Lie groups as 3-dimensional almost paracontact almost paracomplex Riemannian manifolds

Abstract

Almost paracontact almost paracomplex Riemannian manifolds of the lowest dimension 3 are considered. Such structures are constructed on a family of Lie groups and the obtained manifolds are studied. Curvature properties of these manifolds are investigated. An example is commented as support of obtained results.1991 Mathematics Subject Classification. 53C15, 53C25.

Para-Ricci-like Solitons on Almost Paracontact Almost Paracomplex Riemannian Manifolds

“…Proof. The characterization of the basic classes F i by the components of F , given in [18], and the assertions (iii) and (iv) in Proposition 4.2, complete the proof of the corollary.…”

“…Taking into account ( 6) and ( 7), equalities (17) imply θ * (ξ) = −2n f and θ(ξ) = ω = 0. Therefore, bearing in mind the components of F in the basic classes F i , given in [18], we get the statement.…”

“…As a result of the latter proposition, a manifold (M, φ, ξ, η, g) ∈ F 5 with torseforming ξ is determined by (18) (∇ x φ) y = −f {g(x, φy)ξ + η(y)φx}.…”

“…where I is the identity transformation on T M ( [22], [18]). Consequently, we get the following equations:…”

“…Here, we introduce our generalization in terms of apapR manifolds as follows. , φ, ξ, η, g) is called a para-Ricci-like soliton with potential vector field ξ and constants (λ, µ, ν) if its Ricci tensor ρ satisfies: (19) and (20) (…”