Spacetimes with continuous linear isotropies I: spatial rotations

The weakest known criterion for local rotational symmetry (LRS) in spacetimes of Petrov type D is due to Goode and Wainwright (1986). Here it shown, using methods related the Cartan-Karlhede procedure, be equivalent spatial rotation invariance Riemann tensor its first derivatives. Conformally flat are similarly studied shown that almost all cases same ensures LRS. Only conformally accelerated perfect fluids three curvature derivatives required ensure LRS, showing Ellis's original condition case necessary as well sufficient.