Toponogov triangle comparison

Setup and comparison data. [F1, given] Let ${\gamma }_{xy},{\gamma }_{xz},{\gamma }_{yz}$ be the minimizing geodesic segments of the triangle, so that $c=d_g(x,y)$, $b=d_g(z,x)$ and $a=d_g(y,z)$ are the lengths of the three sides, all positive. The angles $\alpha ,\beta ,\gamma \in [0,\pi ]$ are defined at $x,y,z$ by the metric, and for $k>0$ the side lengths satisfy $a,b,c<\pi \mathrm{/}\sqrt{k}$ and $a+b+c<2\pi \mathrm{/}\sqrt{k}$ by the admissibility hypothesis. By [F1] a comparison triangle with side lengths $(a,b,c)$ exists in $M_k^2$, is unique up to isometry, and has angles $\bar{\alpha },\bar{\beta },\bar{\gamma }\in (0,\pi )$ given by the model cosine law: $\bar{\alpha }={\mathrm{\Phi }}_{b,c}(a),\bar{\beta }={\mathrm{\Phi }}_{c,a}(b),\bar{\gamma }={\mathrm{\Phi }}_{a,b}(c)\mathrm{.}$ The strict triangle inequalities give $\mathrm{\mid }b-c\mathrm{\mid }<a<b+c$ and its cyclic permutations, and for $k>0$ the perimeter bound gives $a<2\pi \mathrm{/}\sqrt{k}-b-c$; with the notation $m$ of [F2] this says that the opposite side lies in the admissible interval of its two legs, for instance $a\in (\mathrm{\mid }b-c\mathrm{\mid },m(b,c))\mathrm{.}$

The angle at $x$. [F1, F2, step 1.1, given] The sides from $x$ to $y$ and from $x$ to $z$ are unit-speed minimizing geodesics of lengths $c$ and $b$ with included angle $\alpha$, and the distance between their endpoints is $a$. Suppose first that $\alpha =0$. Then the hinge comparison [F2] applies with legs of lengths $b,c$, included angle $0$ and opposite side $a$, giving $a\le c_k(b,c,0)=\mathrm{\mid }b-c\mathrm{\mid }$ by the endpoint value of [F2], contradicting the strict triangle inequality $\mathrm{\mid }b-c\mathrm{\mid }<a$ from step 1.1. Hence $\alpha \in (0,\pi ]$, and applying [F2] at this angle gives $a\le c_k(b,c,\alpha )\mathrm{.}$ On the other hand step 1.1 gives $\bar{\alpha }={\mathrm{\Phi }}_{b,c}(a)$ with $a\in (\mathrm{\mid }b-c\mathrm{\mid },m(b,c))$, so the inverse identity of [F2] gives $c_k(b,c,\bar{\alpha })=c_k(b,c,{\mathrm{\Phi }}_{b,c}(a))=a\mathrm{.}$ Combining the two displays, $c_k(b,c,\bar{\alpha })\le c_k(b,c,\alpha )$; since $\bar{\alpha }\in (0,\pi )$, $\alpha \in (0,\pi ]$ and $c_k(b,c,\cdot )$ is strictly increasing on $[0,\pi ]$ by [F2], it follows that $\bar{\alpha }\le \alpha$.

The angle at $y$. [F1, F2, step 1.1, given] The same argument at the vertex $y$ applies [F2] to the minimizing sides from $y$ to $z$ and from $y$ to $x$, of lengths $a$ and $c$, whose included angle is $\beta$ and whose endpoint distance is $b$. If $\beta =0$, the endpoint value gives $b\le \mathrm{\mid }a-c\mathrm{\mid }$, contradicting $\mathrm{\mid }a-c\mathrm{\mid }<b$ from step 1.1; hence $\beta \in (0,\pi ]$, the hinge gives $b\le c_k(a,c,\beta )$, and $b=c_k(a,c,\bar{\beta })$ by step 1.1 and the inverse identity of [F2]. Strict increase of $c_k(a,c,\cdot )$ on $[0,\pi ]$ gives $\bar{\beta }\le \beta$.

The angle at $z$. [F1, F2, step 1.1, given] Likewise, at $z$ the minimizing sides to $x$ and $y$ have lengths $b$ and $a$, included angle $\gamma$ and endpoint distance $c$; if $\gamma =0$ then $c\le \mathrm{\mid }a-b\mathrm{\mid }$, contradicting $\mathrm{\mid }a-b\mathrm{\mid }<c$ from step 1.1, so $\gamma \in (0,\pi ]$; the hinge comparison gives $c\le c_k(a,b,\gamma )$, while $c=c_k(a,b,\bar{\gamma })$ by step 1.1 and [F2], and strict increase of $c_k(a,b,\cdot )$ on $[0,\pi ]$ gives $\bar{\gamma }\le \gamma$.

Conclusion and boundary cases. [F1, F2, step 1.1, step 2.1, step 2.2, step 2.3, given] Steps 2.1, 2.2 and 2.3 prove $\bar{\alpha }\le \alpha$, $\bar{\beta }\le \beta$ and $\bar{\gamma }\le \gamma$: every actual vertex angle is at least its comparison angle, so fixed-side triangles of a manifold with $K\ge k$ are at least as fat as their constant-$k$ model triangles. The boundary cases are accounted for: a zero side length is excluded by the positivity requirement of the comparison-triangle definition, so the three vertices are distinct and the angles are defined; a zero vertex angle is impossible, as shown at each vertex via the endpoint value $c_k(\cdot ,\cdot ,0)=\mathrm{\mid }A-B\mathrm{\mid }$ and the strict triangle inequalities; an angle equal to $\pi$ is allowed and is covered, because the hinge comparison is stated for $\theta \in [0,\pi ]$ and $c_k(\cdot ,\cdot ,\cdot )$ is strictly increasing on the closed interval $[0,\pi ]$, with $\bar{\alpha },\bar{\beta },\bar{\gamma }\in (0,\pi )$; and for $k>0$ the endpoint configurations $a,b,c=\pi \mathrm{/}\sqrt{k}$ and $a+b+c=2\pi \mathrm{/}\sqrt{k}$, where the comparison triangle would degenerate, are excluded by the admissibility hypothesis. The argument uses no choice of comparison triangle: its angles are determined by the side lengths through the model cosine law of [F1], and the model side function $c_k(A,B,\theta )$ is independent of the choices in its construction by [F2]; the only choice principle used is the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ of [A1], carried by the hinge comparison and the comparison-triangle interface.