Left-invariant para-sasakian structures on Lie groups

Paracontact structures on manifolds are currently being studied quite actively; there are several different approaches to the definition of the concepts of paracontact and para-Sasakian structures. In this paper, the paracontact structure on a contact manifold $(M^{2n+1},\eta)$ is determined by an affinor $\varphi$ which has the property $\varphi^2=I-\eta\otimes\xi$, where $\xi$ is the Reeb field and $I$ is the identity automorphism. In addition, it is assumed that $d\eta(\varphi X,\varphi Y)=-d\eta(X,Y)$. This allows us to define a pseudo-Riemannian metric by the equality $g(X,Y) = d\eta(\varphi X,Y) + \eta(X)\eta(Y)$. In this paper, Sasaki paracontact structures are determined in the same way as conventional Sasaki structures in the case of contact structures. A paracontact metric structure $(\eta, \xi, \varphi, g)$ on $M^{2n+1}$ is called para-Sasakian if the almost para-complex structure $J$ on $M^{2n+1}\times\mathbf{R}$ defined by the formula $J(X, f\partial_t) = (\varphi X - f\xi, -\eta(X)\partial_t)$, is integrable. In this paper, we obtain tensors whose vanishing means that the manifold is para-Sasakian. In the case of Lie groups, it is shown that left-invariant para-Sasakian structures can be obtained as central extensions of para-Kähler Lie groups. In this case, the relations between the curvature of the para-Kähler Lie group and the curvature of the corresponding para-Sasakian Lie group are found.