2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/229173The maximal complexifications adapted to the Levi Civita connection for a distinguished one-parameter family of left-invariant metrics on a real semisimple Lie group G are determined. For G = SL(2,R) their realizations as invariant Riemann domains into SL(2,C) are carried out and their complex-geometric properties are investigated.34 pages, 6 figures. v2: statement of Prop. 5.4 has changed, minor corrections. v3: revised version, appendix addedComplex Variables53C30; 53C30; 32C09; 32Q99; 32M05text