Introduction Research Article Certain results on Ricci solitons in · 2019-01-25 · 11 Certain...

5
11 Certain results on Ricci solitons in α-Kenmotsu manifolds Rajesh Kumar 1 , Ashwamedh Mourya 2 1 Department of Mathematics, Pachhunga University College, Aizawl 796001, India 2 Department of Mathematics, Ashoka Institute of Technology and Management, Varanasi 221007, India Article Research In this paper, we study some curvature problems of Ricci solitons in α-Kenmotsu manifold. It is shown that a symmetric parallel second order-covariant tensor in a α-Kenmotsu manifold is a constant multiple of the metric tensor. Using this result, it is shown that if (L v g + 2S) is parallel where V is a given vector field, then the structure (g, V, λ) yield a Ricci soliton. Further, by virtue of this result, Ricci solitons for n-dimentional α-Kenmotsu manifolds are obtained. In the last section, we discuss Ricci soliton for 3-dimentional α-Kenmotsu manifolds. 2010 Mathematical Subject Classification: 53C25, 53C10, 53C44 Key words: Ricci soliton, α-Kenmotsu manifold, Einstein manifold. Received 07 November 2017 Accepted 06 January 2018 *For correspondence : [email protected] Contact us : [email protected] https://doi.org/10.33493/scivis.18.01.02 ISSN (print) 0975-6175/(online) 2229-6026. 2018 The Mizo Academy of Sciences. CC BY-SA 4.0 International. OPEN ACCESS Sci Vis 18 (1), 1115 (2018) Available at www.sciencevision.org Introduction , , , +2 , +2 , =0 , () , 0 = 0

Transcript of Introduction Research Article Certain results on Ricci solitons in · 2019-01-25 · 11 Certain...

Page 1: Introduction Research Article Certain results on Ricci solitons in · 2019-01-25 · 11 Certain results on Ricci solitons in α-Kenmotsu manifolds Rajesh Kumar1, Ashwamedh Mourya2

11

Certain results on Ricci solitons in α-Kenmotsu manifolds

Rajesh Kumar1, Ashwamedh Mourya2

1Department of Mathematics, Pachhunga University College, Aizawl 796001, India 2Department of Mathematics, Ashoka Institute of Technology and Management, Varanasi 221007, India

Article Research

In this paper, we study some curvature problems of Ricci solitons in α-Kenmotsu manifold. It

is shown that a symmetric parallel second order-covariant tensor in a α-Kenmotsu manifold

is a constant multiple of the metric tensor. Using this result, it is shown that if (Lvg + 2S) is

parallel where V is a given vector field, then the structure (g, V, λ) yield a Ricci soliton. Further,

by virtue of this result, Ricci solitons for n-dimentional α-Kenmotsu manifolds are obtained.

In the last section, we discuss Ricci soliton for 3-dimentional α-Kenmotsu manifolds.

2010 Mathematical Subject Classification: 53C25, 53C10, 53C44

Key words: Ricci soliton, α-Kenmotsu manifold, Einstein manifold.

Received 07 November 2017 Accepted 06 January 2018 *For correspondence : [email protected] Contact us : [email protected] https://doi.org/10.33493/scivis.18.01.02

ISSN (print) 0975-6175/(online) 2229-6026. 2018 The Mizo Academy of Sciences. CC BY-SA 4.0 International.

OPEN ACCESS

Sci Vis 18 (1), 11—15 (2018)

Available at

www.sciencevision.org

Introduction

𝑀 𝑔, 𝑉, 𝜆

𝑔 𝑉𝜆

ℒ𝑉𝑔 𝑋, 𝑌 + 2 𝑆 𝑋, 𝑌 + 2𝜆 𝑔 𝑋, 𝑌 = 0𝑋,𝑌 𝜒(𝑀) 𝑆

ℒ𝑉

𝑉

𝜆

𝛼 𝛼

𝛼

𝑘, 𝜇

𝜉 ℒ𝑉𝑔 ≠ 0

𝜉 ℒ𝑉𝑔 =0

𝑓

𝛼

𝛼 𝛼

𝛼

𝑘, 𝜇

𝜉 ℒ𝑉𝑔 ≠ 0

𝜉 ℒ𝑉𝑔 =0

𝑓

𝛼

𝛼

𝛼

𝛼 𝛼

𝛼

𝛼𝜂

𝛼

Page 2: Introduction Research Article Certain results on Ricci solitons in · 2019-01-25 · 11 Certain results on Ricci solitons in α-Kenmotsu manifolds Rajesh Kumar1, Ashwamedh Mourya2

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𝛼

𝛼

𝛼 𝛼

𝛼

𝛼𝜂

𝛼

𝜶-Kenmotsu manifold

𝑛 𝐶∞ 𝑀 𝜑, 𝜉, 𝜂

𝜑 𝜉𝜂

𝜂 𝜉 = 1 𝜑2𝑋 = −𝑋 + 𝜂 𝑋 𝜉

𝜑 𝜉 = 0 𝜂 𝜑𝑋 = 0𝑋, 𝑌 𝜒 𝑀 𝜒(𝑀)

𝐶∞ 𝑀 𝑛𝐶∞ 𝑀

𝜑, 𝜉, 𝜂

𝑀𝑔

𝑔 𝜑𝑋, 𝜑𝑌 = 𝑔 𝑋, 𝑌 − 𝜂 𝑋 𝜂 𝑌

𝑔 𝑋, 𝜉 = 𝜂 𝑋

𝑀 𝜑, 𝜉, 𝜂, 𝑔

𝑀𝛼

𝑑𝜂 = 0 𝑑Φ = 2𝛼 𝜂 ∧ ΦΦ

Φ 𝑋, 𝑌 = 𝑔 𝜑𝑋, 𝑌 𝛼𝛼

𝛻𝑋𝜑 𝑌 = −𝛼 𝑔 𝑋, 𝜑𝑌 𝜉 + 𝜂 𝑌 𝜑𝑋

𝛻𝑋𝜉 = 𝛼 𝑋 − 𝜂 𝑋 𝜉

𝛼𝛼 𝑀

𝑅 𝑋, 𝑌 𝜉 = 𝛼2 𝜂 𝑋 𝑌 − 𝜂 𝑌 𝑋

𝑅 𝜉,𝑋 𝑌 = 𝛼2 𝜂 𝑌 𝑋 − 𝑔 𝑋,𝑌 𝜉

𝜂 𝑅 𝑋, 𝑌 𝑍 = 𝛼2 𝑔 𝑋, 𝑌 𝜂 𝑍 − 𝑔 𝑌, 𝑍 𝜂 𝑋

𝑆 𝑋, 𝜉 = −𝛼2 𝑛 − 1 𝜂 𝑋

𝑆 𝜉, 𝜉 = −𝛼2 𝑛 − 1

𝑄𝜉 = −𝛼2 𝑛 − 1 𝜉 ∇𝑋𝜂 𝑌 = 𝛼 𝑔 𝑋, 𝑌 − 𝜂 𝑋 𝜂 𝑌

𝑋, 𝑌,𝑍 𝜒(𝑀) 𝑅𝑆

𝑄𝑆 𝑋, 𝑌 = 𝑔 𝑄𝑋, 𝑌

𝑆 𝜉, 𝜉 = −𝛼2 𝑛 − 1

𝑄𝜉 = −𝛼2 𝑛 − 1 𝜉 ∇𝑋𝜂 𝑌 = 𝛼 𝑔 𝑋, 𝑌 − 𝜂 𝑋 𝜂 𝑌

𝑋, 𝑌,𝑍 𝜒(𝑀) 𝑅𝑆

𝑄𝑆 𝑋, 𝑌 = 𝑔 𝑄𝑋, 𝑌

Parallel symmetric second order tensors

and Ricci solitons in 𝜶-Kenmotsu

manifolds

𝑕𝛻

𝛻𝑕 = 0𝛻2𝑕 𝑋, 𝑌; 𝑍, 𝑊 − 𝛻2𝑕 𝑋, 𝑌; 𝑊, 𝑍 = 0

𝑕 𝑅 𝑋, 𝑌 𝑍, 𝑊 + 𝑕 𝑍, 𝑅 𝑋, 𝑌 𝑊 = 0𝑍 = 𝑊 = 𝜉

𝛼2 𝜂 𝑋 𝑕 𝑌, 𝜉 − 𝜂 𝑦 𝑕 𝑋, 𝜉 = 0

𝛼 𝑋 = 𝜉𝑕

𝑕 𝑌, 𝜉 = 𝜂 𝑌 𝑕 𝜉, 𝜉

𝑋 𝛻𝑋𝑕 𝑌, 𝜉 + 𝑕 𝛻𝑋𝑌, 𝜉 + 𝑕 𝑌, 𝛻𝑋𝜉

= 𝛻𝑋𝜂 𝑌 𝑕 𝜉, 𝜉 + 𝜂 𝛻𝑋𝑌 𝑕 𝜉, 𝜉 +𝜂 𝑌 𝛻𝑋𝑕 𝜉, 𝜉 + 2𝜂 𝑌 𝑕 𝛻𝑋𝜉, 𝜉 .

𝛻𝑕 = 0𝑕 𝑋, 𝑌 = 𝑔 𝑋, 𝑌 𝑕 𝜉, 𝜉

𝑕 𝜉, 𝜉

𝛼

𝛻𝑆 = 0

𝛼𝛼

𝑅 ∙ 𝑆 = 0 𝑖 → 𝑖𝑖 → (𝑖𝑖𝑖)

𝑖𝑖𝑖 → (𝑖)𝛼

𝑅 ∙ 𝑆 = 0 𝑕𝑆

𝑅(𝑋, 𝑌) ∙ 𝑆 𝑈, 𝑉 = −𝑆 𝑅(𝑋, 𝑌)𝑈, 𝑉 −𝑆(𝑈,𝑅(𝑋, 𝑌)𝑉)

𝑅 ∙ 𝑆 = 0 𝑋 = 𝜉

𝑆 𝑅(𝜉, 𝑌)𝑈, 𝑉 + 𝑆 𝑈, 𝑅 𝜉, 𝑌 𝑉 = 0𝛼 ≠ 0

𝜂 𝑈 𝑆 𝑌,𝑉 − 𝑔 𝑌,𝑉 𝑆 𝜉, 𝑉 + 𝜂 𝑉 𝑆 𝑈, 𝑌 −𝑔 𝑌,𝑉 𝑆 𝑈, 𝜉 = 0

Sci Vis 18 (1), 11—15 (2018)

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𝑖 → 𝑖𝑖 → (𝑖𝑖𝑖) 𝑖𝑖𝑖 → (𝑖)

𝛼𝑅 ∙ 𝑆 = 0 𝑕

𝑆 𝑅(𝑋, 𝑌) ∙ 𝑆 𝑈, 𝑉 = −𝑆 𝑅(𝑋, 𝑌)𝑈, 𝑉 −

𝑆(𝑈,𝑅(𝑋, 𝑌)𝑉)𝑅 ∙ 𝑆 = 0 𝑋 = 𝜉

𝑆 𝑅(𝜉, 𝑌)𝑈, 𝑉 + 𝑆 𝑈, 𝑅 𝜉, 𝑌 𝑉 = 0𝛼 ≠ 0

𝜂 𝑈 𝑆 𝑌,𝑉 − 𝑔 𝑌,𝑉 𝑆 𝜉, 𝑉 + 𝜂 𝑉 𝑆 𝑈, 𝑌 −𝑔 𝑌,𝑉 𝑆 𝑈, 𝜉 = 0

𝑈 = 𝜉

𝑆 𝑌, 𝑉 = −𝛼2 𝑛 − 1 𝑔(𝑌,𝑉)

𝛼

𝛼 ℒ𝑉𝑔 + 2𝑆 𝑔,𝑉, 𝜆

𝛼 ℒ𝑉𝑔 + 2𝑆

𝛼

ℒ𝑉𝑔 + 2𝑆

𝑔 ℒ𝑉𝑔 + 2𝑆 𝑋, 𝑌 = 𝑔 𝑋, 𝑌 𝑕(𝜉, 𝜉) 𝑕(𝜉, 𝜉)

𝑔 𝛼𝑉 = 𝜉 𝜂

𝑉 = 𝜉

ℒ𝜉𝑔 𝑋, 𝑌 + 2𝑆 𝑋, 𝑌 + 2𝜆 𝑔 𝑋,𝑌 = 0

ℒ𝜉𝑔 𝑋, 𝑌 = 𝑔 𝛻𝑋𝜉, 𝑌 + 𝑔 𝑋, 𝛻𝑌𝜉

= 2𝛼 𝑔 𝑋,𝑌 − 𝜂 𝑋 𝜂 𝑌

𝑆 𝑋, 𝑌 = − 𝛼 + 𝜆 𝑔 𝑋, 𝑌 + 𝛼 𝜂 𝑋 𝜂 𝑌

𝑔, 𝜉, 𝜆 𝑛𝛼

𝑉 ∈ 𝑆𝑝𝑎𝑛 𝜉 𝑉 ⊥ 𝜉

𝑉 =𝜉

ℒ𝑉𝑔 + 2𝑆

ℒ𝜉𝑔 𝑋, 𝑌 = 0

𝑕 𝑋,𝑌 = −2𝜆 𝑔 𝑋, 𝑌

𝑋 = 𝑌 = 𝜉

𝑕 𝜉, 𝜉 = −2𝜆

𝑕 𝜉, 𝜉 = ℒ𝜉𝑔 𝜉, 𝜉 + 2𝑆 𝜉, 𝜉

𝑕 𝜉, 𝜉 = −2𝜆

𝑕 𝜉, 𝜉 = ℒ𝜉𝑔 𝜉, 𝜉 + 2𝑆 𝜉, 𝜉

𝑕 𝜉, 𝜉 = 2𝛼2 𝑛 − 1

𝜆 = −𝛼2 𝑛 − 1

𝛼𝜆 ≠ 0 𝑛

𝛼𝜆 < 0

𝑛 𝛼𝜂 𝛼

𝑔, 𝜉, 𝜆 𝜆 =−𝛼2 𝑛 − 1

𝛼 𝜂

𝛼

𝛼

𝛼𝜂 𝑔 𝜂

𝑎 𝑏𝑔

𝑆 𝑋, 𝑌 = 𝑎𝑔 𝑋, 𝑌 + 𝑏𝜂 𝑋 𝜂 𝑌

𝑒𝑖 , (𝑖 = 1, 2,…𝑛)

𝑋 = 𝑌 = 𝑒𝑖𝑖

𝑟 = 𝑎 𝑛 + 𝑏

𝑋 = 𝑌 = 𝜉

𝑎 + 𝑏 = −𝛼2 𝑛 − 1

𝑎 = 𝛼2 +𝑟

𝑛−1 𝑏 = − 𝑛 𝛼2 +

𝑟

𝑛−1

𝑎 𝑏

𝑆 𝑋, 𝑌 = 𝛼2 +𝑟

𝑛−1 𝑔 𝑋, 𝑌 − 𝑛 𝛼2 +

𝑟

𝑛−1 𝜂 𝑋 𝜂 𝑌

𝛼𝜂

𝑟𝑛 𝛼

𝑕(𝑋, 𝑌)

𝑕 𝑋,𝑌 = ℒ𝜉𝑔 𝑋, 𝑌 + 2𝑆(𝑋, 𝑌)

𝑕 𝑋,𝑌 = 2 𝛼 𝛼 + 1 +𝑟

𝑛−1 𝑔 𝑋, 𝑌 − 2 𝛼 𝑛𝛼 +

1 +𝑟

𝑛−1 𝜂 𝑋 𝜂 𝑌

𝑍

Sci Vis 18 (1), 11—15 (2018)

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𝑟𝑛 𝛼

𝑕(𝑋, 𝑌)

𝑕 𝑋,𝑌 = ℒ𝜉𝑔 𝑋, 𝑌 + 2𝑆(𝑋, 𝑌)

𝑕 𝑋,𝑌 = 2 𝛼 𝛼 + 1 +𝑟

𝑛−1 𝑔 𝑋, 𝑌 − 2 𝛼 𝑛𝛼 +

1 +𝑟

𝑛−1 𝜂 𝑋 𝜂 𝑌

𝑍

𝛻𝑍𝑕 𝑋, 𝑌 = 2 𝑍𝛼 𝛼 + 1 + 𝛼 𝑍𝛼 +𝛻𝑍𝑟

𝑛 − 1 𝑔 𝑋, 𝑌

−2 𝑍𝛼 𝑛𝛼 + 1 + 𝑛 𝛼 𝑍𝛼 +𝛻𝑍𝑟

𝑛−1 𝜂 𝑋 𝜂 𝑌

−2 𝛼 𝑛𝛼 + 1 +𝑟

𝑛−1 𝛼{𝑔 𝑍, 𝑋 −

𝜂 𝑍 𝜂 𝑋 +𝑔 𝑍, 𝑌 − 𝜂 𝑍 𝜂 𝑌 }

𝑍 = 𝜉 𝑋 = 𝑌 ∈ 𝑆𝑝𝑎𝑛 ⊥

𝛻𝑕 = 0𝛻𝜉𝑟 = − 𝑛 − 1 𝛻𝜉 𝛼 𝛼 + 1

𝑟 = − 𝑛 − 1 𝛼 𝛼 + 1 + 𝑐,𝑐

𝑟

𝛼

𝑕 𝑋,𝑌 − 2𝜆 𝑔 𝑋,𝑌

𝑋 = 𝑌 = 𝜉𝑕 𝜉, 𝜉 = −2𝜆

𝑋 = 𝑌 = 𝜉𝑕 𝜉, 𝜉 = −2 𝑛 − 1 𝛼2

𝜆 = 𝑛 − 1 𝛼2

𝜆 > 0, ∀ 𝑛 > 1𝛼

𝑔, 𝜉, 𝜆 𝜆 =2𝛼2 3 𝛼

3 𝛼𝜂

𝛼

𝛼

𝛼𝑅 𝑋, 𝑌 𝑍 = 𝑔 𝑌, 𝑍 𝑄𝑋 − 𝑔 𝑋, 𝑍 𝑄𝑌 + 𝑆 𝑌, 𝑍 𝑋 −

𝑆 𝑋, 𝑍 𝑌 −𝑟

2 𝑔 𝑌, 𝑍 𝑋 − 𝑔(𝑋, 𝑍)𝑌

𝑍 = 𝜉

𝛼2 𝜂 𝑋 𝑌 − 𝜂 𝑌 𝑋 = 𝜂 𝑌 𝑄𝑋 − 𝜂 𝑋 𝑄𝑌 −

2𝛼2 +𝑟

2 𝜂 𝑌 𝑋 − 𝜂 𝑋 𝑌

𝛼𝑅 𝑋, 𝑌 𝑍 = 𝑔 𝑌, 𝑍 𝑄𝑋 − 𝑔 𝑋, 𝑍 𝑄𝑌 + 𝑆 𝑌, 𝑍 𝑋 −

𝑆 𝑋, 𝑍 𝑌 −𝑟

2 𝑔 𝑌, 𝑍 𝑋 − 𝑔(𝑋, 𝑍)𝑌

𝑍 = 𝜉

𝛼2 𝜂 𝑋 𝑌 − 𝜂 𝑌 𝑋 = 𝜂 𝑌 𝑄𝑋 − 𝜂 𝑋 𝑄𝑌 −

2𝛼2 +𝑟

2 𝜂 𝑌 𝑋 − 𝜂 𝑋 𝑌

𝑌 = 𝜉

𝑄𝑋 = 𝛼2 +𝑟

2 𝑋 − 3𝛼2 +

𝑟

2 𝜂 𝑋 𝜉

𝑌

𝑆 𝑋, 𝑌 = 𝛼2 +𝑟

2 𝑔 𝑋, 𝑌 − 3𝛼2 +

𝑟

2 𝜂 𝑋 𝜂 𝑌

𝛼𝜂

𝑟 𝑟

𝑕 𝑋,𝑌 = ℒ𝜉𝑔 𝑋, 𝑌 + 2𝑆 𝑋, 𝑌

𝑕 𝑋, 𝑌 = 2 𝛼 𝛼 + 1 +𝑟

2 𝑔 𝑋, 𝑌 − 2 𝛼 3𝛼 +

1 +𝑟

2 𝜂 𝑋 𝜂 𝑌

𝑍

𝛻𝑍𝑕 𝑋, 𝑌 = 2 𝑍𝛼 𝛼 + 1 + 𝛼 𝑍𝛼 +𝛻𝑍𝑟

2 𝑔 𝑋, 𝑌

−2 𝑍𝛼 3𝛼 + 1 + 𝛼 3𝑍𝛼 +𝑟

2 𝜂 𝑋 𝜂 𝑌

−2 𝛼 3𝛼 + 1 +𝑟

2 𝛻𝑍𝜂 𝑋 𝜂 𝑌 +

𝜂 𝑋 𝛻𝑍𝜂 𝑌

𝑍 = 𝜉 𝑋 = 𝑌 ∈ 𝑆𝑝𝑎𝑛 ⊥

𝛻𝑕 = 0𝛻𝜉𝑟 = −2𝛻𝜉 𝛼 𝛼 + 1

𝑟 = −2𝛼 𝛼 + 1 + 𝑐𝑐

𝑟

𝑔, 𝜉, 𝜆 𝛼

𝑕 𝑋,𝑌 − 2𝜆 𝑔 𝑋,𝑌

𝑋 = 𝑌 = 𝜉𝑕 𝜉, 𝜉 = −2𝜆

𝑋 = 𝑌 = 𝜉𝑕 𝜉, 𝜉 = −4𝛼2

𝜆 = 2𝛼2

𝜆 > 0 𝛼

Sci Vis 18 (1), 11—15 (2018)

Page 5: Introduction Research Article Certain results on Ricci solitons in · 2019-01-25 · 11 Certain results on Ricci solitons in α-Kenmotsu manifolds Rajesh Kumar1, Ashwamedh Mourya2

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𝜆 = 2𝛼2

𝜆 > 0 𝛼

Sci Vis 18 (1), 11—15 (2018)