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2 Preliminaries

Let M be an n-dimensional Riemannian manifold with Riemannian metric g. A linear connection ∇˜ on a Riemannian manifold M is called a semi-symmetric connection if the torsion tensor T˜ of the connection ∇˜ given by

T˜(X,Y )= ∇˜ X Y − ∇˜ Y X − [X,Y ] (2.1)

satisfies T˜(X,Y )= π(Y )X − π(X)Y, (2.2) where π is a 1-form associated with the vector field P on M defined by

π(X)= g(X,P ). (2.3)

A semi-symmetric connection ∇˜ is called a semi-symmetric metric connection if ∇˜ g = 0. Let ∇ be the Levi-Civita connection of a Riemannian manifold, then the unique semi-symmetric metric connection ∇˜ given by Yano [26] is

∇˜ X Y = ∇X Y + π(Y )X − g(X,Y )P. (2.4)

A relation between the curvature tensors R and R˜ of the Levi-Civita connection ∇ and the semi-symmetric connection ∇˜ is given by

R˜(X,Y )Z = R(X,Y )Z + g(Z, ∇XP )Y − g(Z, ∇Y P )X

+g(X,Z)∇Y P − g(Y,Z)∇XP +π(P )(g(X,Z)Y − g(Y,Z)X) +(g(Y,Z)π(X) − g(X,Z)π(Y )) P +π(Z)(π(Y )X − π(X)Y ) , X,Y,Z ∈ X(M), (2.5)

where X(M) is the set of smooth vector fields on M [26]. Yano proved that a Riemannian manifold is conformally flat if and only if it admits a semi-symmetric metric connection whose curvature tensor vanishes identically. This result was generalized for vanishing Ricci tensor of the semi-symmetric metric connection by T. Imai (see [13], [14]). For a general survey of different kinds of connections (see Tripathi [24]).

3 Doubly Warped Product Manifolds

Doubly warped product manifolds were introduced as a generalization of warped product manifolds. Let (B,gB) and (F,gF ) be two Riemannian manifolds with real dimension n1

2 + + and n2, respectively. Let h : B → R and f : F → R be two smooth functions. Consider the product manifold B × F with its projections ρ : B × F → B and σ : B × F → F . The doubly warped product Bf × hF is the product manifold B × F furnished with the metric tensor 2 ∗ 2 ∗ g =(f ◦ σ) ρ (gB)+(h ◦ ρ) σ (gF ) , where ∗ denotes pullback. If X is tangent to B × F at (p, q), then

2 2 g(X,X)= f (q)gB(dρ(X), dρ(X))+ h (p)gF (dσ(X), dσ(X)).

Thus, we have 2 2 g = f gB + h gF . (3.1) The functions h and f are called the warping functions of the doubly warped product. The manifold B is known as the base of (M,g) and the manifold F is known as the fibre of (M,g). If the warping function f or h is constant, then the doubly warped product Bf × hF reduces to a warped product B × hF˜ (or B˜f × F ), where the fibre F˜ is just F with metric 1 ˜ 1 g˜F given by h2 gF (or where the base B is just B with metricg ˜B given by f 2 gB). If the warping function f or h is equal to 1, then the doubly warped product Bf × hF reduces to a warped product B × hF (or Bf × F ). If both f and h are constant, then it is simply a ˜ ˜ ˜ 1 product manifold B × F , where the fibre F is just F with metricg ˜F given by h2 gF and the ˜ 1 base B is just B with metricg ˜B given by f 2 gB. If both f and h are equal to 1, then it is simply a product manifold B × F . The set of all smooth and positive valued functions h : B → R+ and f : F → R+are denoted by F(B)= C∞(B) and F(F )= C∞(F ), respectively. The lift of h and f to M are defined by h˜ = h◦ρ ∈F(M) and f˜ = f ◦σ ∈F(M), respectively. If Xp ∈ Tp(B) and q ∈ F , then the lift X˜(p,q) of Xp to M is the unique tangent vector in T(p,q)(B ×{q}) such that dρ(p,q)(X˜(p,q))= Xp and dσ(p,q)(X˜(p,q)) = 0. The set of all such horizontal tangent vector lifts will be denoted by L(p,q)(B). Similarly, we can define the set of all vertical tangent vector lifts L(p,q)(F ). Let X ∈ X(B), where X(B) is the set of smooth vector fields on B. The lift X˜ of X to M is the unique element of X(M) whose value at each (p, q) is the lift of Xp to (p, q). The set of such lifts will be denoted by L(B). In a similar manner, we can define L(F ). Throughout the paper, we assume that X,Y,Z ∈L(B) and U, V, W ∈L(F ). The connections ∇, B∇ and F ∇ are Levi-Civita connections on M, B and F , respectively. We will denote by R, BR and F R the Riemann curvatures on M, B and F , respectively; S, BS and F S the Ricci tensors for the connections ∇, B∇ and F ∇, respectively; r,Br and F r are the scalar curvatures for the connections ∇,B∇ and F ∇, respectively. We need the following lemmas for later use. For more details see [1, 11, 19, 25].

Lemma 3.1 Let M = Bf × hF be a doubly warped product manifold. If X,Y ∈ X(B) and V, W ∈ X(F ), then gradf tan∇ Y = − g(X,Y ), X f

nor∇X Y is the lift of ∇X Y on B, Xh V f ∇ V = ∇ X = V + X, X V h f

3 gradh nor∇ W = − g(V, W ), V h tan∇V W is the lift of ∇V W on F, where tan and nor stand for tangent part to F and normal part to B, respectively.

Lemma 3.2 Let M = Bf × hF be a doubly warped product manifold. If X,Y,Z ∈ X(B) and U, V, W ∈ X(F ), then

kgrad fk2 R(X,Y )Z = BR(X,Y )Z + (g(X,Z)Y − g(Y,Z)X), f 2 Hh (X,Y ) 1 R(X,V )Y = B V + g(X,Y )F ∇ (grad f), h f V

V f Y h Xh R(X,Y )V = X − Y , f h h   Hf (V, W ) 1 R(X,V )W = − F X − g(V, W )B∇ (grad h), f h X Xh W f V f R(V, W )X = V − W , h f f   kgradhk2 R(V, W )U = FR(V, W )U + (g(V, U)W − g(W, U)V ), h2

h f where HB and HF are the Hessian of h and f, respectively.

Lemma 3.3 Let M = Bf × hF be a doubly warped product manifold. If X,Y ∈ X(B) and V, W ∈ X(F ), then n S(X,Y ) = BS(X,Y ) − 2 Hh (X,Y ) h B kgrad fk2 ∆ f g X,Y n F , − ( ) ( 1 − 1) 2 + f f ! (n − 2)(Xh)(V f) S(X,V ) = , hf n S(V, W ) = F S(V, W ) − 1 Hf (V, W ) f F kgrad hk2 ∆ h g V, W n B , − ( ) ( 2 − 1) 2 + h h !

where ∆F f and ∆Bh are the Laplacian of f on F and h on B, respectively.

Lemma 3.4 Let M = Bf × hF be a doubly warped product manifold. Then 1 1 ∆ f ∆ h r = Br + F r − 2n F − 2n B f 2 h2 1 f 2 h kgrad fk2 kgrad hk2 −n (n − 1) − n (n − 1) . (3.2) 1 1 f 2 2 2 h2

4 4 Doubly Warped Product Manifolds Endowed with a Semi-symmetric Metric Connection

In this section, we consider doubly warped product manifolds with respect to the semi- symmetric metric connection and find new expressions concerning with curvature tensor, Ricci tensor and the scalar curvature admitting the semi-symmetric metric connection where the associated vector field P ∈ X(B) or P ∈ X(F ). We have the following results.

B Proposition 4.1 Let M = Bf × hF be a doubly warped product manifold and let ∇˜ , ∇˜ and F ∇˜ be the semi-symmetric metric connections on M, B and F , respectively. If X,Y ∈ X(B),V,W ∈ X(F ) and P ∈ X(B), then 1 tan∇˜ Y = − g(X,Y )grad f, (4.1) X f

B nor∇˜ X Y is the lift of ∇˜ X Y, (4.2) Xh V f ∇˜ V = V + X, (4.3) X h f Xh V f ∇˜ X = V + X + π(X)V, (4.4) V h f grad h nor∇˜ W = − + P g(V, W ), (4.5) V h   F tan∇˜ V W is the lift of ∇˜ V W on F, (4.6) where tan and nor stand for tangent part to F and normal part to B, respectively.

Proof. Consider P ∈ X(B) and using (2.4), first equation of Lemma 3.1, we get (4.1). Simailarly, we can easily find the others results by using (2.4) and Lemma 3.1. 

B Proposition 4.2 Let M = Bf × hF be a doubly warped product manifold and let ∇˜ , ∇˜ and F ∇˜ be the semi-symmetric metric connections on M, B and F, respectively. If X,Y ∈ X(B),V,W ∈ X(F ) and P ∈ X(F ), then grad f tan∇˜ Y = − + P g(X,Y ), (4.7) X f   B nor∇˜ X Y is the lift of ∇˜ X Y, (4.8) Xh V f ∇˜ V = V + X + π(V )X, (4.9) X h f Xh V f ∇˜ X = V + X, (4.10) V h f grad h nor∇˜ W = − g(V, W ), (4.11) V h F tan∇˜ V W is the lift of ∇˜ V W on F. (4.12)

5 Proof. Consider P ∈ X(F ) and using (2.4), first equation of Lemma 3.1, we get (4.7). Simailarly, we can easily find the others results by using (2.4) and Lemma 3.1.. 

B Proposition 4.3 Let M = Bf × hF be a doubly warped product manifold and let R,˜ R˜ and F R˜ be the Riemannian curvature tensors with respect to the semi-symmetric metric connections ∇˜ , B∇˜ and F ∇˜ , respectively. If X,Y,Z ∈ X(B), U, V, W ∈ X(F ) and P ∈ X(B), then

kgrad fk2 R˜(X,Y )Z = BR˜(X,Y )Z + (g(X,Z)Y − g(Y,Z)X) f 2 grad f +(g(Y,Z)π(X) − g(X,Z)π(Y )) , f Hh (X,Y ) V f R˜(V,X)Y = B +π(X)π(Y ) − g(Y,B ∇ P ) V + π(Y )X h X f   V f P h 1 F − P + V + π(P )V − ∇ grad f g(X,Y ), f h f V   (V f)(Y h) (V f) R˜(X,Y )V = + π(Y ) X hf f   (V f)(Xh) (V f) − + π(X) Y, hf f   (W f)(Xh) (W f) R˜(V, W )X = − π(X) V hf f   (V f)(Xh) (V f) − − π(X) W, hf f   Hf (V, W ) W f R˜(X,V )W = − F X − π(X)V f f B∇ gradh P h −g(V, W ) X + X +B∇ P h h X  gradf +π(P )X − π(X)P − π(X) , f  Uf V f R˜(U, V )W = F R(U, V )W − g(V, W )P + g(U, W )P f f kgradhk2 2P h π P g V, W U g U, W V. − 2 + + ( ) ( ( ) − ( ) h h !

Proof. Assume that M = Bf × hF is a doubly warped product and R and R˜ denote the curvature tensors with respect to the Levi-Civita connection and the semi-symmetric metric connection, respectively. In view of the (2.5), Lemma 3.1 and Lemma 3.2, we can write

kgrad fk2 R˜(X,Y )Z = BR(X,Y )Z + (g(X,Z)Y − g(Y,Z)X) f 2 B B +g(Z, ∇X P )Y − g(Z, ∇Y P )X

6 B +g(X,Z)( ∇Y P + π(P )Y − π(Y )P ) B −g(Y,Z)( ∇X P + π(P )X − π(X)P ) grad f +(g(Y,Z)π(X) − g(X,Z)π(Y )) f +π(Z)(π(Y )X − π(X)Y ). By using equation (2.5), we can write

R˜(V,X)Y = R(V,X)Y + g(Y, ∇V P )X − g(Y, ∇XP )V

−g(X,Y )[∇V P + π(P )V − π(V )P ] (4.13) +π(Y )[π(X)V − π(V )X]. Since P ∈ X(B) and by making use of Lemma 3.1 and Lemma 3.2, we get Hh (X,Y ) V f R˜(V,X)Y = B +π(X)π(Y ) − g(Y,B ∇ P ) V + π(Y )X h X f   V f P h 1 F − P + V + π(P )V − ∇ grad f g(X,Y ). f h f V   Replacing Z by V in equation (2.5), we get

R˜(X,Y )V = R(X,Y )V + g(V, ∇XP )Y − g(V, ∇Y P )X. (4.14) Using Lemma 3.1 and Lemma 3.2, we get (V f)(Y h) (V f) R˜(X,Y )V = + π(Y ) X hf f   (V f)(Xh) (V f) − + π(X) Y. hf f   By making use of (2.5) and Lemma 3.1 Lemma 3.2, we get (W f)(Xh) (W f) R˜(V, W )X = − π(X) V hf f   (V f)(Xh) (V f) − − π(X) W. (4.15) hf f   From the equation (2.5), we find

R˜(X,V )W = R(X,V )W + g(W, ∇X P )V − g(W, ∇V P )X

−g(V, W )(∇X P + π(P )X − π(X)P ). (4.16) Using Lemma 3.1 and Lemma 3.2 in (4.16), we have Hf (V, W ) W f R˜(X,V )W = − F X − π(X)V f f B∇ gradh P h −g(V, W ) X + X +B∇ P h h X  gradf +π(P )X − π(X)P − π(X) . f  7 In view of the equation (2.5), we have

R˜(U, V )W = R(U, V )W + g(W, ∇U P )V − g(W, ∇V P )U

+g(U, W )∇V P − g(V, W )∇U P +π(P )[g(U, W )V − g(V, W )U]. (4.17) By making use of Lemma 3.1 and Lemma 3.2 in (4.17), we obtain Uf V f R˜(U, V )W = F R(U, V )W − g(V, W )P + g(U, W )P f f kgradhk2 2P h π P g V, W U g U, W V. − 2 + + ( ) ( ( ) − ( ) h h ! Hence, the proof is completed.  B Proposition 4.4 Let M = Bf × hF be a doubly warped product manifold and let R,˜ R˜ and F R˜ be the Riemannian curvature tensors with respect to the semi-symmetric metric connections ∇˜ , B∇˜ and F ∇˜ , respectively. If X,Y,Z ∈ X(B), U, V, W ∈ X(F ) and P ∈ X(F ), then Y h Xh gradf R˜(X,Y )Z = BR(X,Y )Z + g(X,Z) − g(Y,Z) P − h h f     kgradfk2 2P f π P g X,Z Y g Y,Z X , + 2 + + ( ) ( ( ) − ( ) ) f f ! Hh (X,Y ) P f (Y h) R˜(V,X)Y = − B + g(X,Y )+ π(P )g(X,Y ) V − π(V )X h f h   1 1 F +g(X,Y ) π(V )gradh + π(V )P − F ∇ P − ∇ gradf , h V f V   (V f)(Y h) (Y h) R˜(X,Y )V = − π(V ) X hf h   (V f)(Xh) (Xh) − − π(V ) Y, hf h   (W f)(Xh) (Xh) R˜(V, W )X = + π(W ) V hf h   (V f)(Xh) (Xh) − + π(V ) W, hf h   1 B P f (Xh) R˜(X,V )W = −g(V, W ) ∇ gradh + X + π(P )X + P h X f h   f HF (V, W ) F (Xh) − − π(V )π(W )+ g(W, ∇V P ) X + π(W )V, f ! h R˜(U, V )W = F R˜(U, V )W kgradhk2 − (g(V, W )U − g(U, W )V ) h2 gradh +(π(U)g(V, W ) − g(U, W )π(V )) . h

8 Proof. Assume that the associated vector field P ∈ X(F ). Then the equation (2.5) can be written as

R˜(X,Y )Z = R(X,Y )Z + g(Z, ∇XP )Y − g(Z, ∇Y P )X

+g(X,Z)∇Y P − g(Y,Z)∇XP +π(P )(g(X,Z)Y − g(Y,Z)X).

By the use of Lemma 3.1 and Lemma 3.2, the above equation gives us

R˜(X,Y )Z = B R(X,Y )Z kgrad fk2 2P f π P g X,Z Y g Y,Z X + 2 + + ( ) ( ( ) − ( ) ) f f ! grad f P +(g(Y,Z)(Xh) − g(X,Z)(Y h)) − . hf h   By (4.13), Lemma 3.1 and Lemma 3.2, we obtain

Hh (X,Y ) P f (Y h) R˜(V,X)Y = − B + g(X,Y )+ π(P )g(X,Y ) V − π(V )X h f h   1 1 F +g(X,Y ) π(V )gradh + π(V )P − F ∇ P − ∇ gradf , h V f V   Replacing Z by V in equation (2.5), we get

R˜(X,Y )V = R(X,Y )V + g(V, ∇XP )Y − g(V, ∇Y P )X.

Using Lemma 3.1 and Lemma 3.2, we obtain (V f)(Y h) Y h R˜(X,Y )V = − π(V ) X hf h   (V f)(Xh) Xh − − π(V ) Y. hf h   From equation (2.5), we get

R˜(V, W )X = R(V, W )X + g(X, ∇V P )W − g(X, ∇W P )V.

By Lemma 3.1 and Lemma 3.2, we get (W f)(Xh) (Xh) R˜(V, W )X = + π(W ) V hf h   (V f)(Xh) (Xh) − − π(V ) W. hf h   From equation (2.5), we get

R˜(X,V )W = R(X,V )W + g(W, ∇X P )V − g(W, ∇V P )X

−g(V, W )∇X P − π(P )g(V, W )X + π(V )π(W )X.

9 Using Lemma 3.1 and Lemma 3.2, we get 1 B P f (Xh) R˜(X,V )W = −g(V, W ) ∇ gradh + X + π(P )X + P h X f h   f HF (V, W ) F (Xh) − − π(V )π(W )+ g(W, ∇V P ) X + π(W )V. f ! h From equation (2.5), we have

R˜(U, V )W = R(U, V )W + g(W, ∇U P )V − g(W, ∇V P )U

+g(U, W )∇V P − g(V, W )∇U P +π(P )(g(U, W )V − g(V, W )U) +(g(V, W )π(U) − g(U, W )π(U))P +π(W )(π(V )U − π(U)V ). (4.18) By use of Lemma 3.1 and Lemma 3.2in above equation, we obtain R˜(U, V )W = F R˜(U, V )W kgradhk2 − (g(V, W )U − g(U, W )V ) h2 gradh +(π(U)g(V, W ) − g(U, W )π(V )) . h Thus, we complete the proof.  As a consequence of Proposition 4.3 and Proposition 4.4, by a contraction of the curva- ture tensors we obtain the Ricci tensors of the doubly warped product with respect to the semi-symmetric metric connection as follows:

B Corollary 4.5 Let M = Bf × hF be a doubly warped product manifold and let S,˜ S˜ and F S˜ be the Ricci tensors with respect to the semi-symmetric metric connections ∇˜ , B∇˜ and F ∇˜ , respectively. If X,Y ∈ X(B),V,W ∈ X(F ) and P ∈ X(B), then n S˜(X,Y ) = BS˜(X,Y ) − 2 Hh (X,Y )+ n π(X)π(Y ) − n g(Y,B ∇ P ) h B 2 2 X kgradfk2 P h 1 n n π P n f g X,Y , − ( 1 − 1) 2 + 2 ( )+ 2 + △ ( ) f h f ! (V f)(Xh) (V f) S˜(X,V ) = (n − 1) +(n − 2) π(X), 1 hf f (V f)(Xh) (V f) S˜(V,X) = (n − 1) − (n − 2) π(X), 2 hf f n S˜(V, W ) = F S(V, W ) − 1 Hf (V, W ) f F 1 P h − divP + △h + n +(n − 1)π(P ) g(V, W ) h 1 h 1   kgradhk2 2P h n π P g V, W . −( 2 − 1) 2 + + ( ) ( ) h h !

10 B Corollary 4.6 Let M = Bf × hF be a doubly warped product manifold and let S,˜ S˜ and F S˜ be the Ricci tensors with respect to the semi-symmetric metric connections ∇˜ , B∇˜ and F ∇˜ , respectively. If X,Y ∈ X(B),V,W ∈ X(F ) and P ∈ X(F ), then

Hh (X,Y ) S˜(X,Y ) = BS(X,Y ) − n B 2 h kgradfk2 2P f n g X,Y −( 1 − 1) 2 + ( ) f f ! 1 P f − F △ f +(n − 2)π(P )+ n + divP g(X,Y ), f 2 f   (V f)(Xh) Xh S˜(X,V ) = (n − 1) − (n − 2) π(V ), 1 hf h S˜(V, W ) = F S˜(V, W ) n −n g(W,F ∇ P ) − 1 Hf (V, W )+ n π(V )π(W ) 1 V f F 1 1 B P f kgradhk2 h n n n π P g V, W . − △ + 1 +( 2 − 1) 2 + 1 ( ) ( ) h f h !

As a consequence of Corollary 4.5 and Corollary 4.6, by a contraction of the Ricci tensors we get scalar curvatures of the doubly warped product with respect to the semi-symmetric metric connection as follows:

Corollary 4.7 Let M = Bf × hF be a doubly warped product manifold and P ∈ X(B). Let r,˜ Br˜ andF r˜ be the scalar curvatures with respect to the semi-symmetric metric connections ∇˜ , B∇˜ and F ∇˜ , respectively. Then

Br˜ F r kgradfk2 kgradhk2 r˜ = + − n (n − 1) − n (n − 1) f 2 h2 1 1 f 2 2 2 h2 P h F ∆f B∆h −2n (n − 1) − 2n − 2n − 2n divP 2 h 1 f 2 h 2

−n2(n + n1 − 3)π(P ).

B Corollary 4.8 Let M = Bf × hF be a doubly warped product and P ∈ X(F ). Let r,˜ r˜ and F r˜ be the scalar curvatures with respect to the semi-symmetric metric connections ∇˜ , B∇˜ and F ∇˜ , respectively. Then

Br F r˜ kgradfk2 kgradhk2 r˜ = + − n (n − 1) − n (n − 1) f 2 h2 1 1 f 2 2 2 h2 P f F ∆f B∆h −2n (n − 1) − 2n − 2n − 2n divP 1 f 1 f 2 h 1

−n1(n + n2 − 3)π(P ).

Remark 4.9 Doubly warped product manifolds with the semi-symmetric metric connection has been also studied by Sular [21], where the author has taken the associated vector field P ∈ X(M) as P = PB + PF , PB and PF are the components of P on B and F , respectively.

11 5 Einstein doubly warped product manifolds endowed with the semi-symmetric metric connection

In this section, we consider Einstein doubly warped products and Einstein-like doubly warped product of class A endowed with the semi-symmetric metric connection.

Theorem 5.1 Let M = Bf × hF be a doubly warped product manifold (n > 2) and P ∈ X(B). Then (M, ∇˜ ) is an Einstein manifold with Einstein constant µ if and only if n BS˜(X,Y ) = 2 Hh (X,Y ) − n π(X)π(Y )+ n g(Y,B ∇ P ) h B 2 2 X kgradfk2 P h 1 n n π P n f µ g X,Y , + ( 1 − 1) 2 + 2 ( )+ 2 + △ + ( ) (5.1) f h f !

n F S(V, W ) = 1 Hf (V, W ) f F 1 P h + divP + △h + n +(n − 1)π(P ) g(V, W ) h 1 h 1   kgradhk2 2P h n π P µ g V, W . +( 2 − 1) 2 + + ( )+ ( ) (5.2) h h ! Proof. The proof follows from Corollary 4.5 and Corollary 4.6. 

Theorem 5.2 Let (M, ∇˜ ) be an Einstein doubly warped product manifold with Einstein constant µ and P ∈ X(B). Suppose that B is compact Riemannian manifold, F is complete f Riemannian manifold, n1 , n2 ≥ 2, and HF (V, W ) is a constant multiple of gF . If µ ≤ 0, P h ≥ 0, π(P ) ≤ 0, divP ≤ 0 then M is a warped product manifold.

f Proof. Since HF (V, W ) is a constant multiple of gF , so by using the result of Tashiro [23, Theorem 2], we can say that F is a Euclidean space, then Ricci tensor of F is zero. By (5.2), we get cn 0 = 1 +(n − 1) kgradhk2 +2hP h + h2π(P )+ µh2 f 2  2 2 + h divP + h△h+ n1hP h +(n1 − 1)h π(P ) gF (V,
W ). So
cn 0 = 1 +(n − 1) kgradhk2 +2hP h + h2π(P )+ µh2 f 2  2 2 + h divP + h△h+ n1hP h +(n1 − 1)h π(P ) ,
(5.3) Let x ∈ B such that h(x) is maximum of h on B. Therefore gradh(
x) = 0 and △h(x) ≤ 0. Then P h(x)= g(gradh(x),P ) = 0. So the eq(5.3) at the point x is

2 2 0 = (n2 − 1) h (x)π(P )+ µh (x) 2 2 + h (x)divP + h(x)△h(x)+(n
1 − 1)h (x)π(P ) . (5.4)
12 By eq(5.3) and eq(5.4), we have

cn 0 = 1 +(n − 1) kgradhk2 +2hP h +(h2 − h2(x))π(P )+ µ(h2 − h2(x)) f 2  2 2 +(h − h (x))divP + h△h − h(x)△h(x)+ n1hP h
2 2 +(n1 − 1)(h − h (x))π(P ) . (5.5)

Since µ ≤ 0, P h ≥ 0, π(P ) ≤ 0, divP ≤
0, so equation (5.5) implies that h△h ≤ 0, which shows that Laplacian has constant sign and hence h is constant. 

Theorem 5.3 Let (M,g) be an Einstein doubly warped product manifold M = If × hF with respect to the semi-symmetric metric connection, where I is an open interval, dimI =1 and dimF = n − 1(n ≥ 3). If P ∈ X(I), then f is constant on F or P h = hπ(P ).

Proof. Let (M,g) be a doubly warped product M = If × hF , where I is an open interval, dimI = 1 and dimF = n − 1(n ≥ 3). By Corollary 4.5, we have

V f S˜(X,V )=(n − 2) π(X), f

(V f) Xh S˜(V,X)=(n − 2) − π(X) . (5.6) f h   Since M is an Einstein manifold with respect to the semi-symmetric metric connection, we can write S˜(P,V )= αg(P,V ), where α is constant. But g(P,V ) = 0, therefore S˜(P,V ) = 0. By (5.6), we have either P h V f =0 or = π(P ). Therefore either f is constant or P h = hπ(P ).  h

Definition 5.4 [22] A Riemannian manifold (M,g) is said to admit a cyclic-Ricci parallel tensor or is Einstein-like of class A if

(∇X S)(Y,Z)+(∇Y S)(Z,X)+(∇ZS)(X,Y )=0

for any vector fields X,Y,Z ∈ X(M) or equivalently (∇X S)(X,X) = 0.

Proposition 5.5 Let M = Bf × hF be an Einstein-like doubly warped product manifold of class A with respect to the semi-symmetric metric connection ∇˜ and P ∈ X(B). The Riemannian manifold B is an Einstein-like manifold of class A with respect to the semi- ˜ B Xh B symmetric metric connection ∇ if and only if ∇X Y = 2h Y , ∇X P =0 and

Xh kgradhk2 n π X π X g X,X 0 = 2 ( ) ( ) − 2 ( ) h 2h ! Xh kgradfk2 1 g X,X n n π P n f . + ( ) ( 2 − 1) 2 − 2 ( ) − 2 △ h f f !

13 Proof. By using (4.1), (4.2) and Corollary4.5, we have

(∇˜ X S˜)(X,X) = XS˜(X,X) − S˜(∇˜ X X,X) − S˜(X, ∇˜ X X) n = X BS˜(X,X) − 2 Hh (X,X)+ n π(X)π(X) − n g(X,B ∇ P ) h B 2 2 X  kgradfk2 P h 1  X n n π P n f g X,X − ( 1 − 1) 2 + 2 ( )+ 2 + △ ( ) f h f ! ! 1 1 −S˜(∇˜ B X − g(X,X)gradf,X) − S˜(X, ∇˜ B X − g(X,X)gradf), X f X f

n n (∇˜ S˜)(X,X) = (∇˜ B S˜)(X,X)+ 2 XhHh (X,X) − 2 X Hh (X,X) X X h2 B h B 2 B +n2X(π(X)) − n2Xg(X, ∇X P )
kgradfk2 P h 1 n n π P n f Xg X,X − ( 1 − 1) 2 + 2 ( )+ 2 + △ ( ) f h f ! kgradfk2 P h 1 n X n Xπ P n X X f g X,X − ( 1 − 1) 2 + 2 ( )+ 2 + △ ( ) f h f ! 1 1 + g(X,X)S˜(gradf,X)+ g(X,X)S˜(X, gradf). f f

B Xh B Since ∇X Y = 2h Y and ∇X P = 0, we have

Xh kgradhk2 ˜ S˜ X,X ˜ B S˜ X,X n π X π X g X,X (∇X )( ) = (∇X )( )+ 2 ( ) ( ) − 2 ( ) h 2h ! Xh kgradfk2 1 g X,X n n π P n f . + ( ) ( 2 − 1) 2 − 2 ( ) − 2 △ h f f !

The proof is completed. 

Remark 5.6 Einstein-like manifolds are natural extension of Einstein manifolds. Einstein- like manifolds admitting different curvature conditions were considered by Calvaruso [6]. Einstein-like manifolds of dimension 3 and 4 are studied in [3, 5]. Projective spaces and spheres furnished with class A or class B Einstein-like metrics were classified in [18]. An interesting study in [15] shows that Einstein-like Generalized Robertson-Walker spacetimes are perfect fluid space-times except one class of Gray’s decomposition.

Acknowledgement: We thank the reviewer for valuable suggestions and corrections.

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Department of Mathematics & Statistics School of Mathematical & Physical Sciences Dr. Harisingh Gour University Sagar-470003, Madhya Pradesh punam [email protected] and Universit´eAlioune Diop de Bambey UFR SATIC, D´epartement de Math´ematiques ER-ANLG B.P. 30, Bambey, S´en´egal [email protected]

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