Comparison triangle in the two dimensional space form

Fix $k\in \mathbb{R}$ and let $M_k^2$ denote the complete, simply connected two-dimensional Riemannian manifold of constant sectional curvature $k$, the two-dimensional space form: the round sphere of radius $1\mathrm{/}\sqrt{k}$ when $k>0$, the Euclidean plane when $k=0$, and the hyperbolic plane of curvature $k$ when $k<0$ (Constant sectional curvature and space form). Write $d_k$ for its Riemannian distance.

A comparison triangle with side lengths $(a,b,c)$ is a triple of points $(\bar{x},\bar{y},\bar{z})$ of $M_k^2$ together with the three minimizing geodesic segments joining them, such that $d_k(\bar{y},\bar{z})=a,d_k(\bar{z},\bar{x})=b,d_k(\bar{x},\bar{y})=c\mathrm{.}$ The comparison angles $\bar{\alpha },\bar{\beta },\bar{\gamma }$ are the Riemannian angles at $\bar{x},\bar{y},\bar{z}$ between the two comparison sides meeting there (Pointwise norm and angle from a riemannian metric).

The side lengths are part of the data and are required to be positive: $a,b,c>0$, satisfying the strict triangle inequalities $a<b+c,b<c+a,c<a+b\mathrm{.}$ When $k>0$ the following two further restrictions are part of the definition: $a,b,c<\frac{\pi }{\sqrt{k}},a+b+c<\frac{2\pi }{\sqrt{k}}\mathrm{.}$ The first keeps every side shorter than a diameter of the sphere, so that no two comparison vertices are antipodal; the second rules out the degenerate configuration in which the three sides already exhaust two half-circles. No upper restriction is imposed when $k\le 0$, where the positive comparison sine ${\mathrm{sn}}_k$ has no positive zero (Comparison sine, cosine and cotangent functions).

The defining angle at $\bar{x}$ is the unique $\bar{\alpha }\in (0,\pi )$ determined by the model cosine law, namely $\cos \bar{\alpha }=\frac{\cos (\sqrt{k}a)-\cos (\sqrt{k}b)\cos (\sqrt{k}c)}{\sin (\sqrt{k}b)\sin (\sqrt{k}c)}$ for $k>0$, by $\cos \bar{\alpha }=\frac{b^2+c^2-a^2}{2bc}$ for $k=0$, and by $\cos \bar{\alpha }=\frac{\cosh (\sqrt{-k}b)\cosh (\sqrt{-k}c)-\cosh (\sqrt{-k}a)}{\sinh (\sqrt{-k}b)\sinh (\sqrt{-k}c)}$ for $k<0$; the two further angles are given by the same formulas with the roles of the sides cycled, and the three angles always sum to more than $\pi$, equal to $\pi$, or less than $\pi$ according as $k>0$, $k=0$ or $k<0$. The stated side hypotheses make the right-hand sides lie in $(-1,1)$, so the model angle exists and is unique in $(0,\pi )$. For $k>0$ the displayed identity is the spherical law of cosines for the angular side lengths $\sqrt{k}a,\sqrt{k}b,\sqrt{k}c\in (0,\pi )$, whose sum is less than $2\pi$.

Uniqueness is asserted up to isometry of $M_k^2$, including orientation-reversing isometries: if $(\bar{x},\bar{y},\bar{z})$ and $({\bar{x}}^{\mathrm{\prime }},{\bar{y}}^{\mathrm{\prime }},{\bar{z}}^{\mathrm{\prime }})$ are two comparison triangles with the same side lengths, then there is an isometry of $M_k^2$ carrying one labelled triple onto the other. Existence is part of this definition as a supplied model configuration; the comparison theorems that use it only invoke the displayed side lengths, the angle bounds $0<\bar{\alpha },\bar{\beta },\bar{\gamma }<\pi$ and this uniqueness. For $k>0$ a side length equal to $\pi \mathrm{/}\sqrt{k}$ or a perimeter equal to $2\pi \mathrm{/}\sqrt{k}$ gives a different, degenerate spherical configuration and is excluded here; the terminal model endpoint is likewise excluded from the domain of the comparison cotangent.