# riemannian geometry – What is the minimal length of a “Diagonal” in a Torus?

Given a Riemannian torus $$(T,d)$$ with fundamental group $$pi_1(T)=langle a,b mid ab=ba rangle$$. Denote for any $$gamma in pi_1(T)$$ the infimum length of all representatives of $$gamma$$ by $$L(gamma)$$.

Is there a way to get a general upper bound on $$L(ab)$$ in terms of $$L(a),L(b)$$ and the volume of $$T$$?

An obvious one is $$L(ab) leq L(a) + L(b)$$, but i guess, there are better results.

This is inspired by Loewner’s torus inequality.