Conformal Ricci soliton in an indefinite trans-Sasakian manifold

In 1982 Hamilton [3] discovered that the Ricci solitons move under the Ricci flow simply by diffeomorphisms of the initial metric; that is, they are stationary points of the Ricci flow given by

= -28. ∂t

(1.1)

In 2004 Fischer [4] introduced the concept of conformal Ricci flow which is a variation of the classical Ricci flow equation. In Ricci flow equation the unit volume constraint plays a important role but in conformal Ricci flow equation scalar curvature r is considered as constraint.

dg + 2 (S + g) = -pg, ∂t          n

(1.2)

where p is a scalar non-dynamical field and n is the dimension of the manifold.

In the year 2015, Basu and Bhattacharyya [1] introduced the notion of conformal Ricci soliton equation as:

L v g + 2S = 2A

-

(p+1).

g.

(1.3)

In 1985 J. A. Oubina [5] introduced a new class of almost contact manifold namely trans-Sasakian manifold.

(c) 2021 Girish Babu, S., Reddy, P. S. K. and Somashekhara, G.

2.    Preliminaries

A smooth manifold (M ⁿ , g) is said to be indefinite almost contact metric manifold, if there exists a a (1,1) tensor field y , structure vector field e , a 1-form n and an indefinite metric g such that (see [2]):

y2Xi = —Xi + n(Xi)e, ye = 0, n(yXi) = 0, n(0 = 1, g^) = e, (2.1) n(Xi)= eg(e,Xi), g(y(Xi),y(Yi)) = g(Xi,Yi) — en(Xi)n(Yi),(2.2)

g(yXi,Yi) = —g(Xi,yYi), g(yXi,Xi) =0,(2.3)

for all vector fields X i , Y i on manifold M , where e = ± 1 accordingly as £ is space like vector field and rank y is n — 1 .

If dn(Xi ,Yi) = g(Xi,yY),(2.4)

then M n (y,£,,n, g) is called an indefinite contact metric manifold.

Indefinite almost contact metric manifold is called an indefinite trans-Sasakian manifold if it is of the form

Vxi yY = a(g(Xi,Yi)^ — en(Yi)Xi) + в(g(yX1 , Yi)e — en(Yi)yXi),(2.5)

for any X i Y i G Г(Т M ) , where V is a metric connention of indefinite metric g , a and в are smooth function on a manifold M ⁿ .

On using (2.1), (2.2), (2.3), (2.4) and (2.5), we get

Vxi£ = e [—ayXi + в№ - nXi)<\ ,(2.6)

(Vxin) Yi = —ag(yXi,Yi) + в[g(X1,Y1) — en(Xi)n(Yi)].(2.7)

The indefinite trans-Sasakian manifold M ⁿ , the following relation holds:

R(Xi,Yi)e = (a2 — в2) (n(Yi)Xi — n(Xi)Yi) + 2ав (n(Yi)yXi — n(Xi)y(Yi))8)

+ e ((Yia)yXi — (Xia)yYi + (Y1в)y2X1 — (X1в)y2Y1) ,

R«, Yi)Zi = (a2 — в2) (eg(Yi, Zi)e — n(Zi)Yi) + 2ав (eg(yYi, yZi)e + n(Zi)yYi)9)

+ e(Z i a)yY i + eg(Y i , yZ i )(grada) — eg(yY i , yZ i )(gradв) + e(Z^) (Y i — n(Z i )£),

S(Z i , e) = ((n — 1)(a ² — в ² ) — е^в)) n(Z i ) — e(n — 2)(Z i e),          (2.10)

S (e, e) = (n — 1)(a ² — в ² ) — e(n — 1)(ев),                     (2.11)

Qe = e(n — 1)(a ² — в 2 )е — (ев)е + ey(grada) — e(n — 2)(grad в),       (2.12)

where R is the Riemannian curvature tensor, S is the Ricci tensor and Q is the Ricci operator. Then we have that S(X _i , Y _i ) = g(QX _i , Y _i ) ( V X _i , Y _i G r(TM)).

Now from equation 1.3, we have

S (X i ,Y i ) = A i g(X i ,Y i )+ A 2 n(X i )n(Y i ),                   (2.13)

where A i = 2 (2A — (p + П ) — ев) , A 2 = ев

QX i = A i X i + A 2 n(X i )e,                         (2.14)

S (X i ,e) = A 4 n(X i ),                                 (2.15)

where A 4 = (eA i + A 2 )

Qe = A 3 e,

(2.16)

where A 3 = A i + A 2 .

3.    Conformal Ricci Soliton in an Indefinite Trans-SasakianManifold Satisfying R(^,X1 ).C = 0

Let a n -dimensional conformal Ricci soliton in an indefinite trans-Sasakian manifold satisfying R(^,X i ).C = 0 , where C is quasi conformal curvature tensor on a manifold M and is defined by

C(X 1 ,Y 1 )Z 1 = aR(X i ,Y i )Z i + b(S (Y i , Z i )X i — S (X i , Z i )Y i + g(Y i , Z i )QX i

— g(Xi, Zi)QYi) — (7—тугf —+ 2b) (g(Yi, Zi)Xi — g(Xi, Z^Yi), \2n + 1 / \2n / where r is scalar curvature.

Substituting Z i = £ , we get

C (X 1 ,Y1K = aR(X i ,Y i )e + b(S(Y i ,C)X i — S(X i , ОТ + g(Y i ,0QX i

— g(X i ,^)QY i ) — () f + 2b) (g(Y i , № — g(X i , №)• \ 2 iL ^—+ 1 у \ 2 n      /

Using equation (2.2), (2.8), (2.13) and (2.14) in (3.2), we get

C (X i ,Y i )e = A 5 (n(Y i )X i — n(X i )Y i ),                       (3.3)

where A 5 = (a(a ² — в ² ) + bA 4 + ЬеА 1 — e equation (3.3) becomes

2п+т) ( 2n + 2b/) . Taking inner product with Z i

—n(C (X 1 ,Y 1 ),Z 1 )= A 5 E (n(Y i )g(X i ,Z 1 ) — n(X i )g(Y i ,Z 1 )).           (3.4)

We assume that R(^,X i ).C = 0 , which implies that

R(e,Xi)(C (Yi,Zi)Z2) — C (R(^,Xi)Yi,Zi)Z2 — C (Yi ,R(e,Xi )Z1)Z2 — C (Y1,Z1 )R(^,X1 )Z2 = 0, for all vector fields X1 , Y1 , Z1 , Z2 on a manifold M .

Putting Z 2 = ^ and using (2.9) in (3.5), we get

E(a ² — e ² )g(X i , CC(Y 1 , Z 1 )^)^ — E(a ² — e 2 )g(X i , Y i )C(e, Zi)C + (a ² — в 2 )n(Y i )C(X i , Z i )€ — e(a² — e 2 )g(X i ,Z i )C(Y i ,0^ + (a ² — e ² )n(Z i )C(Y i , Z i )€ — (a ² — в 2 )п(Х 1 )C(Y i , Zi)^ + (a ² — e ² )(5(Y i ,Z i )X i = 0,

Taking inner product with ξ and using (2.2), (3.3), equation (3.6) reduces to

g(X i , C(Y i , Z i X) + n(C (Y i , Z i )X i ) = 0,                       (3.7)

provided (a ² — в ² ) = 0.

Substituting Z i = ^ and using (3.3) in (3.7), we obtain

(3.8)

A 5 g(X i , Y i ) — A5^n(X i )n(Y) + n(C (Y i ,^)X i ) = 0.

Again substituting Y i — £ in (3.1), we get

C (X i ,^)Z i — aR(X i ,OZ i + b(S(£, Z i )X i — S(X i , Z i )£ + g«, Z i )QX i

^{— g(}X i , ZJQC) ^— f5 tt^ (9—^{+ 2b})^(g(^’ Z i ^)X i ^{— g(X} i ’ Z i D, 2n + 1   2n

Taking inner product with ξ and using (2.1), (2.2), (2.9), (2.10), (2.11), (2.12), equation (3.9) reduces to

(3.10)

n(C(Xi,e)Zi) — Абд(Х1, Zi) + A7n(Xi)n(Zi) — bS(Xi, Zi), where

А б — ^—Еа(а ² — в ² ) - Ье ( А 1 + А 2 ) + А 7 — ^а(а ² — в 2 ) + Ье ( А 1 + A 3 + А 4 )

(  .+D),

replacing X 1 with Y 1 and Z 1 with X 1 in (3.10), we obtain

n(C(Y i ,№) — А б д(Х 1 ,Y i ) + A 7 n(X i )n(Y i ) — bS(X i ,Y i ).

• (3.11)

Substituting (3.11) in (3.8), we get

S (X 1 ,Y 1 ) = A 8 g(X i ,Y 1 ) + A 9 n(X i )n(Y i ),

• (3.12)

4. Conformal Ricci Soliton in an Indefinite Trans-Sasakian

Manifold Satisfying R(^,X _i ).S — 0

where A 8 — A 5 + А б , A 9 — A 7 — ЕА 5 .

Hence we can state the following theorem

Theorem 3.1. A conformal Ricci soliton in an indefinite trans-Sasakian manifold satis fying R(£,X i )C — 0 is an п-Einstein manifold.

Leta a n -dimensional conformal Ricci soliton in an indefinite trans-Sasakian manifold satisfying R(£, X i ).S — 0 , which implies that

S (R(e,X i )Y i ,Z i ) + S(Y i ,R(^,X i )Z i ) — 0.

Using (2.1), (2.2), (2.9) and (2.13) in (4.1), we get

A i ^((a 2 ^{— e} 2 ^)Eg(X i , Y i ⁾n^(Z i ) ^{— (a} 2 ^{— в}2'⁾п⁽У 1 ^)д(х 1 , Z i )) ⁺ A i ^((a 2 ^{— e} 2 ^)Eg(X i , ^Z i ⁾n^(Y i ^{) — (a} 2 ^{— e 2 )}n^(Z i ^)g(X i , YD) ⁺ A 2 ^(a 2 ^{— e} 2 ⁾⁽g⁽X i ,Y i ⁾n^(Z i ) ^{— E}n^(X i )n^(Y i )n^(Z i ) ⁺ g⁽X i .ZiMY i ) ^{— E}n⁽X i )n^(Y i ⁾n⁽Z i ^{)) —} 0.

Substituting Z i — ^ and using (2.1), (2.2) in (4.2), we get

g(Xi,Yi) — Еn(Xi)n(Yi), provided A2(a2 — в2) — 0.

Hence, we state the following theorem

Theorem 4.1. A conformal Ricci soliton in an indefinite trans-Sasakian manifold satisfying R«,X i )S — 0, then g(X i , Y i ) — En(X i )n(Y i ).

(4.1)

• (4.2)

• (4.3)

5. Conformal Ricci Soliton in an Indefinite Trans-Sasakian

Manifold Satisfying R(£, X _i ).P = 0

Let a n -dimensional conformal Ricci soliton in an indefinite trans-Sasakian manifold satisfying R(^, X i ).P = 0 , where P is projective curvature tensor on a manifold M and is defined by

P (X i ,Y i )Z i = R(X i ,Y i )Z i - ^(S(Y i ,Z i )X i - S (X i ,Z i )Y i ) . 2n

• (5.1)

We assume that R(£,X i ).P = 0 , which implies that

-

R(e,X i )(P(Y i ,Z i )Z 2 ) - P (R(£,X i )Y i ,Z i )Z 2

P(Y i , R(£, X i )Z i )Z 2 - P (Y i ,Z i )R(£,X i )Z 2 = 0,

• (5.2)

for all vector fields X 1 , Y 1 , Z 1 and Z 2 on M .

Putting Z i = £ and using equation (2.9) in (5.2), we get

_Eg(X i ,P (Y i ,C)Z 2 )£ - n((P (Y i ,£)Z 2 )X i ) - i ,Y )P (£,№ + n(Y )P (X i ,£ )Z 2 - en(X i )P(Y i ,C)Z 2 + P (Y i ,X i )Z 2 - eg(X i ,Z 2 )P(Y i ,^)^ + n(Z 2 )P(Y i ,^)X i = 0.

• (5.3)

Substituting Y i = £ in equation (5.1), we get

P (X i ,e)Z i = R(X i ,^)Z i - ^(S(^,Z i )X i - S(X i ,Z i )e), 2n

• (5.4)

using equation (2.9) and (2.15) in (5.4), we get

P(X i ,0Z i = - e(a ² - e ² )g(X i ,Z i )€

+ (^(a 2 ^{- e} 2 )

-

-A⁴-) n(Z i )X i + - n - 1           n

-

1 S (X i ,Z i )£.

(5.5)

Replacing X 1 with Y 1 and Z 1 with Z 2 in (5.5), we get

P(Y i ,C)Z 2 = - e(a ² - e ² )g(Y i ,Z 2 )e

+ (^(a 2 - ^e 2 )

-

-A⁴-) n(Z 2 )Y i + - n - 1           n

-

1 S (Y i ,Z 2 )^.

(5.6)

Now substituting Z 2 = £ and using (5.6) in (5.3), we get

4^47) g(X i ,Y i X + An(Y i )X i - A ⁴ n(X i )Y i + у n — 1/            2n          2n

S(X i ,Y i )£ = 0. n - 1

(5.7)

Taking inner product with ξ and using (2.1), (2.2), equation (5.7) becomes

S (X i ,Y i ) = - A io g(X i ,Y i ),                             (5.8)

where A io = EA 4 Hence we can state the following theorem

Theorem 5.1. A conformal Ricci soliton in an indefinite trans-Sasakian manifold satis fying R(£,X i )P = 0 is an Einstein manifold.

6. Conformal Ricci Soliton in an Indefinite Trans-SasakianManifold Satisfying R(^,X1 )P = 0

Let a n -dimensional conformal Ricci soliton in an indefinite trans-Sasakian manifold

.“^

satisfying R(^,X i )P = 0 , where P is pseudo projective curvature tensor on a manifold M and is defined by

P(X i ,Y i )Z i = aR(X i ,Y i )Z i + b(S(Y i ,Z i )X i - S(X i ,Z i )Y i )

- r ) (g(Y i ,Z 1 )X 1 - g(X i ’Z i )Y i ).

n n - 1

We assume that R(^,X i )P = 0 , which implies that

(6.1)

-

^^

^^

R(^X i )(P ( Y i ,Z i ) Z 2 ) - P (R(^,X i )Y i ,Z i )Z 2

^^

^^

P (Y i ,R(^,X i )Z 1 )Z 2 - P (Y i ,Z i )RU,X i )Z 2 = 0,

(6.2)

for all vector field X 1 , Y 1 , Z 1 and Z 2 on M .

Putting Z 2 = ^ and using (2.9) in (6.2), we get

A ii g(X i , Y i )Z i + A i2 g(X i , Z i )Y i + P (Y i , Z i )X i = 0, provided (a ² — в ² ) = 0 and where

(6.3)

A ii = ^aE + bA₄E^~

a

-

1+ b))’ A 12 = (aE — bA 4 E + Г(п

a

-

1+b))-

Substituting Z i = £ in (6.3), we get

A ii g(X i , Y i К + A i2 g(X i , £) Y i + P (Y i ,<)X i = 0.

Taking inner product with ξ and using (2.1), (2.2), equation (6.4) becomes

A i4 g(X i ,Y i ) + A i3 n(X i )n(Y i ) + n(P (Y i ,^)X i ) = 0, where A 13 = EA 12 ,  A 14 = еАц. In the view of (6.1) and (6.5) we have

S (X i , Y i ) = A i5 g(X i , Y i ) + A i6 n(X i )n(Y i ),

(6.4)

(6.5)

(6.6)

where

A 11

A 15 =  ---

-

a(a 2 - в 2) • bε

-

ε ^r n

,

A 16 =

A13 + aE(a2 - в2) + bA4E + n bε

Hence we can state the following theorem

Theorem 6.1. A conformal Ricci soliton in an indefinite trans-Sasakian manifold satis fying R(^,X i )P = 0 is an п-Einstein manifold.

Acknowledgment. The authors would like to thank the anonymous referee for his comments that helped us improve this article.