Dynamics on Bi-Lagrangian Structures and Cherry maps

The paper’s Theorem 4.1 is stated precisely as the three claims about lifting a bi‑Lagrangian structure to T∗M (with dθ and with ω̃ = π∗ω + dθ) and to TM via the complete lift, including the statement about affineness and the Hess connection lifting to ∇c (see the statement in section 4.1 ). The proof uses the conormal foliation N∗F and an explicit coordinate description to establish that N∗F1 and N∗F2 are transverse Lagrangian foliations for (T∗M, dθ) and for (T∗M, ω̃), and then applies Hess’s adapted-coordinate criterion to conclude affineness (Lemma 4.2 and its calculation; the coordinate spans (4.2) and the local form (4.3) of dθ; Theorem 3.3) . For TM, the paper invokes properties of complete lifts (identities (2.3)–(2.6)) to show the lifted foliations are Lagrangian and that ∇c preserves them and ωc, hence is the Hess connection (point 3) . However, the paper does not explicitly justify that the complete lift ωc is a symplectic (closed, nondegenerate) 2‑form on TM; the proof implicitly assumes this standard fact without proof or citation. This is a small but real gap in the argument of point (3). By contrast, the candidate solution supplies this missing ingredient by citing standard lifting results ensuring that ωc is symplectic and by giving a clean coordinate proof for T∗M with explicit foliations D1, D2 and a flat (zero‑Christoffel) local form of the Hess connection. Thus the model’s solution is correct and slightly more complete, while the paper needs a minor fix to document the symplecticity of ωc.