sectional curvature with distance between two points

Question

I am wondering if the following statement is true or not:

Let $ M $ be a complete Riemannian manifold with sectional curvature $ K $. Let $ \tilde{M} $ be the simply connected complete Riemannian manifold of constant curvature $ \kappa $ with $ K\geq \kappa $ and $ \dim M = \dim \tilde{M} = n $. Let $ p,q \in M $ and $ \gamma $ a shortest geodesic from $ p $ to $ q $ with $ L(\gamma) = r $ i.e. dist$ (p,q) = r $. Does it hold that for arbitrary $\tilde{p} \in \tilde{M} $, there exists $\tilde{q} \in \tilde{M} $ with dist$ ( \tilde{p} , \tilde{q} ) \geq r$? thanks a lot

Answer 1

It seems to me that the statement is true. Indeed, if $\kappa \leq 0$ then for any $r > 0$ and any point $\tilde{p} \in \tilde{M}$, where $\tilde{M}$ is the simply connected and complete space form $\tilde{M}$ of constant sectional curvature $\kappa$, there is $\tilde{q} \in \tilde{M}$ such that $dist(\tilde{p},\tilde{q}) \geq r$. So we can assume $\kappa > 0$. In this case the diameter of $M$ is bounded by $\pi K^{-1/2}$ as follows from the classica Bonnet-Myers Theorem. On the other hand the space form of constant sectional curvatue $\kappa$ has diameter $\pi \kappa^{-1/2}$. So $$r \leq \pi K^{-1/2} \leq \pi \kappa^{-1/2} = dist(\tilde{p},\tilde{q}) \, .$$ Since space forms are homogeneous spaces the point $\tilde{p}$ is arbitrary. This answer positively the OP question.