Conformally Flat Almost Kenmotsu 3-Manifolds

Almost Kenmotsu metric as a conformal Ricci soliton

“…For further details on almost Kenmotsu manifolds, we refer the reader to go through the references ( [15]- [18]).…”

Almost Kenmotsu 3-manifolds satisfying some generalized nullity conditions

“…Now putting X = ξ and Y = Z = e in (15) and using (16) give (∇ V Q)ξ = 0 for any vector field V, where we have used (28). Putting V = ξ in the above relation and using (17) give ξ(κ) = 0. In view of (9) and µ = 0 we have κ = −1.…”

“…where {e i } is a local orthogonal basics of the tangent space at certain point. Applying Using (17) and 9 Definition 3.12. The Ricci tensor of an almost contact metric manifold is said to be of Codazzi type if…”

“…The classification of conformally flat almost Kenmotsu manifolds was rarely studied. Very recently, Y. Wang in [17] proved that any CR-integrable almost Kenmotsu manifold of dimension > 3 with scalar curvature invariant along the Reeb vector field is conformally flat if and only if it is of constant sectional curvature −1. In [11,Section 6], it was shown that M 3 is a generalized (k, µ) -almost Kenmotsu 3-manifold with k = −1 − 1 4z 2…”

“…As an extension of the well-known Kenmotsu manifolds (see [10]) and an analogy of almost Hermitian manifolds for manifolds of odd dimension, almost Kenmotsu manifolds defined in [9] are becoming an important research object in differential geometry of almost contact metric manifolds. For some recent results regarding such manifolds we refer the reader to [6,7], [11][12][13] and also [16,17]. Almost Kenmotsu manifolds satisfying the (κ, µ) and (κ, µ) -nullity conditions were firstly introduced and studied by Dileo and Pastore [7], where κ and µ both are constants.…”