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 (M2n+1, eta) is determined by an affinor phi which has the property phi(2) = I - eta circle times xi, where xi is the Reeb field and I is the identity automorphism. In addition, it is assumed that d eta(phi X, phi Y) = - d eta(X,Y). This allows us to define a pseudo-Riemannian metric by the equality g(X,Y) = d eta(phi 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, phi, g) on M2n+1 is called para-Sasakian if the almost para-complex structure J on M(2n+1)xR defined by the formula J(X, f partial derivative(t)) = (phi X - f xi, -eta(X)partial derivative(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-Kahler Lie groups. In this case, the relations between the curvature of the para-Kahler Lie group and the curvature of the corresponding para-Sasakian Lie group are found.