Potential Carroll Structures and Special Carrollian Manifolds

It is well-known that unlike space-like and time-like hypersurfaces, null hypersurfaces in Lorentzian manifolds do not naturally inherit an affine connection from the spacetime in which they are embedded. On the other hand, recent developments in flat-space holography motivate the study of the intrinsic geometry of null hypersurfaces such as null infinity and black hole event horizons. Here we initiate the study of potential Carroll structures, a candidate for an intrinsic description of null hypersurfaces which may be particularly useful in settings where conformal isometries are of interest, and we explore their relationship to another such candidate intrinsic geometry, the special Carrollian manifolds.

💡 Research Summary

The paper addresses a long‑standing geometric issue: null hypersurfaces in Lorentzian manifolds do not inherit a natural affine connection from the ambient spacetime. Motivated by recent developments in flat‑space holography—celestial holography and Carrollian holography—the authors seek an intrinsic description of such hypersurfaces, which appear as null infinity or black‑hole event horizons. They introduce a new geometric structure called a “potential Carroll structure” and compare it systematically with the already known “special Carrollian manifold” (SCM).

A Carrollian structure consists of a smooth manifold (M) equipped with a degenerate metric (g) of signature ((0,+,\dots,+)) and a distinguished null vector field (\ell) satisfying (g(\ell,\cdot)=0). The metric alone does not determine a connection, so an Ehresmann 1‑form (\omega) (with (\omega(\ell)=1)) and an affine connection (\nabla) are introduced. Lemma 2.1 shows that for any connection preserving both (g) and (\ell) ((\nabla g=\nabla\ell=0)), the torsion tensor (T) encodes the Lie derivative of the metric along (\ell): \