Reflective projective billiards in the plane: odd periods, period five, and metric rigidity

Abstract: We study polygonal projective billiards whose reflection law is constant on each side. We give a single explicit family with an open set of periodic trajectories for every period at least three, including every odd period, and identify the previously known centrally projective examples as a special parameter value. For period five we describe the full scalar-monodromy variety, prove its irreducibility, and show that every configuration preserves a nonzero quadratic form. Under the stated distinct-centre and non-collinearity hypotheses that form is nondegenerate; this yields an explicit unirational classification of the convex five-reflective locus. We also relate the definite members to spherical billiards and prove that, among analytic simply connected constant-curvature surfaces with geodesic walls, only the sphere can carry an open family of periodic trajectories. For general complete analytic ambients with global wall reflections we prove compactness and a common period for the geodesic flow. The arguments reduce the billiard return map to products of harmonic homologies and separate the algebraic closure condition from the strict first-impact inequalities.