Comparison triangles, the CAT(0) and CAT(1) inequalities, local CAT, local geodesics and round circles

Fix the following definitions and conventions for this page.

(1) Models. ${\mathbb{E}}^2$ is ${\mathbb{R}}^2$ with the Euclidean metric $d_2$ (${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it). The comparison sphere is $S^2\subset {\mathbb{R}}^3$ with the round metric $d_S(x,y):=\arccos (x\cdot y)$ (Spherical Gram simplices and angular links of Euclidean faces, Principal inverse sine and inverse cosine); more generally $d_S(x,y):=\arccos (x\cdot y)$ is defined on every sphere $S^{n-1}$ (Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$).

(2) Geodesic triangles and comparison. A geodesic triangle in a metric space $X$ consists of three points $p,q,r\in X$ and a choice of geodesic segments $[p,q]$, $[q,r]$, $[r,p]$ joining them (Geodesics and geodesic metric spaces); its perimeter is $d(p,q)+d(q,r)+d(r,p)$. A comparison triangle for it in ${\mathbb{E}}^2$, or in $S^2$ when its perimeter is $<2\pi$, is a triangle $(\bar{p},\bar{q},\bar{r})$ in that model with the same three side lengths; it is unique up to an isometry of the model (Comparison triangles in the Euclidean plane and the round sphere, model spaces, and CAT(0) and CAT(1) consequences ↗). For an occurrence of a point $x$ on a specified chosen side, the comparison point $\bar{x}$ is the point of the corresponding side of the comparison triangle at the same distance from the corresponding vertex; a vertex corresponds to itself. If a point belongs to more than one side, each side occurrence has its own comparison point, and the CAT inequalities quantify over every pair of side occurrences.

(3) The CAT inequalities. A metric space $X$ is CAT(0) if it is geodesic and for every geodesic triangle in $X$ and all points $x,y$ of that triangle, $d(x,y)\le d_2(\bar{x},\bar{y})$. It is CAT(1) if every pair of points of $X$ at distance $<D_1:=\pi$ is joined by a geodesic segment in $X$, and every geodesic triangle in $X$ of perimeter $<2\pi$ satisfies $d(x,y)\le d_S(\bar{x},\bar{y})$ for all points $x,y$ of the triangle. Thus for CAT(1) only triangles of perimeter $<2\pi$ are tested, and geodesic segments are demanded only for pairs at distance $<\pi$; a triangle of perimeter $<2\pi$ has all sides $<\pi$, so its sides are available by hypothesis. A metric space is locally CAT(0), equivalently of curvature $\le 0$, if every point has a closed ball $\bar{B}(x,r)$, $r>0$, such that the induced metric on $\bar{B}(x,r)$ is CAT(0); locally CAT(1) is defined in the same way.

(4) Truncated angular metrics on links. Let $F$ be a face of an isometric polyhedral gluing with its chain metric (Abstract isometric polyhedral gluings and the chain metric) and let $L:={\mathrm{Lk}}_X(F)$ be its angular link with the auxiliary extended componentwise path metric $d_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}$ and the finite angular metric $d_{\pi }:=\min \{\pi ,d_{\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{h}}\}$ (The angular path metric, the Euclidean cone and spherical joins, Gram realisations, radial normalisation, finite spherical complexes and link Gram formulas). Then every $d_{\pi }$-triangle of perimeter $<2\pi$ has all sides $<\pi$: if one side were $\pi$, the triangle inequality would make the perimeter at least $2\pi$. Thus its vertices lie in one intrinsic component of $L$, and its side lengths equal the untruncated intrinsic path distances. It also holds between any two side points: the shorter of the two boundary routes has length at most half the perimeter, hence $<\pi$, so their truncated distance is $<\pi$ and equals their intrinsic path distance. Hence the CAT(1) tests in $(L,d_{\pi })$ of perimeter $<2\pi$ agree with componentwise intrinsic tests (The cone and join metrics and the local product chart of a polyhedral gluing). The empty metric space carries no triangles and satisfies the CAT(0) and CAT(1) tests vacuously; a one-point space is CAT(0) and CAT(1). With the conventions $C(\mathrm{\varnothing })=\{o\}$ and $L\ast \mathrm{\varnothing }=L$ (The angular path metric, the Euclidean cone and spherical joins), the empty link satisfies the CAT(1) tests vacuously and its cone is a point.

(5) Local geodesics. Let $I\subseteq \mathbb{R}$ be an interval. A map $c:I\to X$ is a constant-speed local geodesic if there is a fixed $\lambda \ge 0$ such that for every $t\in I$ some $\epsilon >0$ satisfies $d(c(t^{\mathrm{\prime }}),c(t^{\mathrm{\prime }\mathrm{\prime }}))=\lambda \mathrm{\mid }{t}^{\mathrm{\prime }}-t^{\mathrm{\prime }\mathrm{\prime }}\mathrm{\mid }$ whenever $t^{\mathrm{\prime }},t^{\mathrm{\prime }\mathrm{\prime }}\in (t-\epsilon ,t+\epsilon )\cap I$. Here $\lambda$ is its speed; $\lambda =1$ is the unit-speed convention and $\lambda =0$ gives the constant paths. In this chapter “local geodesic” includes these linear reparametrizations. It is a minimizing geodesic precisely when the same distance equality holds for every pair $t^{\mathrm{\prime }},t^{\mathrm{\prime }\mathrm{\prime }}\in I$.

(6) Length. A continuous path $\gamma :[a,b]\to X$ has length $L(\gamma )\in [0,\mathrm{\infty }]$, the supremum of its polygonal sums, and is rectifiable if $L(\gamma )<\mathrm{\infty }$ (Length in a metric target: lower semicontinuity and arc-length reparametrization, Upper bound, least upper bound, and strict upper bound). $X$ is a length space if for all $x,y\in X$ and every $\epsilon >0$ there is a path from $x$ to $y$ of length $<d(x,y)+\epsilon$.

(7) Round circles. For $\mathrm{\ell }>0$ let $S_{\mathrm{\ell }}^1:=\mathbb{R}\mathrm{/}\mathrm{\ell }\mathbb{Z}$ be the circle of circumference $\mathrm{\ell }$, with $d_{\mathrm{\ell }}(x,y):=\min \{\mathrm{\mid }x-y+k\mathrm{\ell }\mathrm{\mid }:k\in \mathbb{Z}\}$; for $\mathrm{\ell }=2\pi$ this is the unit circle. An isometrically embedded circle of length $\mathrm{\ell }$ in a metric space $X$ is an isometric embedding $S_{\mathrm{\ell }}^1\to X$ (Isometry, isometric embedding, and the subspace metric on a subset); its image is a subset of $X$ isometric to $(S_{\mathrm{\ell }}^1,d_{\mathrm{\ell }})$.

The definition asserts no property of the objects it names beyond the conventions recorded. The metric axioms for $d_S$ and $d_{\mathrm{\ell }}$, the existence and uniqueness of comparison triangles under the stated perimeter restrictions, the description of geodesic segments in the models as minimal great arcs and round arcs, and the facts that ${\mathbb{E}}^2$ and $S^2$ are CAT(0), respectively CAT(1), are all proved in the recorded justifier Comparison triangles in the Euclidean plane and the round sphere, model spaces, and CAT(0) and CAT(1) consequences ↗, which depends on this definition; The agreement of the tests is derived in (4), and the local product chart is established by The cone and join metrics and the local product chart of a polyhedral gluing, rather than by the comparison lemma.