A classification of certain almost α-Kenmotsu manifolds

We study $\mathscr{D}$-homothetic deformations of almost α-Kenmotsu structures. We characterize almost contact metric manifolds which are CR-integrable almost α-Kenmotsu manifolds, through the existence of a canonical linear connection, invariant under $\mathscr{D}$-homothetic deformations. If the canonical connection associated to the structure (φ, ξ, η, g) has parallel torsion and curvature, then the local geometry is completely determined by the dimension of the manifold and the spectrum of the operator h′ defined by 2αh′ = ($\mathscr{L}$ξφ) $\circ$ φ. In particular, the manifold is locally equivalent to a Lie group endowed with a left invariant almost α-Kenmotsu structure. In the case of almost α-Kenmotsu (κ, μ)′-spaces, this classification gives rise to a scalar invariant depending on the real numbers κ and α