AI 中文总结
本文证明 Poly-Raby 定理在无限维希尔伯特空间中分裂：平方距离函数的 C^k 光滑性刻画弱 C^k 子流形，而 C^k 子流形需附加最高阶导数的逐点等度连续性条件。
AI 中文摘要
Poly 和 Raby 的一个经典结果（见 \\(\cite{Poly-Raby:1984}\\)）指出，在有限维欧几里得空间中，闭集上一点附近的平方距离函数的 \\(\mathcal{C}^k\\) 光滑性（其中 \\(k\geq2\\)）刻画了该集合作为子流形的光滑性，且光滑阶数相同。本文表明，这一刻画在无限维希尔伯特空间中发生分裂：平方距离函数的 \\(\mathcal{C}^k\\) 光滑性刻画了弱 \\(\mathcal{C}^k\\) 子流形，而 \\(\mathcal{C}^k\\) 子流形则由该光滑性加上平方距离最高阶导数的逐点等度连续性条件共同刻画。Poly 和 Raby 的微分同胚、其相关的图表示及其逆的描述仍然是这两种刻画的共同几何基础。本工作的发展得到了 GPT-6 Astra 的协助。
英文摘要
A classic result of Poly and Raby \cite{Poly-Raby:1984} states that, in finite-dimensional Euclidean spaces, $\mathcal{C}^k$-smoothness (with $k\geq2$) of the squared distance function near a point of a closed set characterizes smoothness of the set as a submanifold, with the same order of differentiability. In this work, we show that this characterization splits in infinite-dimensional Hilbert spaces: $\mathcal{C}^k$-smoothness of the squared distance function characterizes weakly $\mathcal{C}^k$-submanifolds, while $\mathcal{C}^k$-submanifolds are characterized by this smoothness together with an additional pointwise equicontinuity condition on the highest-order derivative of the squared distance. The diffeomorphism of Poly and Raby, its associated graph representation, and the description of its inverse remain the common geometric basis of both characterizations. The development of this work was assisted by GPT-6 Astra.
