Abstract:In 1910, Élie 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.
| Subjects: | Differential Geometry (math.DG); Optimization and Control (math.OC) |
| MSC classes: | 58A30, 58A17, 37J60, 53A55, 17B25, 17B66, 17A75, 53C17, 49K15 |
| Cite as: | arXiv:2605.25910 [math.DG] |
| (or arXiv:2605.25910v1 [math.DG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25910 arXiv-issued DOI via DataCite (pending registration) |
From: Nicklas Day [view email]
[v1]
Mon, 25 May 2026 14:46:06 UTC (48 KB)
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