$ \ast $-Ricci tensor on $ (\kappa, \mu) $-contact manifolds

Rongsheng Ma; Donghe Pei; Rongsheng Ma; Donghe Pei

doi:10.3934/math.2022642

AIMS Mathematics

2022, Volume 7, Issue 7: 11519-11528. doi: 10.3934/math.2022642

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Research article

$ \ast $-Ricci tensor on $ (\kappa, \mu) $-contact manifolds

Rongsheng Ma ,
Donghe Pei ^,

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

Received: 30 December 2021 Revised: 07 March 2022 Accepted: 10 March 2022 Published: 13 April 2022
MSC : 53C21, 53D10, 53D15

We introduce the notion of semi-symmetric $ \ast $-Ricci tensor and illustrate that a non-Sasakian $ (\kappa, \mu) $-contact manifold is $ \ast $-Ricci semi-symmetric or has parallel $ \ast $-Ricci operator if and only if it is $ \ast $-Ricci flat. Then we find that among the non-Sasakian $ (\kappa, \mu) $-contact manifolds with the same Boeckx invariant $ I_M $, only one is $ \ast $-Ricci flat, so we can think of it as the representative of such class. We also give two methods to construct $ \ast $-Ricci flat $ (\kappa, \mu) $-contact manifolds.
- $ \ast $-Ricci flat,
- $ \ast $-Ricci semi-symmetric,
- parallel $ \ast $-Ricci operator,
- $ (\kappa, \mu) $-contact manifolds
Citation: Rongsheng Ma, Donghe Pei. $ \ast $-Ricci tensor on $ (\kappa, \mu) $-contact manifolds[J]. AIMS Mathematics, 2022, 7(7): 11519-11528. doi: 10.3934/math.2022642

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Abstract

We introduce the notion of semi-symmetric $ \ast $-Ricci tensor and illustrate that a non-Sasakian $ (\kappa, \mu) $-contact manifold is $ \ast $-Ricci semi-symmetric or has parallel $ \ast $-Ricci operator if and only if it is $ \ast $-Ricci flat. Then we find that among the non-Sasakian $ (\kappa, \mu) $-contact manifolds with the same Boeckx invariant $ I_M $, only one is $ \ast $-Ricci flat, so we can think of it as the representative of such class. We also give two methods to construct $ \ast $-Ricci flat $ (\kappa, \mu) $-contact manifolds.

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