完整空间
Holonomic Spaces
摘要：
完整空间$(V,H,L)$是赋范向量空间$V$，是$Aut(V, \|\cdot\|)$的子群$H$和具有凸性的群范数$L$。证明了在度量$d_L(u,v)=\inf_{a\in H}\{\sqrt{L^2(a)+\|u-av\|^2}\}$下，$V$是局部等距于欧氏球的度量空间.给定黎曼流形上向量丛$E$上的Sasaki型度量，我们证明了三元组$(E_p,Hol_p,L_p)$是完整空间，其中$Hol_p$是完整群，$L_p$是定义在其中的长度范数.由$L_p$给出的$Hol_p$上的拓扑比子空间拓扑更精细，同时仍然保留许多期望的性质。利用这些概念，我们引进了黎曼流形的完整性半径的概念，并证明了它是正的。这些结果适用于黎曼流形的Gromov-Hausdorff收敛性。
英文摘要：
A holonomic space $(V,H,L)$ is a normed vector space, $V$, a subgroup, $H$, of $Aut(V, \|\cdot\|)$ and a group-norm, $L$, with a convexity property. We prove that with the metric $d_L(u,v)=\inf_{a\in H}\{\sqrt{L^2(a)+\|u-av\|^2}\}$, $V$ is a metric space which is locally isometric to a Euclidean ball. Given a Sasaki-type metric on a vector bundle $E$ over a Riemannian manifold, we prove that the triplet $(E_p,Hol_p,L_p)$ is a holonomic space, where $Hol_p$ is the holonomy group and $L_p$ is the length norm defined within. The topology on $Hol_p$ given by the $L_p$ is finer than the subspace topology while still preserving many desirable properties. Using these notions, we introduce the notion of holonomy radius for a Riemannian manifold and prove it is positive. These results are applicable to the Gromov-Hausdorff convergence of Riemannian manifolds.
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