CONFORMAL RICCI SOLITON IN AN INDEFINITE TRANS-SASAKIAN MANIFOLD Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2021, Volume 23, Issue 3, P. 45-51

YAK 517.7

DOI 10.46698/t3715-2700-6661-v

CONFORMAL RICCI SOLITON IN AN INDEFINITE TRANS-SASAKIAN MANIFOLD

S. Girish Babu1, P. S. K. Reddy2 and G. Somashekhara2

1 Acharya institute of Technology, Bengaluru 560107, Karnataka, India; 2 JSS Science and Technology University, Mysuru 570006, Karnataka, India; 3 Ramaiah University of Applied Sciences, Bengaluru 560054, Karnataka, India E-mail: [email protected], [email protected], somashekhara.mt,[email protected]

Abstract. Conformal Ricci solitons are self similar solutions of the conformal Ricci flow equation. A new class of n-dimensional almost contact manifold namely trans-Sasakian manifold was introduced by Oubina in 1985 and further study about the local structures of trans-Sasakian manifolds was carried by several authors. As a natural generalization of both Sasakian and Kenmotsu manifolds, the notion of trans-Sasakian manifolds, which are closely related to the locally conformal Kahler manifolds introduced by Oubina. This paper deals with the study of conformal Ricci solitons within the framework of indefinite trans-Sasakian manifold. Further, we investigate the certain curvature tensor on indefinite trans-Sasakian manifold. Also, we have proved some important results.

Key words: indefinite trans-Sasakian manifold, trans-Sasakian manifold, Ricci flow, conformal Ricci flow. Mathematical Subject Classification (2010): 53C15, 53C20, 53C25, 53C44.

For citation: Girish Babu, S., Reddy, P. S. K. and Somashekhara, G. Conformal Ricci Soliton in an Indefinite Trans-Sasakian Manifold, Vladikavkaz Math. J., 2021, vol. 23, no. 3, pp. 45-51. DOI: 10.46698/t3715-2700-6661-v.

1. Introduction

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

I = <">

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.

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:

Lvg + 2S= 2A- (p + -

n

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

g. (1.3)

© 2021 Girish Babu, S., Reddy, P. S. K. and Somashekhara, G.

2. Preliminaries

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

= -Xi + n(Xi)e, = 0,n(^Xi) = o, n(0 = i,g(£-0= e, (2.1)

n(Xi) = eg(£,Xi), 0(p(Xi),^(Y1)) = g(Xi,Yi) - en(Xi)n(Yi), (2.2)

g(^Xi,Y1) = -g(Xi,^Yi), g(^Xi,Xi) = 0, (2.3)

for all vector fields Xi, Yi on manifold M, where e = ±1 accordingly as £ is space like vector field and rank ^ is n — 1. If

dn(Xi, Yi) = g(Xi,^Yi), (2.4)

then Mn(^,£,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 ^Yi = a(g(Xi, Yi)£ — en(Y)Xi) + P(g(pXi, Yi)£ — en(Yi)pXi), (2.5)

for any Xi Yi e r(TM), where V is a metric connention of indefinite metric g, a and P are smooth function on a manifold Mn.

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

Vxi£ = e [—a^Xi + p(Xi — nXi)£], (2.6)

(VXln) Yi = —ag(^Xi,Yi) + P[g(Xi, Yi) — en(Xi)n(Y)]. (2.7) The indefinite trans-Sasakian manifold Mn, the following relation holds:

R(Xi, Yi)£ = (a2 — P2) (n(Yi)Xi — n(Xi)Yi) + 2aP (n(Y )^Xi — n№MYi)) (2 g)

+ e ((Yia)pXi — (Xia)^Yi + (YiP)^2Xi — (XiP)^2Yi) , ( . )

R(£, Yi)Zi = (a2 — P2) (eg(Yi, Zi)£ — n(Z)Yi) + 2aP (egfrY, + n(Zi)^i) (2 g) + e(Zia)^Yi + eg(Yi,^Zi)(grada) — eg(^Yi)(gradP) + e(ZiP) (Yi — n(Z)£), ( . ) S(Zi,£) = ((n — 1)(a2 — P2) — e(£P)) n(Z) — e(n — 2)(ZiP), (2.10)

S (£, £) = (n — 1)(a2 — P2) — e(n — 1)(£P), (2.11)

Q£ = e(n — 1)(a2 — P2)£ — (£P)£ + e^(grada) — e(n — 2)(gradP), (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(Xi, Yi) = g(QXi, Yi) (VXi, Yi e r(TM)). Now from equation 1.3, we have

S (Xi, Yi) = Aig(Xi ,Yi)+ A2n(Xi)n(Yi), (2.13)

where A1 = ± (2A - (p + §) - e/3), A2 = efj

QXi = AiXi + A2n(Xi )£, (2.14)

S (Xi,£)= A4n(Xi), (2.15)

where A4 = (eAi + A2)

Q£ = As£, (2.16)

where = Ai + A2.

3. Conformal Ricci Soliton in an Indefinite Trans-Sasakian Manifold Satisfying R(£,X1 ).C = 0

Let a n-dimensional conformal Ricci soliton in an indefinite trans-Sasakian manifold satisfying R(£,X1).<C = 0, where C is quasi conformal curvature tensor on a manifold M and is defined by

C(Xl,Yl)Zl = aR(Xi,Yi)Zi + b(S(Yi, Zi)Xi - S(Xi, Z1)Y1 + g(Yi, Zl)QXl

r \ / a 2n + l ) V2n

-g(Xu Zi)QY\) - (¿pï) + 2b) (9(Xi, Zi)Xi - g(Xu Z^Y,),

(3.1)

where r is scalar curvature. Substituting Z1 = £, we get

C(Xi,Yi)£ = aR(Xi,Yi)Ç + b(S(Yu£)Xi - S(Xu£)Yi + g(Yi,£)QXi -g(Xl,0QYl)-(^T^l (¿+26) (g(Yl,£)Xl-g(Xl,£)Yl).

(3.2)

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

(J(X1,Y1 )£ = A5(n(Y1)X1 - n(Xi)Yi), (3.3)

where A5 = (a(a2 - (32) + M4 + beAi - e (^Fi) + • Taking inner product with Z\ equation (3.3) becomes

-n(C(Xi,Yi), Zi) = A5£ (n(Yi)g(Xi, Zi) - n(Xi)g(Yi ,Zi)). (3.4)

We assume that R(£,Xi).C = 0, which implies that

R(Ç,Xi)(C(Yi,Zi)Z2) - C(R(£,Xi)Yi,Zi)Z2 - C(Yi,R(£,Xi)Zi)Z2 - C(Yi,Zi)R(£,Xi)Z2 = 0,

(3.5)

(3.6)

for all vector fields Xi,Yi, Zi, Z2 on a manifold M. Putting Z2 = £ and using (2.9) in (3.5), we get

£(a2 - $2)g(Xi,C(Yi, Zi)£)£ - e(a2 - $2)g(Xi,Yi)C(£, Zi)£ + (a2 - $2)n(Yi)C(Xi, Zi)£ - e(a2 - $2)g(Xi, Zi)C(Yi,£)£ + (a2 - f32)V(Zi)C(Yi, Zi)£ - (a2 - $2)n(Xi)C(Yi, Zi)£ + (a2 - f32)C(Yi,Zi)Xi =0,

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

g(Xi,C(Yi, Zi)£) + n(C(Yi, Zi)Xi) = 0, (3.7)

provided (a2 - $2) = 0.

Substituting Zi = £ and using (3.3) in (3.7), we obtain

A5g(Xi,Yi) - A5£n(Xi)n(Yi) + n(C(Yi,£)Xi) = 0. (3.8)

Again substituting Yi = £ in (3.1), we get

C(Xi, £)Zi = aR(Xi, £)Zi + b(S(£, Zi)Xi - S(Xi, Zi)£ + g(£, Zi)QXi

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

n(C(Xi,£)Zi) = A6g(Xi,Zi)+ Arn(Xi)n(Zi) - bS(Xi,Zi), (3.10)

where

* = (-«.(o» - f) - + + + 26) e) ,

* = ("(Q'2 - "2)++-43+~ (¿n) (£+2i>))

replacing Xi with Yi and Zi with Xi in (3.10), we obtain

n(C(Yi,£)Xi) = A6g(Xi,Yi)+ A7n(Xi)n(Yi) - bS(Xi,Yi). (3.11)

Substituting (3.11) in (3.8), we get

S (Xi, Yi) = Asg(Xi, Yi) + Aon(Xi)n(Yi), (3.12)

where A8 = A5 + A6, A9 = A7 - eA5.

Hence we can state the following theorem

Theorem 3.1. A conformal Ricci soliton in an indefinite trans-Sasakian manifold satisfying R(£,Xi)C = 0 is an n-Einstein manifold.

4. Conformal Ricci Soliton in an Indefinite Trans-Sasakian Manifold Satisfying R(£,Xi).S = 0

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

S (R(£,Xi )Yi,Zi) + S(Yi,R(£,Xi)Zi) = 0. (4.1)

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

Ai((a2 - P2)eg(Xi,Yi)n(Zi) - (a2 - p2)n(Yi)g(Xi,Zi)) + Ai ((a2 - p2)eg(Xi, Zi)n(Yi) - (a2 - p2)n(Zi )g(Xi, Yi)) (4 2)

+ A2(a2 - P2)(g(Xi, Yi)n(Zi) - en(Xi)n(Yi)n(Zi) .

+ g(Xi, Zi)n(Yi) - en(Xi)n(Yi)n(Zi)) = 0.

Substituting Zi = £ and using (2.1), (2.2) in (4.2), we get

g(Xi,Yi)= en(Xi)n(Yi), (4.3)

provided A2(a2 - P2) = 0.

Hence, we state the following theorem

Theorem 4.1. A conformal Ricci soliton in an indefinite trans-Sasakian manifold satisfying R(£,Xi)S = 0, then g(Xi, Yi) = en(Xi)n(Yi).

5. Conformal Ricci Soliton in an Indefinite Trans-Sasakian Manifold Satisfying R(£,X^.P = 0

Let a n-dimensional conformal Ricci soliton in an indefinite trans-Sasakian manifold satisfying R(£,X1).P = 0, where P is projective curvature tensor on a manifold M and is defined by

P(X1,Y1)Z1 = R{X\, Y\)Z\ - i- (S(Yi, Zi)Xi - S(Xi, ZJYJ . (5.1)

+ (a2 - ß2)--r1(Z1)X1 +--S(X1} Z1)C-

n — 1 n — 1

Replacing X1 with Y1 and Z1 with Z2 in (5.5), we get

P №,£)Z = —e(a2 — ß2)g(Y1,Z2)£

(5.2)

(5.3)

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

R(£, Xi)(P(Yi, Zi)Z2) — P(R(£,Xi)Yi, Zi)Z — P (Yi,R(£,Xi)Zi)Z2 — P (Yi ,Zi)R(£,Xi)Z2 = 0,

for all vector fields Xi, Yi, Zi and Z2 on M.

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

eg(Xi ,P (Yi,£)Z2 )£ — n((P (Yi ,£ )^)Xi) — eg(Xi ,Yi )P (£,£)Z + n(Yi )P (Xi ,£ )Z — en(Xi)P(Yi, £)Z + P(Yi, Xi)Z — eg(Xi, Z)P(Yi, £)£ + n(Z)P(Yi, £)Xi = 0.

Substituting Yi = £ in equation (5.1), we get

P{Xl,i)Zl = R{Xl,i)Zl - ^(S(C,Z1)X1-S(X1,Z1)0, (5.4)

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

P(Xi, £)Zi = —e(a2 — P2)g(Xi, Zi)£

2 A4 V./^N^ , 1 ry ^ (5.5)

2 a2\ A4 \ ^ , 1 ^ ^ (5.6)

+ (a2 - ß--V(Z2)Y1 +--S(Yi, Z2)£

n— 1 n— 1

Now substituting Z2 = £ and using (5.6) in (5.3), we get

/ A \ A A 1

£ (^l) K*'.™ + - + —- 0. (5.7)

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

S (X1 ,Y1) = — A10g(X1,Y1), (5.8)

where A10 = eA4 Hence we can state the following theorem

Theorem 5.1. A conformal Ricci soliton in an indefinite trans-Sasakian manifold satisfying R(£,X1 )P = 0 is an Einstein manifold.

6. Conformal Ricci Soliton in an Indefinite Trans-Sasakian Manifold Satisfying R(£,X1 )P = 0

Let a n-dimensional conformal Ricci soliton in an indefinite trans-Sasakian manifold satisfying R(£,X1)P = 0, where P is pseudo projective curvature tensor on a manifold M and is defined by

P(Xi,Yi)Zi = aR(Xi, Yi )Zi + b(S (Yi, Zi)Xi - S (Xi, Zi)Yi)

(g(Yi,Zi)Xi - g(Xi,Zi)Yi).

where

/ An-a(a2-ß2)-e^ + b Al5 = ^-te-

A13 + ae(a2 - ß2) + bÄAe + £ + ft)

Air =

(6.1)

(6.2)

n \n - 1

We assume that R(£,Xi)P = 0, which implies that

R(£,Xi)(P(Yi, Zi)Z2) - P(R(£, Xi)Yi, Zi)Z2 - P(Yi,R(£,Xi)Zi)Z2 - P(Yi,Zi)R(£,Xi)Z2 = 0

for all vector field Xi , Yi , Zi and Z2 on M.

Putting Z2 = £ and using (2.9) in (6.2), we get

Aiig(Xi ,Yi)Zi + Ai2g(Xi,Zi)Yi + P (Yi,Zi)Xi =0, (6.3)

provided (a2 - $2) = 0 and where

An = [ae + bAAe -- (—+ 6) ) , A12 = ( ae - bAAe + - ( ——- + b \ n \n - 1 J J \ n \n - 1

Substituting Zi = £ in (6.3), we get

Aiig(Xi,Yi)£ + Ai2g(Xi,£)Yi + P (Yi,£)Xi = 0. (6.4)

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

AiAg(Xi,Yi)+ Ai3n(Xi)n(Yi) + n(P (Yi,£)Xi ) = 0, (6.5)

where Ai3 = eAi2, Ai4 = eAii. In the view of (6.1) and (6.5) we have

S (Xi ,Yi) = Ai5g(Xi ,Yi) + Ai6n(Xi)n(Yi), (6.6)

be

Hence we can state the following theorem

Theorem 6.1. A conformal Ricci soliton in an indefinite trans-Sasakian manifold satisfying R(£,X\)P = 0 is an n-Einstein manifold.

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

References

1. Basu, N. and Bhattacharyya, A. Conformal Ricci Soliton in Kenmotsu Manifold, Glob. J. Adv. Res. Class. Mod. Geom., 2015, vol. 4, no. 1, pp. 15-21.

2. Blair, D. E. Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics, Vol. 509, Springer Verlag, 1976.

3. Hamilton, R. S. The Ricci Flow on Surfaces, Mathematics and General Relativity, Santa Cruz, CA, 1986, pp. 237-262. Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1988.

4. Fischer, A. E. An Intorduction to Conformai Ricci Flow, Class. Quantum Grav., 2004, vol. 21, pp. S171-S218.

5. Oubina, J. A. New Classes of Almost Contact Metric Structures, Publ. Math. Debrecen, 1985, vol. 32, pp. 187-193.

Received December 13, 2019

Shivanna Girish Babu Acharya institute of Technology, Bengaluru 560 107, India Research Scholar

E-mail: [email protected],

Polaepalli Siva Kota Reddy JSS Science and Technology University, Mysuru 570 006, Karnataka, India, Professor

E-mail: [email protected]

Ganganna Somashekhara Ramaiah University of Applied Sciences, Bengaluru 560 054, India, Associate Professor

E-mail: somashekhara.mt ,mp@msruas .ac.in

Владикавказский математический журнал 2021, Том 23, Выпуск 3, С. 45-51

КОНФОРМНЫЕ СОЛИТОНЫ РИЧЧИ НА НЕОПРЕДЕЛЕННОМ ТРАНССАСАКИЕВОМ МНОГООБРАЗИИ

Гириш Бабу С.1, Редди П. С. К.2, Сомашекхара Г.3

1 Технологический институт Ачарьи, Бангалор 560107, Карнатака, Индия;

2 Научно-технический университет JSS, Майсур 570006, Карнатака, Индия;

3 Университет прикладных наук Рамая, Бангалор 560054, Карнатака, Индия

E-mail: [email protected], [email protected], somashekhara.mt,[email protected]

Аннотация. Конформные солитоны Риччи являются автомодельными решениями конформного уравнения потока Риччи. Новый класс n-мерных почти контактных многообразий, а именно транссаса-киевы многообразия, был введен Обиной в 1985 г. Дальнейшее изучение локальной структуры транссаса-киевых многообразий было проведено несколькими авторами. Транссасакиевы многообразия, являющиеся естественным обобщением как сасакиевых многообразий, так и многообразий Кенмоцу, тесно связаны с локально конформными келеровыми многообразиями. В настоящей статье изучаются конформные со-литоны Риччи в контексте неопределенного транссасакиева многообразия. Исследован тензор кривизны на неопределенном транссасакиевом многообразии и доказаны некоторые важные результаты.

Ключевые слова: неопределенное транссасакиево многообразие, поток Риччи, конформный поток Риччи.

Mathematical Subject Classification (2010): 53C15, 53C20, 53C25, 53C44.

Образец цитирования: Girish Babu, S., Reddy, P. S. K. and Somashekhara, G. Conformal Ricci Soliton in an Indefinite Trans-Sasakian Manifold // Владикавк. мат. журн.—2020.—Т. 23, № 3.—С. 45-51 (in English). DOI: 10.46698/t3715-2700-6661-v.