On Weyl tensor of ACR-manifolds of class \(C_{12}\) with applications. (English) Zbl 1503.53090

Abass, Mohammed Yousif; Al-Zamil, Qusay S.

Izv. Inst. Mat. Inform., Udmurt. Gos. Univ. 59, 3-14 (2022).

Summary: In this paper, we determine the components of the Weyl tensor of almost contact metric (ACR-) manifold of class \(C_{12}\) on associated G-structure (AG-structure) space. As an application, we prove that the conformally flat ACR-manifold of class \(C_{12}\) with \(n>2\) is an \(\eta \)-Einstein manifold and conclude that it is an Einstein manifold such that the scalar curvature \(r\) has provided. Also, the case when \(n=2\) is discussed explicitly. Moreover, the relationships among conformally flat, conformally symmetric, \( \xi \)-conformally flat and \(\Phi \)-invariant Ricci tensor have been widely considered here and consequently we determine the value of scalar curvature \(r\) explicitly with other applications. Finally, we define new classes with identities analogously to Gray identities and discuss their connections with class \(C_{12}\) of ACR-manifold.

MSC:

53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
53D10	Contact manifolds (general theory)
53D15	Almost contact and almost symplectic manifolds

Keywords:

almost contact metric manifold of class \(C_{12}\); \(\eta \)-Einstein manifold; Weyl tensor

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