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I´m trying to think of some examples of CAT(0)-spaces that are not manifolds. I haven´t found any examples in the book of Bridson and Haefliger, so I have tried to come up with some own examples. I suppose there are some involving Graph Theory, but since I´ver never studied graph theory, I try to avoid it for now.

The only example I could think that migth work is the following: Let $A=\{(x,y) \in \mathbb{R}^2 \;\vert\; x \cdot y =0\}$ equipped with the taxicab metric $d_1$. We have shown in my Differential Geometry 1 course that this is not a manifold by the usual argument involving connected components. When thinking and drawing about it, it feels like a CAT(0)-space, but I was not able to prove it. I can't find any connection between a triangle in $A$ and a corresponding comparison triangle in $\mathbb{R}^2$, except the length of the sides.

$\textbf{EDIT:}$ I have come to the conclusion that the above $(A,d_1)$ is in fact an $\mathbb{R}$-tree and this answer Geometric realisations of trees are Cat(0) proved that it is a CAT(0)-space. I´m still interested in different examples, if anyone knows something!

Does anyone know some other examples of such spaces?

Saimel
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Exercise - Euclidean geometry : In $\mathbb{E}^2$, consider a triangle $[ABC]$. If $\theta=\angle BAC$, then $|B-C|^2=|A-B|^2+|A-C|^2-2|A-B||A-C|\cos\theta$. Hence when $|A-B|,\ |A-C|$ are fixed, then $\frac{d}{d\theta}|B-C|\geq 0$.

Example : $Y=\bigcup_i\ Y_i$, union of closed half planes, where $Y_1=\{ (x,y,z)| z=0,\ x\geq 0\},\ Y_2= \{ (x,y,z) | z=0,\ x\leq 0\} ,\ Y_3= \{(x,y,z)|x=0,\ z\geq 0\} $

Proof : To show that $Y$ is ${\rm CAT}[0]$ space, define a triangle $ \Delta = [y_1y_2y_3],\ y_i\in Y_i$

If $[\overline{y}_1\overline{y}_2\overline{y}_3]$ is a comparison triangle, we have to show that $|y_3-p|\leq |\overline{y}_3-\overline{p}|$ for all $p\in [y_1y_2]$.

If $P$ is an intersection point between a line segment $[y_1y_2]$ and $y$-axis, and if we rotate a point $y_3$ $-\frac{\pi}{2}$ degree along $y$-axis so that we have $Y_3$, then we consider the triangle $[y_1PY_3]$. If $p\in [y_1P]$, then we have $ |p-Y_3|\leq |\overline{p}-\overline{y}_3|$ by Exercise. Remainings are trivial.

Example : $ \{ (x,y,z)| z\leq 0\} \bigcup \{ (x,y,z)| y=0\}$. The proof is almost same to the above.

Remark : Circle is not a ${\rm CAT}[0]$ space since a shortest path between two points is not unique.

HK Lee
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