Bumpy Metric Theorem for Co-Rank 1 Sub-Riemannian and Reversible Sub-Finsler Metrics

The paper proves a bumpy-metric-type statement for co-rank 1 sub-Riemannian/reversible sub-Finsler settings on supercritical (contact-type) energy levels: for a residual set of parameters (K,U), every D-regular closed orbit is non-degenerate (indeed, with no root-of-unity eigenvalues) . The core local ingredient (Theorem C) shows that, after a preliminary kinetic perturbation T near the orbit, the map u ↦ dP of restricted transition maps is a submersion at u=0; this requires a neat and D-regular time and uses a careful normal form and a Mane-type controllability argument . The global, parametric transversality/projection step then yields genericity, with explicit handling of the root-of-unity spectrum via sets P1–P5 and a meager Fσ analysis . By contrast, the model’s key step asserts that varying the potential U alone, supported near a single D-regular time, generically removes 1-eigenvectors of the return map; it neither proves the required surjectivity onto the cokernel nor addresses the known difficulties at non-neat times (and the need for reversibility/supercritical levels to avoid round trips) that the paper treats explicitly . In short, the model claims a stronger lemma (potential-only transversality) that the paper does not assert and even poses as an open question in related form, so the model’s proof is incomplete/incorrect for the stated problem.