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 .
@article{ASNSP_2006_5_5_2_171_0, author = {Halverscheid, Stefan and Iannuzzi, Andrea}, title = {On naturally reductive left-invariant metrics of ${\rm SL}(2,\mathbb {R})$}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 5}, number = {2}, year = {2006}, pages = {171-187}, zbl = {1150.53015}, mrnumber = {2244697}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2006_5_5_2_171_0} }
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, Serie 5, Volume 5 (2006) no. 2, pp. 171-187. http://www.numdam.org/item/ASNSP_2006_5_5_2_171_0/
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