Metric tensors invariant under some subgroups of the conformal group on space-time

General metric tensors invariant under some maximal subgroups of the conformal group of space-time are determined through an infinitesimal method based on conformal point transformations (discussed by Fulton, Rohrlich and Witten). The cases of the following maximal subgroups are successively studied: SO(3) ⊗ SO(2, 1), O(3, 2) and O(2) ⊗ O(4).