On global rigidity of transversely holomorphic Anosov flows

The paper establishes the dichotomy for topologically transitive, transversely holomorphic Anosov flows on closed 5–manifolds (suspension of a hyperbolic automorphism of a complex torus, or—up to finite covers—geodesic flow on a compact hyperbolic 3–manifold) via a transverse (PSL(2,C)×PSL(2,C), P1×P1) structure and a careful developing map/holonomy analysis (Theorem 4.20) , building on earlier results that strong (un)stable leaves are complex and that in dimension five the weak foliations become transversely holomorphic/projective under transitivity . In the suspension branch the paper invokes Plante to obtain a global section and a holomorphic return map, then Ghys to conclude the section is a complex torus ; in the other branch, the representation-theoretic and Haefliger arguments yield C∞ orbit equivalence to a hyperbolic geodesic flow after a finite cover . By contrast, the model’s proof crucially (and incorrectly) asserts that the joint distribution Es ⊕ Eu is smooth because it “equals the smooth normal bundle,” and then applies Fang’s UQC classification; however, Es ⊕ Eu is generally only Hölder as a subbundle of TM in Anosov flows, so the key smoothness hypothesis required to invoke Fang is not justified. The model’s conclusion matches the paper’s, but its logical route is invalid.