Definition 1. A differentiable (2n+1)-dimensional manifold M of the class C-infinity is called a contact manifold if there exists a differential 1-form eta on M2n+1, such that (eta<^>d eta)(n) not equal 0. The form eta is called a contact form. Definition 2. If M2n+1 is a contact manifold with a contact form eta, then a contact metric structure is the quadruple (eta,xi,phi,g), where is a Reeb's field, xi is a Riemannian metric, and phi is an affinor on M2n+1, for which the following properties are valid: 1) phi(2) = -I+eta circle times xi, 2) d eta(X,Y)=g(X,phi Y), 3) g(phi X,phi Y) = g(X,Y) - eta(X)eta(Y). We consider a non-unimodular Lie group G; its Lie algebra has a basis e(1),e(2),e(3) such that [e(1),e(2)] = alpha e(2)+beta e(3), [e(1),e(3)] =gamma e(2)+delta e(3),[e(2),e(3)] = 0, and matrix A = [GRAPHICS] has a trace alpha + delta = 2. The left invariant 1-form eta = a(1)theta(1)+a(2)theta(2)+a(3)theta(3 )defines a contact structure on the group G if (delta - alpha)a(2)a(3 )- beta a(3)(2)+gamma a(2)(2) not equal 0 As a contact form, we choose the simplest one, eta = theta(3),phi(0 )= [GRAPHICS] , and consider other 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 (eta,xi,phi,g), where [GRAPHICS] .
