A classification of certain almost $\alpha$-Kenmotsu manifolds

We study $\mathcal{D}$-homothetic deformations of almost $\alpha$-Kenmotsu structures. We characterize almost contact metric manifolds which are $CR$-integrable almost $\alpha$-Kenmotsu manifolds, through the existence of a canonical linear connection, invariant under $\mathcal{D}$-homothetic deformations. If the canonical connection associated to the structure $(\phi ,\xi ,\eta ,g)$ has parallel torsion and curvature, then the local geometry is completely determined by the dimension of the manifold and the spectrum of the operator $h^{\mathrm{\prime }}$ defined by $2\alpha h’=({\mathcal L}_\xi\varphi)\circ\varphi$. In particular, the manifold is locally equivalent to a Lie group endowed with a left invariant almost $\alpha$-Kenmotsu structure. In the case of almost $\alpha$-Kenmotsu $(\kappa ,\mu {)}^{\mathrm{\prime }}$-spaces, this classification gives rise to a scalar invariant depending on the real numbers $\kappa$ and $\alpha$.