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 
@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},
     pages = {171--187},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {2},
     year = {2006},
     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, Série 5, Tome 5 (2006) no. 2, pp. 171-187. http://www.numdam.org/item/ASNSP_2006_5_5_2_171_0/

[A] M. Abate, “Iteration Theory of Holomorphic Mapping on Taut Manifolds”, Mediterranean Press, Commenda di Rende, Italy. | Zbl 0747.32002

[B] L. Bianchi, Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti, Mem. Soc. It. delle Sc. (dei XL), (3) 11 (1898), 267-252 (also in Luigi Bianchi, Opere, Vol. IX, Edizioni Cremonese, Roma, 1958). | JFM 29.0415.01

[BTV] J. Berndt, F. Tricerri, L. Vanhecke, “Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces”, LNM 1598, Springer-Verlag, 1995. | MR 1340192 | Zbl 0818.53067

[C] E. Cartan, “Leçons sur la Géométrie des Espaces de Riemann”, Gauthier-Villars, Paris, 1951. | MR 44878 | Zbl 0044.18401

[DZ] J. E. D'Atri, W. Ziller, Naturally reductive metrics and Einstein metrics on compact Lie groups. Mem. Amer. Math. Soc. 215 (1979). | MR 519928 | Zbl 0404.53044

[G] C. S. Gordon, Naturally reductive homogeneous Riemannian manifolds, Canad. J. Math. 37 (1985), 467-487. | MR 787113 | Zbl 0554.53035

[H] S. Helgason, “Differential Geometry, Lie Groups and Symmetric Spaces”, GSM 34, AMS, Providence, 2001. | MR 1834454 | Zbl 0993.53002

[HI] S. Halverscheid, A. Iannuzzi, A family of adapted complexifications for noncompact semisimple Lie groups, ArXiv: math.CV/0503377. | Numdam | Zbl 1180.53053

[KN] S. Kobayashi, K. Nomizu, “Foundations of Differential Geometry”, Vol. II., Interscience, New York, 1969. | MR 238225 | Zbl 0119.37502

[M1] J. Milnor “Morse Theory”, Annals of Math. Studies 51, Princeton University Press, Princeton, N.J. 1963. | MR 163331 | Zbl 0108.10401

[M2] J. Milnor, Curvature of left invariant metrics on Lie groups, Adv. Math. 21 (1976), 293-329. | MR 425012 | Zbl 0341.53030

[O'N] B. O'Neill, “Semi-Riemannian Geometry”, Academic Press, 1983. | MR 719023 | Zbl 0531.53051

[V] G. Vranceanu, “Leçons de Géométrie Différentielle”, Vol. I, Ed. Acad. Rp. Pop. Roumaine, 1957. | MR 124823 | Zbl 0119.17102

[W] J. A. Wolf, “Spaces of Constant Curvature”, New York: McGraw-Hill, 1967. | MR 217740 | Zbl 0162.53304