Vacuum spacetimes with two-parameter spacelike isometry groups and compact invariant hypersurfaces: Topologies and boundary conditions

Spacetimes with closed spacelike hypersurfaces and spacelike two-parameter isometry groups are investigated to determine their possible global structures. It is shown that the two spacelike Killing vectors always commute with each other. Connected group-invariant spacelike hypersurfaces must be homeomorphic to S ¹ ⊗ S ¹ ⊗ S ¹ (three-torus), S ¹ ⊗ S ² (three-handle), S ³ (three-sphere), or to a manifold which is covered by one of these. The spacetime metric and Einstein equations are simplified in the absence of nongravitational sources and used to establish the impossibility of topology change as well as other limitations on global structure. Regularity conditions for spacetimes of this type are derived and shown to be compatible with Einstein's equations.