Some Results on Almost Paracontact Metric Manifolds

Abstract

We use the tensor \({h=(1/2){\mathcal{L}_{\xi}{\varphi}}}\) to investigate the geometry of an almost paracontact metric manifold \({(M, \varphi, \xi, \eta, g)}\), 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 \({(\varphi, \xi, \eta)}\) is normal if and only if h = 0 and the corresponding almost para-CR structure \({(\mathcal{H} = \ker \eta, J = \varphi_{|\mathcal{H}})}\) is a para-CR structure.