Positively Curved Combinatorial 3-Manifolds

Abstract

We present two theorems in the "discrete differential geometry" of positively curved spaces. The first is a combinatorial analog of the Bonnet-Myers theorem:

$\bullet$ A combinatorial 3-manifold whose edges have degree at most five has edge-diameter at most five.

When all edges have unit length, this degree bound is equivalent to an angle-deficit along each edge. It is for this reason we call such spaces positively curved.

Our second main result is analogous to the sphere theorems of Toponogov and Cheng:

$\bullet$ A positively curved 3-manifold, as above, in which vertices $v$ and $w$ have edge-distance five is a sphere whose triangulation is completely determined by the structure of $Lk(v)$ or $Lk(w)$.

In fact, we provide a procedure for constructing a maximum diameter sphere from a suitable $Lk(v)$ or $Lk(w)$.

The compactness of these spaces (without an explicit diameter bound) was first proved via analytic arguments in a 1973 paper by David Stone. Our proof is completely combinatorial, provides sharp bounds, and follows closely the proof strategy for the classical results.