Positively Curved Combinatorial 3-Manifolds

We present two theorems in the “discrete differential geometry” of positively curved spaces. The first is a combinatorial analog of the Bonnet-Myers theorem: • A combinatorial 3-manifold whose edges have degree at most five has edgediameter 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 [12] and Cheng [2]: • A positively curved 3-manifold, as above, in which vertices v and w have edgedistance 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.