MAGNETIC FIELDS ON SUB-RIEMANNIAN MANIFOLDS

The paper proves exactly the three claims: (a) normal extremals for the lifted structure project to magnetic geodesics (Proposition 10), via the conservation of the vertical covector component and identification with the magnetic Hamiltonian on M ; (b) abnormal extremals project to horizontal curves satisfying ι_{γ̇}β = 0 (Proposition 15), obtained by restricting the canonical symplectic form to D^⊥ and identifying its characteristic distribution ; and (c) on {β ≠ 0} the lifted distribution has growth vector (2,3,4), hence step 3, and there is a unique abnormal through each point, because the kernel of β on D is a line field (Theorem 2 and Lemma 13) . The candidate solution reproduces the same Hamiltonian reduction in (a), the same restricted-symplectic computation in (b) (with τ ≡ θ = dw − A), and the same bracket argument in (c). Differences are cosmetic (e.g., using A vs A′ in τ and not explicitly invoking Goh’s condition), not substantive.