Contact metric structures on 3-dimentional non-unimodular Lie groups

Definition 1. A differentiable (2n+1)-dimensional manifold М of the class С is called a contact manifold if there exists a differential 1-form n on M + , such that (nAdn) ф 0. The form n is called a contact form. Definition 2. If M + is a contact manifold with a contact form n, then a contact metric structure is the quadruple (n,^,9,g), where Z is a Reeb's field, g is a Riemannian metric, and ф is an affinor on M + , for which the following properties are valid: 1) ф =-I +n®Z, 2) dn(X,Y)=g(XwY), 3) g^^Y) = g(X,Y) - n(X)n(Y). We consider a non-unimodular Lie group G; its Lie algebra has a basis е _ье ₂,е ₃ such that к.,] = a*^ [^3] = У^ . 3 = nx = (; P) + 8 = 2. The left invariant 1-form n = а ₁6 + а ₂0 + а ₃0 defines a contact structure on the group G if (8 - a)a ₂a ₃ - Pa ₃ + Ya ₂ * 0 f0 -1 As a contact form, we choose the simplest one, n = Q ,ф ₀ , and consider other 1 0 0 0 0 0 metrics that also define a contact metric form. We obtain that a contact metric structure on a non-unimodular Lie group can be set by the quadruple (n,Z^,g), where 0 П = б , = ез, ф 1 + 2р cos а ₁ - р' 1 - 0 0

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