Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Positively Curved Combinatorial 3-Manifolds. | Positively Curved Combinatorial 3-Manifolds Aaron Trout Department of Mathematics Chatham University Pittsburgh PA USA atrout@ Submitted Dec 5 2006 Accepted Mar 18 2010 Published Mar 29 2010 Mathematics Subject Classifications 52C99 53A99 57M99 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 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. 0 Introduction The relationship between the curvature of a Riemannian or semi-Riemannian space and its topology is of central interest to differential geometers topologists and physicists. The classical results in this area are numerous beautiful and have inspired an enormous amount of subsequent research. One currently active branch of this venerable tree seeks combinatorial analogs to these classical theorems and concepts. Recent work along these THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R49 1 lines can be found in 1 3 4 6 7 10 and 11 . Here we present combinatorial versions of the Bonnet-Myers theorem and the associated maximum-diameter sphere theorems of Toponogov .
