Einstein's equations simplify considerably in the presence of a second Killing field. Spacetimes with two Killing fields provide the framework for both the theory of colliding gravitational waves and the theory of rotating black holes (Chandrasekhar 1991). Although they deal with different physical objects, the theories are, in fact, closely related from a mathematical point of view. Whereas in the first case both Killing fields are spacelike, there exists an (asymptotically) timelike Killing field in the second situation, since the equilibrium configuration of an isolated system is assumed to be stationary. It should be noted that many stationary and axi-symmetric solutions which have no physical relevance give rise to interesting counterparts in the theory of colliding waves. We refer the reader to Chandrasekhar (1989) for a comparison between corresponding solutions of the Ernst equations. In this chapter we discuss the properties of circular manifolds, that is, asymptotically flat spacetimes which admit a foliation by integrable 2–surfaces orthogonal to the asymptotically timelike Killing field k and the axial Killing field m.

In the first section we argue that the integrability conditions imply that locally M = Σ × Γ and (4)g = σ + g. Here (Σ, σ) and (Γ, g) denote 2–dimensional manifolds where, in an adapted coordinate system, the metrics σ and g do not depend on the coordinates of Σ.

In the second section we discuss the properties of (Σ, σ), the pseudo–Riemannian manifold spanned by the orbits of the 2–dimensional Abelian group generated by the Killing fields.