Some applications of the cosine law in low-dimensional geometry I, II
by
Feng Luo
Rutgers University

In the discrete approach to smooth metrics on surfaces, the basic building blocks are sometimes taken to be triangles in constant curvature spaces. In this setting edge lengths and inner angles of triangles correspond to the metrics and its curvatures. For triangles in hyperbolic, spherical and Euclidean geometries, edge lengths and inner angles are related by the cosine law. Thus cosine law should be considered as the metric-curvature relation. From this point of view, the derivative of the cosine law is an analogy of the Bianchi identity in Riemannian geometry. The derivative cosine law provides unification of many known approaches of constructing constant curvature metrics on surfaces. These include the work of Colin de Vedierer, Greg Leibon, Bragger, Chow and myself on variational approachs and discrete curvature flows on triangulated surfaces.

In the first talk which is aimed for general audience, we will give a survey of these works viewed from the derivative cosine law. In the second talk, we will provide some technical details of the singularity analysis in the case of discrete Ricci flow on surfaces and Teichmuller space of surface with boundary. If time permits, we will also mention briefly a variational approach to find constant curvature metrics for triangulated 3-manifolds.

Date received: February 13, 2006