发表机构
Yale University; New York University; University of Vienna（耶鲁大学; 纽约大学; 维也纳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究黎曼区域到洛伦兹区域延拓的拓扑障碍，给出相对欧拉类障碍消失的充要条件，并构造显式符号变化度量，揭示拓扑变化可局限于洛伦兹区域。
AI 中文摘要
受瞬子延拓问题的启发，我们研究通过光滑横截符号变化将黎曼区域延拓到洛伦兹区域的几何拓扑选择规则。给定光滑流形 $M$ 沿超曲面 $\mathcal{H}$ 的环带粘合 $M=\overline{M_R}\cup_{\mathcal{H}}\overline{M_L}$，我们证明：在 $M_R$ 上为黎曼度量、在 $M_L$ 上为洛伦兹度量的度量存在，当且仅当 $\overline{M_L}$ 上的相对欧拉类障碍消失。该障碍以依赖于黎曼区域哪些边界分量代表大爆炸、哪些代表大挤压的方式组合普通欧拉示性数 $χ$。它同时限制 $M_R$ 的选择；例如，当 $\dim M$ 为偶数时要求 $χ(\overline{M_R})=χ(M)$，当 $\dim M$ 为奇数时要求 $χ(\overline{M_R})=χ(\overline{M_L})$。我们还确定了在此类环带粘合中 $\overline{M_R}$、$\mathcal{H}$ 和 $\overline{M_L}$ 可能出现的紧致类型，并为每种情形构造显式的符号变化度量。当 $M$ 是带边紧流形的内部时，拓扑变化可以完全包含在洛伦兹区域内。
英文摘要
Motivated by problems in instanton continuation, we establish geometro-topological selection rules for continuing a Riemannian region into a Lorentzian region through a smooth transverse change of signature. Given a collar-gluing $M=\overline{M_R} \cup_{\mathcal{H}}\overline{M_L}$ of a smooth manifold $M$ along a hypersurface $\mathcal{H}$, we show that an admissible signature-changing metric of the form $\widetilde g=g-fV^\flat\otimes V^\flat$ with $g$ a Riemannian metric, having Riemannian signature on $M_R$ and Lorentzian signature on $M_L$, exists if and only if a relative Euler class obstruction on $\overline{M_L}$ vanishes. This obstruction combines ordinary Euler characteristics $χ$ in a manner that depends on which components of $\mathcal{H}$ represent big bangs and which represent big crunches. It also restricts the choice of $M_R$; for example, if $M$ is closed and $\dim M$ is even, then it requires$χ(\overline{M_R})=χ(M)$. We also determine and classify the possible compactness types of $\overline{M_R}$, $\mathcal{H}$, and $\overline{M_L}$ occurring in such a collar-gluing construction, and construct explicit signature-changing metrics in each case. Topology change can be entirely contained within the Riemannian region when $M$ is the interior of a compact manifold with boundary.
Comments27 pages, 8 figures; minor revisions
