凸超曲面中的稳定闭测地线与稳定八字形
Stable closed geodesics and stable figure-eights in convex hypersurfaces
AI总结：
本文在奇数维闭凸超曲面中构造指数零的非退化闭测地线，并在各维构造稳定八字形嵌入，说明 Synge 定理具有尖锐性。
摘要：
对于每个奇数 $n \geq 3$，我们构造了 $\mathbb{R}^{n+1}$ 中的一个闭凸超曲面，它包含一条 Morse 指数为零的非退化闭测地线。J. L. Synge 的一个经典定理会在偶数 $n$ 时排除这种构造，因此在某种意义上，我们证明了 Synge 定理是“尖锐的”。我们还构造了稳定的八字形：也就是说，对于每个 $n \geq 3$，我们将八字形图嵌入 $\mathbb{R}^{n+1}$ 的一个闭凸超曲面中，使得该嵌入的充分小变分要么保持其像不变，要么必须增加其长度。这些指数为零的测地线和稳定八字形主要通过在凸多面体中构造具有“受控平行移动”的显式弹子球轨迹得到。
英文摘要：
For each odd $n \geq 3$, we construct a closed convex hypersurface of $\mathbb{R}^{n+1}$ that contains a non-degenerate closed geodesic with Morse index zero. A classical theorem of J. L. Synge would forbid such constructions for even $n$, so in a sense we prove that Synge's theorem is "sharp." We also construct stable figure-eights: that is, for each $n \geq 3$ we embed the figure-eight graph in a closed convex hypersurface of $\mathbb{R}^{n+1}$, such that sufficiently small variations of the embedding either preserve its image or must increase its length. These index-zero geodesics and stable figure-eights are mainly derived by constructing explicit billiard trajectories with "controlled parallel transport" in convex polytopes.
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