CARNOT METRICS, DYNAMICS AND LOCAL RIGIDITY

The paper’s Theorem 1.14 proves exactly the local rigidity statement for higher hyperbolic rank metrics near quaternionic or octonionic hyperbolic metrics, with κ ≥ −1, and upgrades to isometry by Mostow; see the statement in the introduction and the detailed proof in Section 6.2, which rely on Proposition 6.2/Corollary 6.10 and Lemma 6.1 to identify the exponent-1 distribution and keep its dimension, and then apply the geodesic-flow local rigidity criterion (Theorem 1.9) to conclude local symmetry . The model’s solution reproduces these same ingredients: equality in Riccati/Rauch to identify the exponent-1 (−1-curvature) directions, structural stability of the Brin–Pesin asymptotic holonomy to keep the rank (1,3,7), Carnot tangent-cone structure and smoothness along unstable leaves, and the local rigidity criterion, followed by Mostow. The only discrepancies are not substantive: a different curvature normalization is used in the model’s outline (leading to swapped ‘slow/fast’ labels) and some steps (e.g., the permanence argument to get C1-dependence needed for tangent-cone isomorphisms) are only sketched in the model but are provided in the paper (Remark 6.11) .