“…(c) The projection π : SL(2, R), ds 2 → H + , g + is also a Riemannian submersion. To see this, recall that {AE, AF, AH}, with E, F and H defined in (21), spanning the tangent space of SL(2, R) at A. It is known by Patragenaru (see [24]) that all left-invariant metrics on SU(1, 1) are isometric to one of the 3-parameter families of metrics g(c 1 , c 2 , c 3 ) with c 1 ≥ c 2 > 0 > c 3 , and its isometry group has dimension 4 if and only if c 1 = c 2 .…”