In this chapter, we present, before resuming the general theory in all dimensions, a variety of results for oriented 2–dimensional Riemannian manifolds – surfaces (the phrase “Riemann surface” reserved for when the surface is orientable with constant curvature equal to –1). So, in all that follows,

Definition.A surface will be an oriented 2–dimensional Riemannian manifold.

We start with a topic motivated by the concluding one of Chapter IV. Namely, once one knows that in a nontrivial free homotopy class of a compact Riemannian manifold M there is a minimizing closed geodesic, one may ask for geometric estimates on its length, for example, to estimate its length against the volume of the manifold. Or, one may ask such a question for any homology class. Here, for surfaces, we estimate the length of the shortest homotopically nontrivial closed geodesic (among all homotopically nontrivial closed curves) against the area of M. This study was initiated by C. Loewner and P. Pu in the 1950s, almost completely dormant for 30 years, and resuscitated in the 1980s by M. Gromov. Here, we only introduce the subject.

Then, we turn (§V.2) to the celebrated Gauss–Bonnet theorem and formula, followed by (§V.3) B. Randol's collar theorem for compact Riemann surfaces, that is, surfaces of constant curvature –1. The result quite fundamental in the geometry of Riemann surfaces and in analysis on them and the proof is quite beautiful in its own right.

In §V.4, we begin discussion of one of the major themes of the rest of the book, the isoperimetric problem in Riemannian manifolds.