contemp. Math. Amer. Math. soc., Vol 71, 237-262, 1966.

[6] S. E. Meri¸c. Some remarks on Riemannian submersions admitting an almost Yamabe soliton. Adiyaman University Journal of Science, Vol 10(1), 295-306, 2020.

[7] S. E. Meri¸cand E. Kili¸c. Riemannian submersions whose total manifolds admit a Ricci soliton. International Journal of Geometric Methods in Modern Physics, Vol 16, 1950196-1 - 1950196-12, 2019.

[8] G. Perelman. The Entropy formula for the Ricci ﬂow and its geometric applications. arXiv: math/0211159.

[9] S. Pigola, M. Rigoli, M. Rimoldi and A.G. Setti. Ricci almost solitons. Ann. Scuola Norm. Sup. Pisa. Cl. Sci. Vol 10, 757-799, 2011.

[10] B. S¸ahin. Notes on Riemannian maps. U. P. B. Sci. Bull., Series A, Vol 79(3), 131-138, 2017.

[11] B. S¸ahin. A survey on diﬀerential geometry of Riemannian maps between Rie- mannian manifolds. An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si. Mat. (N.S.), Tomul LXIII, f.1, 151-167, 2017.

[12] B. S¸ahin. Invariant and anti-invariant Riemannian maps to K¨ahler manifolds. Int. J. Geom. Methods Mod. Phys. Vol 7(3), 337-355, 2010.

[13] B. S¸ahin. Semi-invariant Riemannian maps from almost Hermitian manifolds. Indag. Math., Vol 23, 80-94, 2012.

[14] B. S¸ahin. Riemannian submersions, Riemannian maps in Hermitian geometry, and their applications. Academic Press, 2017.

[15] B. S¸ahin. Semi-invariant Riemannian maps to K¨ahler manifolds. Int. J. Geom. Methods Mod. Phys., Vol 7, 1439-1454, 2011.

[16] B. S¸ahin. Conformal Riemannian maps between Riemannian manifolds, their harmonicity and decomposition theorems. Acta Appl. Math., Vol 109, 829-847, 2010. 20 Akhilesh Yadav and Kiran Meena

[17] B. S¸ahin. Biharmonic Riemannian maps. Ann. Polon. Math., Vol 102, 39-49, 2011. [18] M. D. Siddiqi and M. A. Akyol. η-Ricci-Yamabe solitons on Riemannian submersions from Riemannian manifolds. arXiv:2004.14124 [math.DG], Vol 1, 1-16, 2020. [19] A. Yadav and K. Meena. Riemannian maps whose total manifolds admit a Ricci soliton. Journal of Geometry and Physics. To Appear. [20] K. Yano and M. Kon. Structure on Manifolds. World Scientiﬁc, Singapore, 1984.

Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-221005, India. E-mail: [email protected]; [email protected]