Lightlike Cartan Geometries

• Lightlike Cartan geometries are Cartan geometries modeled on the future light cone of Lorentz–Minkowski spacetime, characterized by a degenerate metric and a globally defined radical generator.
• They induce lightlike structures on manifolds by defining invariant screen distributions and tractor bundles that encode additional geometric data via compatible connections.
• The theory bridges conformal, Carrollian, and Newton–Cartan geometries, highlighting unique curvature invariants and normalization challenges that differentiate these models.

Lightlike Cartan geometries are Cartan geometries modeled on the future lightlike cone of Lorentz–Minkowski spacetime. In this framework, a manifold is treated as a curved analogue of the cone, and the Cartan connection encodes not only a degenerate metric structure but also additional geometric data. The modern formulation appears explicitly in "Lightlike manifolds and Cartan geometries" (Palomo, 2020) and is developed further in "Cartan geometries with model the future lightlike cone of Lorentz-Minkowski spacetime" (Morón et al., 27 Aug 2025), where the central point is that the resulting geometry carries a lightlike metric with one-dimensional radical, a globally defined radical generator, and further compatible structures not determined by the metric and radical field alone.

1. Homogeneous model and Cartan-geometric definition

The homogeneous model is the future lightlike cone

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_{m+2}>0\},$$

or, in the earlier convention,

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$

inside Lorentz–Minkowski space. In the 2025 formulation the ambient metric is

$$\langle\, ,\, \rangle = \sum_{i=1}^{m+1}dv_i^2-dv_{m+2}^2,$$

while the 2020 formulation writes

$$\langle\ ,\ \rangle=-dx_0^2+\sum_{i=1}^{m+1}dx_i^2.$$

The cone inherits a degenerate symmetric bilinear form from the ambient Lorentzian metric, and its radical is globally spanned by the position vector field $\mathcal Z_v=v$ or $Z(v)=v$. The acting group is written either as $G=O^+(m+1,1)$ or as $G=PO(m+1,1)$, with $H$ the stabilizer of a null vector $\ell$; in both descriptions the action is transitive and

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$0

The subgroup $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$1 is identified with the Euclidean rigid motion group $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$2, and the quotient $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$3 is identified with $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$4 (Morón et al., 27 Aug 2025, Palomo, 2020).

A lightlike Cartan geometry is then a Cartan geometry of type $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$5: a principal $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$6-bundle

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$7

equipped with a $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$8-valued Cartan connection

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$9

such that each $$\langle\, ,\, \rangle = \sum_{i=1}^{m+1}dv_i^2-dv_{m+2}^2,$$0 is an isomorphism, $$\langle\, ,\, \rangle = \sum_{i=1}^{m+1}dv_i^2-dv_{m+2}^2,$$1, and $$\langle\, ,\, \rangle = \sum_{i=1}^{m+1}dv_i^2-dv_{m+2}^2,$$2 for $$\langle\, ,\, \rangle = \sum_{i=1}^{m+1}dv_i^2-dv_{m+2}^2,$$3. Its curvature is

$$\langle\, ,\, \rangle = \sum_{i=1}^{m+1}dv_i^2-dv_{m+2}^2,$$4

A distinctive structural feature of the cone model is that it is a first-order Klein geometry but is neither reductive nor parabolic; in particular, $$\langle\, ,\, \rangle = \sum_{i=1}^{m+1}dv_i^2-dv_{m+2}^2,$$5 admits no reductive complement in $$\langle\, ,\, \rangle = \sum_{i=1}^{m+1}dv_i^2-dv_{m+2}^2,$$6 (Morón et al., 27 Aug 2025).

2. Induced lightlike structures on the base manifold

A lightlike manifold is a pair $$\langle\, ,\, \rangle = \sum_{i=1}^{m+1}dv_i^2-dv_{m+2}^2,$$7 in which $$\langle\, ,\, \rangle = \sum_{i=1}^{m+1}dv_i^2-dv_{m+2}^2,$$8 is a symmetric $$\langle\, ,\, \rangle = \sum_{i=1}^{m+1}dv_i^2-dv_{m+2}^2,$$9-tensor satisfying $$\langle\ ,\ \rangle=-dx_0^2+\sum_{i=1}^{m+1}dx_i^2.$$0 for all tangent vectors and whose radical

$$\langle\ ,\ \rangle=-dx_0^2+\sum_{i=1}^{m+1}dx_i^2.$$1

is one-dimensional at every point. When the radical distribution is orientable, one fixes a global spanning vector field $$\langle\ ,\ \rangle=-dx_0^2+\sum_{i=1}^{m+1}dx_i^2.$$2 and writes $$\langle\ ,\ \rangle=-dx_0^2+\sum_{i=1}^{m+1}dx_i^2.$$3 (Palomo, 2020).

Every lightlike Cartan geometry canonically induces such data. On the model quotient $$\langle\ ,\ \rangle=-dx_0^2+\sum_{i=1}^{m+1}dx_i^2.$$4, the invariant lightlike form is

$$\langle\ ,\ \rangle=-dx_0^2+\sum_{i=1}^{m+1}dx_i^2.$$5

or equivalently Euclidean inner product on the $$\langle\ ,\ \rangle=-dx_0^2+\sum_{i=1}^{m+1}dx_i^2.$$6-factor, and its radical is spanned by $$\langle\ ,\ \rangle=-dx_0^2+\sum_{i=1}^{m+1}dx_i^2.$$7. Via the Cartan connection and the identification

$$\langle\ ,\ \rangle=-dx_0^2+\sum_{i=1}^{m+1}dx_i^2.$$8

these descend to a lightlike metric $$\langle\ ,\ \rangle=-dx_0^2+\sum_{i=1}^{m+1}dx_i^2.$$9 on $\mathcal Z_v=v$0 and to a globally defined vector field $\mathcal Z_v=v$1 spanning $\mathcal Z_v=v$2. The radical generator is also described by the grading element

$\mathcal Z_v=v$3

(Palomo, 2020, Morón et al., 27 Aug 2025).

The 2025 theory shows that $\mathcal Z_v=v$4 do not exhaust the geometric content of the Cartan connection. For each

$\mathcal Z_v=v$5

one has the screen distribution $\mathcal Z_v=v$6 and projection

$\mathcal Z_v=v$7

The Cartan geometry determines a family $\mathcal Z_v=v$8 assigning to each $\mathcal Z_v=v$9 a metric linear connection $Z(v)=v$0 on $Z(v)=v$1, and a family $Z(v)=v$2 assigning to each $Z(v)=v$3 a bundle morphism

$Z(v)=v$4

Equivalently, tractor curvature defines a tensor

$Z(v)=v$5

through

$Z(v)=v$6

and $Z(v)=v$7 determines a Galilean connection $Z(v)=v$8 on $Z(v)=v$9 satisfying

$G=O^+(m+1,1)$0

The two additional compatible structures mentioned in the abstract are therefore either $G=O^+(m+1,1)$1 or, equivalently, $G=O^+(m+1,1)$2 (Morón et al., 27 Aug 2025).

3. Standard tractor bundle and intrinsic vector-bundle characterization

The standard tractor bundle of a lightlike Cartan geometry is constructed from the standard representation of $G=O^+(m+1,1)$3 on $G=O^+(m+1,1)$4: $G=O^+(m+1,1)$5 It carries the canonical Lorentzian tractor metric

$G=O^+(m+1,1)$6

and the Cartan connection induces a metric tractor connection $G=O^+(m+1,1)$7. Because $G=O^+(m+1,1)$8 fixes the null vector $G=O^+(m+1,1)$9, there is a distinguished lightlike section

$G=PO(m+1,1)$0

The map

$G=PO(m+1,1)$1

is a vector bundle monomorphism and an isometry between $G=PO(m+1,1)$2 and $G=PO(m+1,1)$3, and it satisfies

$G=PO(m+1,1)$4

Thus $G=PO(m+1,1)$5 is a Lorentzian rank-$G=PO(m+1,1)$6 extension of the degenerate tangent bundle (Morón et al., 27 Aug 2025).

For each $G=PO(m+1,1)$7, there is a null section $G=PO(m+1,1)$8 with

$G=PO(m+1,1)$9

which yields the splitting

$H$0

In this splitting the tractor metric is

$H$1

and the tractor connection is

$H$2

This makes explicit how the additional data $H$3 are read off from the tractor connection (Morón et al., 27 Aug 2025).

The same paper introduces a purely bundle-theoretic replacement for a lightlike Cartan geometry, called a lightlike extension vector bundle. It consists of a rank $H$4 real vector bundle $H$5, a Lorentzian bundle metric $H$6, a metric linear connection $H$7, and a distinguished lightlike section $H$8 such that

$H$9

defines a vector bundle monomorphism $\ell$0. From this one recovers a lightlike metric

$\ell$1

a radical vector field $\ell$2 with $\ell$3, and conversely one reconstructs a lightlike Cartan geometry. The resulting bijection is

$\ell$4

and, more intrinsically,

$\ell$5

This identifies $\ell$6 as exactly the extra information carried by the Cartan connection beyond $\ell$7 (Morón et al., 27 Aug 2025).

4. Curvature, normalization, and conformal correspondence

The Cartan curvature is

$\ell$8

and the tractor curvature is

$\ell$9

A key invariant relation is

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$00

which extracts the tensor $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$01 from tractor curvature. The paper also observes that the induced lightlike metric and radical vector field depend only on the “soldering” part of the Cartan connection,

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$02

so different Cartan connections can induce the same $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$03. The remaining components $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$04 and $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$05 contain the additional geometry encoded by $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$06 or $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$07 (Morón et al., 27 Aug 2025).

A notable criterion singles out the geometries that locally arise as scale bundles of conformal manifolds. In Cartan form,

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$08

and in tractor form this is equivalent to

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$09

for all vector fields $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$10 and tractor fields $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$11. This places the cone model in a precise relation with conformal Cartan geometry through the inclusions

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$12

where $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$13 is the conformal parabolic stabilizing the projective null line. The cone is the bundle of scales of the Möbius sphere, and likewise the correspondence space $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$14 of a conformal Cartan geometry of type $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$15 is the scale bundle of the induced conformal structure (Morón et al., 27 Aug 2025).

Normalization in the lightlike setting is more limited than in parabolic geometry. The 2025 paper states that it does not impose a general normalization of Cartan curvature analogous to parabolic normality for all lightlike Cartan geometries, but instead develops a partial normalization under strong assumptions. Under the condition that $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$16 is collinear with $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$17, the map $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$18 is uniquely determined, equivalently

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$19

A further condition that $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$20 be collinear with $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$21 is equivalent to

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$22

and for $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$23 the vanishing of a certain Ricci-type contraction is equivalent to the Schouten-like formula

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$24

If $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$25 and $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$26, there exists at most one normalized lightlike extension vector bundle or lightlike Cartan geometry satisfying the listed curvature normalization conditions; existence is equivalent to solvability of the equation for $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$27 (Morón et al., 27 Aug 2025).

The earlier 2020 paper develops a complementary extrinsic curvature picture for lightlike hypersurfaces $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$28. Writing

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$29

for the expansion function and

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$30

for the null second fundamental form, it proves that the pull-back of the ambient Levi-Civita connection form is a lightlike Cartan connection if and only if $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$31 is a vector bundle isomorphism, equivalently if and only if $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$32 is generic and $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$33 is nowhere vanishing. In the properly totally umbilical case $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$34, one obtains

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$35

which is the sense in which the construction essentially returns the original lightlike metric (Palomo, 2020).

5. Relation to Carrollian, conformal, and coisotropic Cartan geometry

Lightlike Cartan geometry belongs to a broader landscape of Cartan-geometric models with degenerate causal data. In the review of non-Lorentzian spacetimes, the most directly null structures are Carrollian geometries, characterized by a nowhere-vanishing vector field $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$36 and a corank-one positive semidefinite metric $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$37 with

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$38

That paper also includes the explicit lightcone Klein pair

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$39

realized as the deleted future or past lightcone, and emphasizes that a null hypersurface in a Lorentzian manifold naturally carries a Carrollian structure. A second null mechanism is Newton–Cartan geometry obtained by null reduction along a null Killing vector. This suggests a useful distinction between intrinsically null models, such as Carrollian and lightcone geometry, and quotient constructions along null directions (Figueroa-O'Farrill, 2022).

The specific cone model of lightlike Cartan geometry is closely related to conformal geometry but is not identical to it. Conformal geometry is parabolic, with $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$40 stabilizing an isotropic ray, whereas the lightlike model uses the smaller subgroup $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$41. The 2025 theory repeatedly emphasizes that the resulting geometries are neither parabolic nor reductive, even though tractor methods remain effective. A plausible implication is that lightlike Cartan geometry occupies an intermediate position: it is close enough to conformal geometry for bundles of scales and tractor constructions to be decisive, but far enough from the parabolic framework that standard normalization machinery does not automatically apply (Morón et al., 27 Aug 2025).

More generally, holonomy reduction theory provides a mechanism by which isotropic tractor data in conformal, projective, and CR Cartan geometries generate null or boundary-type strata carrying induced Cartan geometries. In conformal geometry, the parabolic subgroup already stabilizes an isotropic ray, and parallel tractors produce decompositions into open Einstein regions and closed hypersurface-type or more degenerate isotropic loci. The theory of curved orbit decompositions shows that such strata are locally modeled on the corresponding isotropic orbits in the homogeneous space and inherit canonical Cartan geometries (Cap et al., 2011).

A different adjacent framework is supplied by coisotropic Cartan geometry. For any coisotropic Cartan geometry, including parabolic geometries, the associated tractor bundle carries a canonical twisted Courant algebroid. The central linear algebra is again pseudo-metric and isotropic/coisotropic: the anchor kernel has orthogonal complement

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$42

The paper is not explicitly about lightlike Cartan geometries, but it is relevant whenever Cartan-geometric structures are built from isotropic or coisotropic data, especially in tractor formulations of conformal geometry (Xiaomeng, 2012).

6. Examples, constructions, and neighboring literature

Two explicit classes of examples are given in the 2025 theory. First, if $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$43 is Sasakian, then

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$44

is a lightlike metric with radical spanned by $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$45, and using the generalized Tanaka connection one defines a lightlike-compatible structure by

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$46

Second, for a degenerate hyperplane $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$47 with coordinates $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$48,

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$49

and with $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$50 one sets

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$51

Both yield concrete lightlike-compatible structures and hence lightlike Cartan geometries (Morón et al., 27 Aug 2025).

The 2020 paper provides two complementary construction methods. Extrinsically, a causally oriented lightlike hypersurface in a Lorentzian manifold yields a lightlike Cartan geometry when the pulled-back Levi-Civita form satisfies the Cartan isomorphism condition. Intrinsically, starting from a generic lightlike manifold $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$52 with complete $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$53 whose $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$54-flow is free and proper, one constructs ambient Lorentzian metrics inspired by the Fefferman–Graham ambient metric construction: $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$55 Imposing

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$56

forces $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$57, giving the one-parameter family

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$58

For every spacelike section and every $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$59, the pull-back of the Levi-Civita connection form of $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$60 is a lightlike Cartan geometry on $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$61, and the induced data satisfy

$$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$62

(Palomo, 2020).

An adjacent but distinct line of work is the Cartan-style moving-frame analysis of lightlike submanifolds. For lightlike surfaces in $$\mathcal N^{m+1}=\{v\in \mathbb L^{m+2}:\langle v,v\rangle=0,\ v_0>0\},$$63, Cartan’s method of moving frames yields a complete local classification of constant-type surfaces into planes, cones, and a non-conical family depending on one arbitrary function of one variable. That work is not a theory of lightlike Cartan geometries in the sense of cone-modeled Cartan connections, but it is naturally interpreted in Cartan-geometric language because it proceeds by frame adaptation, structure reduction, extraction of invariants, and canonical coframing (Carlsen et al., 2013).

A source note is also warranted. The document titled "An introduction to Cartan geometries" (McKay, 2023) does not contain mathematical text on Cartan geometry; it consists only of a short TikZ source drawing two curves in a 3D plot. It therefore contributes no definitions, curvature formulas, or results on Cartan geometry or lightlike geometry (McKay, 2023).