2025/06/10 by Nicklas Day, Day, Nicklas, Boris Doubrov +3 · 1 citation
Mathematics · #53A55 #53C05 #58A17 #58A30 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2506.09232
openalex publication_date 2025/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ≥ 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 G2-parabolic geometries); (b) For n ≥ 5, the answer to the first question is negative: unification occurs already at the (n-5)th iterated Cartan prolongation.