  In this work we provide a definition of the constraint tensor of a null
hypersurface data which is completely explicit in the extrinsic geometry of the
hypersurface. The definition is fully covariant and applies for any topology of
the hypersurface. For data embedded in a spacetime, the constraint tensor
coincides with the pull-back of the ambient Ricci tensor. As applications of
the results, we find three geometric quantities on any transverse submanifold
$S$ of the data with remarkably simple gauge behaviour, and prove that the
restriction of the constraint tensor to $S$ takes a very simple form in terms
of them. We also obtain an identity that generalizes the standard near horizon
equation of isolated horizons to totally geodesic null hypersurfaces with any
topology. Finally, we prove that when a null hypersurface has product topology,
its extrinsic curvature can be uniquely reconstructed from the constraint
tensor plus suitable initial data on a cross-section.
