no. 2
On naturally reductive left-invariant metrics of SL (2,ℝ)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 2, pp. 171-187.

On any real semisimple Lie group we consider a one-parameter family of left-invariant naturally reductive metrics. Their geodesic flow in terms of Killing curves, the Levi Civita connection and the main curvature properties are explicitly computed. Furthermore we present a group theoretical revisitation of a classical realization of all simply connected 3-dimensional manifolds with a transitive group of isometries due to L. Bianchi and É. Cartan. As a consequence one obtains a characterization of all naturally reductive left-invariant riemannian metrics of SL(2,ℝ).

Classification : 53C30, 53C50, 53C55 
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     title = {On naturally reductive left-invariant metrics of ${\rm SL}(2,\mathbb {R})$},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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     volume = {Ser. 5, 5},
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Halverscheid, Stefan; Iannuzzi, Andrea. On naturally reductive left-invariant metrics of ${\rm SL}(2,\mathbb {R})$. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 2, pp. 171-187. http://www.numdam.org/item/ASNSP_2006_5_5_2_171_0/

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