SYMPLETIC REDUCTION OF THE SUB-RIEMANNIAN GEODESIC FLOW ON META-ABELIAN CARNOT GROUPS

The paper’s Theorem A and reduction picture are consistent: it defines an a*-valued polynomial 1-form α_G, derives the reduced Hamiltonian H_µ on T*H with both the “magnetic” minimal-coupling term and an additional scalar potential φ(x)=||β_µ(x)||^2, and proves the horizontal-lift/projection correspondence (i)–(ii) via a Poisson-bracket/ODE comparison (Theorem A; see the statement, the α_G and H_µ definitions, and the proof in Sections 1, 2, 4, 5) . The candidate solution’s symplectic reduction argument is close in spirit and correctly identifies α_G via a Maurer–Cartan/BCH construction and the momentum map level set, but it omits the explicit β-part and therefore misses the scalar potential φ(x) in H_µ, writing H_µ(x,p)=1/2||p+α_µ(x)||^2 with the norm on TH. The paper’s correct expression (1.2) shows there is an extra 1/2||β_µ(x)||^2 term that depends on µ and x, arising from the V-component of g_1 and cyclic momenta (pk, prj) on T*G that become parameters after reduction (Proposition 3.1 and (1.2)) . Because the model’s Hamiltonian omits this term, its reduced flow is, in general, not the same as the paper’s α_G-system, so its claim “this proves (i)–(ii)” is incorrect as stated.