The authors study the geometry of a general hypersurface
which is embedded in a spacetime. They choose a transverse vector field
which they call a ‘rigging’ and construct then two (in general different) connections on this hypersurface. The first connection is induced by the splitting
. They find a condition for the volume form on
to be parallel with respect to this condition. They also derive Gauß and Codazzi equations for their setup. The second connection is a metric connection. Here they use the rigging in order to single out a special metric on
which is inverse to the restriction of the contravariant metric
of spacetime to
. They establish how – for a given rigging – these connections relate. Finally, they derive junction conditions for joining pieces of spacetime across general hypersurfaces.