The authors study the geometry of a general hypersurface

${\Sigma}$ which is embedded in a spacetime. They choose a transverse vector field

$l$ which they call a ‘rigging’ and construct then two (in general different) connections on this hypersurface. The first connection is induced by the splitting

$TM=\text{span}\left\{l\right\}\oplus T{\Sigma}$. They find a condition for the volume form on

${\Sigma}$ induced by

$l$ 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

${\Sigma}$ which is inverse to the restriction of the contravariant metric

${g}^{ab}$ of spacetime to

${\Sigma}$. They establish how – for a given rigging – these connections relate. Finally, they derive junction conditions for joining pieces of spacetime across general hypersurfaces.