Some notes on the tangent bundle with a Ricci quarter-symmetric metric connection

Yanlin Li; Aydin Gezer; Erkan Karakaş; Yanlin Li; Aydin Gezer; Erkan Karakaş

doi:10.3934/math.2023886

AIMS Mathematics

2023, Volume 8, Issue 8: 17335-17353. doi: 10.3934/math.2023886

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Research article

Some notes on the tangent bundle with a Ricci quarter-symmetric metric connection

1.
School of Mathematics, Key Laboratory of Cryptography of Zhejiang Province, Hangzhou Normal University, Hangzhou 311121, China
2.
Faculty of Science, Department of Mathematics, Ataturk University, Erzurum 25240, Turkey

Received: 27 February 2023 Revised: 11 May 2023 Accepted: 15 May 2023 Published: 19 May 2023
MSC : 53B20, 53C07, 53C35

Let $ (M, g) $ be an $ n $-dimensional (pseudo-)Riemannian manifold and $ TM $ be its tangent bundle $ TM $ equipped with the complete lift metric $ ^{C}g $. First, we define a Ricci quarter-symmetric metric connection $ \overline{\nabla } $ on the tangent bundle $ TM $ equipped with the complete lift metric $ ^{C}g $. Second, we compute all forms of the curvature tensors of $ \overline{\nabla } $ and study their properties. We also define the mean connection of $ \overline{\nabla } $. Ricci and gradient Ricci solitons are important topics studied extensively lately. Necessary and sufficient conditions for the tangent bundle $ TM $ to become a Ricci soliton and a gradient Ricci soliton concerning $ \overline{\nabla } $ are presented. Finally, we search conditions for the tangent bundle $ TM $ to be locally conformally flat with respect to $ \overline{\nabla } $.
- complete lift metric,
- Ricci quarter-symmetric metric connection,
- Ricci soliton,
- tangent bundle,
- vector field
Citation: Yanlin Li, Aydin Gezer, Erkan Karakaş. Some notes on the tangent bundle with a Ricci quarter-symmetric metric connection[J]. AIMS Mathematics, 2023, 8(8): 17335-17353. doi: 10.3934/math.2023886

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Abstract

Let $ (M, g) $ be an $ n $-dimensional (pseudo-)Riemannian manifold and $ TM $ be its tangent bundle $ TM $ equipped with the complete lift metric $ ^{C}g $. First, we define a Ricci quarter-symmetric metric connection $ \overline{\nabla } $ on the tangent bundle $ TM $ equipped with the complete lift metric $ ^{C}g $. Second, we compute all forms of the curvature tensors of $ \overline{\nabla } $ and study their properties. We also define the mean connection of $ \overline{\nabla } $. Ricci and gradient Ricci solitons are important topics studied extensively lately. Necessary and sufficient conditions for the tangent bundle $ TM $ to become a Ricci soliton and a gradient Ricci soliton concerning $ \overline{\nabla } $ are presented. Finally, we search conditions for the tangent bundle $ TM $ to be locally conformally flat with respect to $ \overline{\nabla } $.

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