  Motivated by the thermodynamics of black hole solutions conformal to
stationary solutions, we study the geometric invariant theory of null
hypersurfaces. It is well-known that a null hypersurface in a Lorentzian
manifold can be treated as a Carrollian geometry. Additional structure can be
added to this geometry by choosing a connection which yields a Carrollian
manifold. In the literature various authors have introduced Koszul connections
to study the study the physics on these hypersurfaces. In this paper we examine
the various Carrollian geometries and their relationship to null hypersurface
embeddings. We specify the geometric data required to construct a rigid
Carrollian geometry, and we argue that a connection with torsion is the most
natural object to study Carrollian manifolds. We then use this connection to
develop a hypersurface calculus suitable for a study of intrinsic and extrinsic
differential invariants on embedded null hypersurfaces; motivating examples are
given, including geometric invariants preserved under conformal
transformations.
