It is well known that in Riemannian geometry, local information can lead to global phenomena. On the other hand, in the category of contact manifolds, we can naturally focus on the so called compatible Riemannian structures. However, it is very little known about how to use compatible global geometry to achieve contact topological information. After reviewing some known results, we will discuss the problem of Ricci curvature realization for Reeb vector fields associated to a contact 3-manifold. These vector fields have significantly helped understanding contact topology since the early 90s. We will use topological tools, namely open book decompositions, to show that any function can be realized as such Ricci curvature for a singular metric which is an honest compatible metric away from a measure zero codimension one set. However, we will see that resolving such singularities depends on contact topological data and is yet to be fully understood.

Zoom link: https://rochester.zoom.us/j/94332745046