论文标题
CAT（0）空间中的内在等级
Intrinsic Rank in CAT(0) Spaces
论文作者
论文摘要
让$ x $成为一个适当的，地理上完整的猫（0）空间，满足陈和Eberlein的双重性状态。我们通过证明平行集$ p_v $的Geodesics $ v $ in $ x $的$ p_v $通常是平坦的，这表明了$ x $的强烈排名概念。更确切地说，让$ gx $是$ x $的参数化单位速度测量学的空间。 $ gx $中有一个唯一的$ k $和一个密集的$g_δ$ set $ \ nathcal {a} $，因此$ p_v $是等于euclidean space $ \ mathbb {r}^k $的均等，对于所有$ v \ in \ mathcal {a} $。因此，$ \ mathbb {r}^k $在gx $中的每个$ v \ in $ p_v $中嵌入。
Let $X$ be a proper, geodesically complete CAT(0) space which satisfies Chen and Eberlein's duality condition. We show the existence of a strong notion of rank for $X$ by proving that the parallel sets $P_v$ of geodesics $v$ in $X$ are generically flat. More precisely, let $GX$ be the space of parametrized unit-speed geodesics in $X$. There is a unique $k$ and a dense $G_δ$ set $\mathcal{A}$ in $GX$ such that $P_v$ is isometric to flat Euclidean space $\mathbb{R}^k$, for all $v \in \mathcal{A}$. It follows that $\mathbb{R}^k$ isometrically embeds in $P_v$ for every $v \in GX$.
