Some Results on Almost Paracontact Metric Manifolds

We use the tensor h=(1/2)Lξφ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h=(1/2){\mathcal{L}_{\xi}{\varphi}}}$$\end{document} to investigate the geometry of an almost paracontact metric manifold (M,φ,ξ,η,g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(M, \varphi, \xi, \eta, g)}$$\end{document}, also in terms of almost para-CR geometry, emphasizing analogies and differences with respect to the contact metric case. In particular, we investigate in the paracontact metric setting, some conditions which, in the contact metric case, characterize K-contact and Sasakian manifolds. We then give examples of paracontact metric manifolds without Riemannian counterparts. Besides, we show that an almost paracontact structure (φ,ξ,η)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\varphi, \xi, \eta)}$$\end{document} is normal if and only if h =  0 and the corresponding almost para-CR structure (H=kerη,J=φ|H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\mathcal{H} = \ker \eta, J = \varphi_{|\mathcal{H}})}$$\end{document} is a para-CR structure.