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, η) is determined by an affinor φ which has the property φ² = I - η0ξ, where ξ is the Reeb field and I is the identity automorphism. In addition, it is assumed that dη(φX, φY) = - dη(X,Y). This allows us to define a pseudo-Riemannian metric by the equality g(X,Y) = dη(φX,Y) + η(X)η(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 (η, ξ, φ, g) on M²ⁿ⁺¹ is called para-Sasakian if the almost para-complex structure J on M²ⁿ⁺¹×R defined by the formula J(X, f∂t) = (φX - fξ, -η(X)∂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. AMS Mathematical Subject Classification: 53C15, 53D10, 53C25, 53C50

Keywords

Authors

References