arXiv subjects

Nicklas Day

Publications and source records attributed to Nicklas Day.

3 recordsLinked to original sources

arXiv2026

$G_2$ and the Maximally Symmetric (3, 8) Distribution with 6-Dimensional Square

In 1910, \'{E}lie Cartan famously realized the split real form of the exceptional Lie group $G_2$ as the symmetry group of the maximally symmetric rank 2 distribution on a 5-dimensional manifold with the small growth vector (2,3,5). In this paper, we discover a new appearance of $G_2$ in the geometric theory of distributions, arising from a rank 3 distribution on an 8-dimensional manifold with the growth vector $(3,6,8)$. The algebra of infinitesimal symmetries of this distribution at any point is 29-dimensional and isomorphic to $(\mathfrak{g}_2 \oplus \mathbb{R}) \ltimes W$, where $\mathfrak{g}_2$ is the Lie algebra of $G_2$ and $W$ is an adjoint module of $\mathfrak{g}_2$. Our model possesses three remarkable properties. First, it is maximally symmetric among all bracket-generating rank 3 distributions with a 6-dimensional square (a family that includes both (3,6,8) and (3,6,7,8) distributions). To the best of our knowledge, this is the first example of a family of distributions defined by a set of prescribed small growth vectors in which maximal symmetry is achieved by a member whose growth vector is not the longest. Second, this model provides the first counterexample to the conjecture that all bracket-generating rank 3 distributions with a 6-dimensional square are of maximal class at a generic point (which is known to hold in dimensions 6 and 7). Third, further analysis yields the control-theoretic consequence that all abnormal extremal trajectories of this model originating at any point of the ambient manifold have a corank of at least 2. To our knowledge, this is the first example with this property among bracket-generating distributions with generic small growth vector for a given rank and ambient dimension. We also give an interpretation of our model in terms of split-octonions, more precisely, in terms of a natural algebraic structure on the tangent bundle to split octonions.

math.DG↗

arXiv2025

Canonical Frames for Bracket Generating Rank 2 Distributions which are not Goursat

We complete a uniform construction of canonical absolute parallelism for bracket generating rank $2$ distributions with $5$-dimensional cube on $n$-dimensional manifold with $n\geq 5$ by showing that the condition of maximality of class that was assumed previously by Doubrov-Zelenko for such a construction holds automatically at generic points. This also gives analogous constructions in the case when the cube is not $5$-dimensional but the distribution is not Goursat through the procedure of iterative Cartan deprolongation. This together with the classical theory of Goursat distributions covers in principle the local geometry of all bracket generating rank 2 distributions in a neighborhood of generic points. As a byproduct, for any $n\geq 5$ we describe the maximally symmetric germs among bracket generating rank $2$ distributions with $5$-dimensional cube, as well as among those which reduce to such a distribution under a fixed number of Cartan deprolongations. Another consequence of our results on maximality of class is for optimal control problems with constraint given by a rank $2$ distribution with $5$-dimensional cube: it implies that for a generic point $q_0$ of $M$, there are plenty abnormal extremal trajectories of corank $1$ (which is the minimal possible corank) starting at $q_0$. The set of such points contains all points where the distribution is equiregular.

math.DG↗

arXiv2025

Symplectification of Rank 2 Distributions, Normal Cartan Connections, and Cartan Prolongations

We study the Doubrov--Zelenko symplectification procedure for rank $2$ distributions with $5$-dimensional cube -- originally motivated by optimal control theory -- through the lens of Tanaka--Morimoto theory for normal Cartan connections. In this way, for ambient manifolds of dimension $ n \geq 5 $, we prove the existence of the normal Cartan connection associated with the symplectified distribution. Furthermore, we show that this symplectification can be interpreted as the $(n-4)$th iterated Cartan prolongation at a generic point. This interpretation naturally leads to two questions for an arbitrary rank $2$ distribution with $5$-dimensional cube: (1) Is the $(n-4)$th iterated Cartan prolongation the minimal iteration where the Tanaka symbols become unified at generic points? (2) Is the $(n-4)$th iterated Cartan prolongation the minimal iteration admitting a normal Cartan connection via Tanaka--Morimoto theory? Our main results demonstrate that: (a) For $n > 5$, the answer to the second question is positive (in contrast to the classical $n = 5$ case from $G_2$-parabolic geometries); (b) For $n \geq 5$, the answer to the first question is negative: unification occurs already at the $(n-5)$th iterated Cartan prolongation.

math.DG↗