平面反射投影台球：奇周期、周期五与度量刚性
Reflective projective billiards in the plane: odd periods, period five, and metric rigidity
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中文总结 AI 辅助
本文研究反射律逐边恒定的多边形投影台球，给出显式单参数族覆盖所有至少三的周期（含奇周期），并完整刻画周期五情形，证明度量刚性，指出仅球面支持周期轨道开族。
中文摘要 AI 辅助
我们研究反射律在每条边上恒定的多边形投影台球。我们给出一个显式单参数族，对于每个至少为三的周期（包括每个奇周期），该族具有一个周期轨道的开集，并将先前已知的中心投影例子识别为特殊参数值。对于周期五，我们描述了完整的标量单值簇，证明其不可约性，并表明每个构型保持一个非零二次型。在所述的不同中心和共线假设下，该形式是非退化的；这给出了凸五反射轨迹的显式有理分类。我们还将正定成员与球面台球联系起来，并证明在具有测地线壁的解析单连通常曲率曲面中，只有球面可以承载周期轨道的开族。对于具有全局壁反射的一般完备解析环境，我们证明了测地流的紧致性和公共周期。这些论证将台球返回映射化简为调和同调的乘积，并将代数闭包条件与严格首次碰撞不等式分开。
英文摘要
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.
