Certain results on almost contact pseudo-metric manifolds

We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields $h:=\frac{1}{2}\pounds _\xi \varphi$ and $\ell := R(\cdot,\xi)\xi$, emphasizing analogies and differences with respect to the contact metric case. Certain identities involving $\xi$-sectional curvatures are obtained. We establish necessary and sufficient condition for a nondegenerate almost $CR$ structure $(\mathcal{H}(M), J, \theta)$ corresponding to almost contact pseudo-metric manifold $M$ to be $CR$ manifold. Finally, we prove that a contact pseudo-metric manifold $(M,\varphi,\xi,\eta,g)$ is Sasakian if and only if the corresponding nondegenerate almost $CR$ structure $(\mathcal{H}(M), J)$ is integrable and $J$ is parallel along $\xi$ with respect to the Bott partial connection.