See the wiki page <https://en.wikipedia.org/wiki/Toponogov%27s_theorem>


One consequence of the Toponogov comparison Theorem is that if the sectional curvature of a manifold $M$ is pinched below by a number $\delta$. Let $pqr$ be a geodesic triangle, i.e. a triangle whose sides are geodesics, in $M$, such that the geodesic $pq$ is minimal. Let $p′q′r′$ be a geodesic triangle in the model space $M\_\delta$, i.e. the simply connected space of constant curvature $\delta$, such that the length of sides $p′q′$ and $p′r′$is equal to that of $pq$ and $pr$ respectively and the angle at $p′$ is equal to that at $p$. 


Then $d(q,r) \le d(q',r').$


The wiki then claim that "When the sectional curvature is bounded from above, a corollary to the Rauch comparison theorem yields an analogous statement, but with the reverse inequality."


Is this true? If so, can someone please provide a reference to this? It seems a bit weird to me since the statement of Rauch Theorem is symmetric, if this reverse statement is a corollary of Rauch Theorem, why isn't Toponogov Theorem also a corollary?


Any comments and references are welcome.



Reverse Toponogov triangle comparison
Rauch's comparison local result, while Toponogov's comparison is global.


For the upper curvature bound an analog of Toponogov's comparison holds only locally and indeed it follows from Rauch's comparison. There is a gloabal version for upper comparison --- the so-called (generalized) Hadamard--Cartan theorem [9.65+9.67 in [our book](https://arxiv.org/abs/1903.08539v5)]: For the curvature bound $\kappa\le 0$ it has an addition assumption that space is simply connected. If $\kappa=1$ then one has to assume that any closed curve shorter than $2\cdot\pi$ can be contracted in the class of closed curves shorter than $2\cdot\pi$. (The case $\kappa>0$ can be reduced to $\kappa=1$ by rescaling.)


