### 2.4 The polarized subcase

The polarized Gowdy solutions constitute an interesting subclass. They are characterized by the
additional condition that the Killing vectors be mutually orthogonal. In the polarized T^{3}-Gowdy case, the
metric can be written
i.e., it corresponds to setting in Equation (2). In the case of S^{3} and S^{2} × S^{1}, the metric takes the
form
see Equations (5) – (7) of [50]. Clearly, the connection between the coordinates, the metric components
and the topology is more complicated in the case of S^{3} and S^{2} × S^{1} than in the case of T^{3}. We shall not
go into a detailed discussion of these issues here, but rather refer the interested reader to [21] and [50] for a
further discussion. A brief explanation of the origin of the name (in terms of polarizations of
gravitational waves) as well as a different characterization is provided at the bottom of [70,
p. 65].