论文标题
三个顽固球的本地和全球同质性
Local and Global Homogeneity for Three Obstinate Spheres
论文作者
论文摘要
在本说明中，我们完成了一项针对全球均匀的Riemannian商的研究，$γ\ Backslash（m，ds^2）$。具体而言，$ m $是一种均质的空间$ g/h $，承认$ g $ invariant的riemannian标准严格积极的截面曲率，$ ds^2 $是$ g $ - $ g $ - 不变的riemannian in $ m $，不一定是正常的，不一定是正常的，并且不一定是正常的。同质性的猜想是，$γ\ backslash（m，ds^2）$在且仅当$（m，ds^2）$都是同质的时（全球）同质的，γ$中的每一个$γ\inγ$inγ$ inγ$ inγ$ inγ$ inγ$ inγ$ inγ$ ind of（m，ds^2）$ compant置换。在较早的论文中，我们验证了所有同质空间的猜想，这些空间承认了一个不变的riemannian正曲率指标 - 除三个例外，所有奇数尺寸球体都没有屈服于早期的方法。在这里，我们开发了一些方法，让我们验证这三个固执的领域的同质性猜想。这完成了对正弯曲中同质性猜想的验证。
In this note we complete a study of globally homogeneous Riemannian quotients $Γ\backslash (M,ds^2)$ in positive curvature. Specifically, $M$ is a homogeneous space $G/H$ that admits a $G$-invariant Riemannian metric of strictly positive sectional curvature, and $ds^2$ is a $G$--invariant Riemannian metric on $M$, not necessarily normal and not necessarily positively curved. The Homogeneity Conjecture is that $Γ\backslash (M,ds^2)$ is (globally) homogeneous if and only if $(M,ds^2)$ is homogeneous and every $γ\in Γ$ is of constant displacement on $(M,ds^2)$. In an earlier paper we verified that conjecture for all homogeneous spaces that admit an invariant Riemannian metric of positive curvature -- with three exceptions, all odd dimensional spheres, which surprisingly did not yield to the earlier approaches. Here we develop some methods that let us verify the Homogeneity Conjecture for those three obstinate spheres. That completes verification of the Homogeneity Conjecture in positive curvature.
