Not self-intersecting loops on a connected closed orientable smooth surface $S$ must have a minimal length not to disconnect it, e.g. the equators of a torus. "Not to disconnect" is - on such surfaces - the same as "not to be contractible", isn't it? So my question is:
Given a metric on a surface $S$ as above, (how) can the minimal length $l$ of a non-contractible loop be calculated? Or at least estimated?
With which other geometrical properties of the surface (say its diameter) might $l$ be related?
The minimal length of a non-contractible loop on a surface is known as the systole. See the Wikipedia articles Systoles of surfaces, Introduction to systolic geometry, and Systolic geometry. (Apparently, the topic is thoroughly covered in Wikipedia thanks to the efforts of Mikhail Katz). To the references listed there I would add Metaphors in systolic geometry by Larry Guth, available on his homepage in two versions.
The systole is usually estimated in terms of the square root of the area ($n$th root of the volume, for $n$-dimensional manifolds). But the diameter can also be brought into play, as in the paper On the diameters of compact Riemann surfaces by Fwu Ranq Chang.
Note that some sources talk about shortest noncontractible loop, others about shortest closed geodesic. This is the same thing, because the shortest noncontractible loop is necessarily a geodesic.