A note on the Bonnet-Myers theorem (Q1915215)

Let \((M,g)\) be a complete Riemannian manifold of dimension \(n\). It is open to study under which conditions this Riemannian manifold is compact. There are some results in this direction. The aim of this article is to contribute the following result to this problem. A complete AP-Riemannian manifold is compact, and its diameter \(d(M)\) is bounded by \[ d(M)\leq e_{k-1} \Bigl(L_{k-1} \Bigl(\exp {\pi\over\nu} \max\bigl\{r_0,e_k (0)\bigr\} \Bigr)\Bigr) \] where \(e_0(x) =x\) and \(e_{m+1} (x)=\exp e_m (x)\) for \(m>0\).