arXiv:1607.00367v2 [math.DG] 8 Jul 2016 RIEMANNIAN GEOMETRY OF TWO FAMILIES OF TANGENT LIE GROUPS F. ASGARI AND H. R. SALIMI MOGHADDAM Abstract. Using vertical and complete lifts, any left invariant Riemannian metric on a Lie group induces a left invariant Riemannian metric on the tangent Lie group. In the present article we study the Riemannian geometry of tangent bundle of two families of Lie groups. The first one is the family of special Lie groups considered by J. Milnor and the second one is the class of Lie groups with one-dimensional commutator groups. The Levi-Civita connection, sectional and Ricci curvatures have been investigated. 1. Introduction Suppose that M is a real m−dimensional differentiable manifold. For any vector field X on M , the infinitesimal generator of the one-parameter group of diffeomorphisms ψ t (y) := y + tX(x), ∀y ∈ T x M , is a vector field on TM which is called the vertical lift of X and is denoted by X v . Let φ t be the (local) one-parameter group of diffeomophisms defined by X on M . The infinitesimal generator of Tφ t : TM −→ TM , is called the complete lift of X and is denoted by X c (see [2, 5, 7]). For any two vector fields X,Y on M the Lie bracket of vertical and complete lifts of them satisfy the following relations (see [5, 7]): (1.1) [X v ,Y v ]=0 , [X c ,Y c ]=[X,Y ] c , [X v ,Y c ]=[X,Y ] v . Let G be a Lie group with multiplication map µ and inversion map ι. Then TG is also a Lie group with multiplication Tµ and inversion map Tι, where Tµ and Tι denote the tangent maps of µ and ι respectively. More precisely, TG is a Lie group with the multiplication (1.2) Tµ(v,w)= T h l g w + T g r h v, ∀g,h ∈ G,v ∈ T g G,w ∈ T h G, where l g and r h denote the left and right translations respectively. In [2], it is shown that the complete and vertical lifts of left invariant vector fields of G are left invariant vector fields of TG. Therefore for any left invariant Riemannian metric g on G we Key words and phrases. Left invariant Riemannian metric, tangent Lie group, complete and vertical lifts, sectional and Ricci curvatures AMS 2010 Mathematics Subject Classification: 53B21, 22E60, 22E15. 1