发表机构
Central China Normal University（华中师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了欧拉示性数至多为3的严格正截面曲率完备可定向爱因斯坦四流形必等距于标准球面或复射影平面，并应用于非平凡Killing场情形。
AI 中文摘要
一个猜想认为，具有严格正截面曲率的完备可定向爱因斯坦四流形必定等距于带Fubini--Study度量的四维球面$S^4$或复射影平面$\nmathbb{CP}^2$。本文在该流形同胚于$S^4$或$\nmathbb{CP}^2$的情形下证实了这一猜想。更精确地说，我们证明了：一个具有严格正截面曲率且欧拉示性数$\chi(M) \le 3$（等价地，由Freedman分类定理，同胚于$S^4$或$\nmathbb{CP}^2$）的完备可定向爱因斯坦四流形$M$，必定等距于标准球面$S^4$或Fubini--Study度量的$\nmathbb{CP}^2$。作为应用，我们得到：一个具有严格正截面曲率且容许非平凡Killing场的完备可定向爱因斯坦四流形，必定等距于标准球面$S^4$或Fubini--Study度量的$\nmathbb{CP}^2$。
英文摘要
It is conjectured that a complete, orientable Einstein four-manifold with strictly positive sectional curvature must be isometric to either the round four-sphere $S^4$ or the complex projective plane $\mathbb{CP}^2$ with the Fubini--Study metric. In this paper, we confirm this conjecture when the manifold is homeomorphic to $S^4$ or $\mathbb{CP}^2$. More precisely, we show that a complete, orientable Einstein four-manifold $M$ with strictly positive sectional curvature and Euler characteristic $χ(M) \le 3$ (equivalently, by Freedman's classification theorem, homeomorphic to $S^4$ or $\mathbb{CP}^2$) is isometric to either the round $S^4$ or the Fubini--Study $\mathbb{CP}^2$. As an application, we obtain that a complete, orientable Einstein four-manifold with strictly positive sectional curvature admitting a nontrivial Killing field is isometric to either the round $S^4$ or the Fubini--Study $\mathbb{CP}^2$.
Comments8 pages
