Classification of Morse–Smale projectively Anosov flows

Classify the closed 3-manifolds that admit Morse–Smale projectively Anosov flows, determine whether any bi-contact homotopy class admits infinitely many distinct examples of such flows, and characterize the relationship between this condition and the contact-topological features of the underlying contact structures.

Background

The paper constructs structurally generic, transversely oriented projectively Anosov flows without saddle periodic orbits and classifies the resulting manifolds as lens spaces L(n,1). It also constructs Morse–Smale examples with any prescribed number of saddle periodic orbits and zero topological entropy.

These constructions motivate broader unresolved classification questions: whether the class of manifolds supporting Morse–Smale projectively Anosov flows is larger than the explicitly constructed examples, whether a fixed bi-contact homotopy class can contain infinitely many distinct flows, and which contact-topological invariants control the existence of such flows. The paper does not resolve these questions.