The diversity of El Ni\~no events is commonly described by two distinct flavors, the Eastern Pacific (EP) and Central Pacific (CP) types. While the remote impacts, i.e. teleconnections, of EP and CP events have been studied for different regions individually, a global picture of their teleconnection patterns is still lacking. Here, we use Forman-Ricci curvature applied on climate networks constructed from 2-meter air temperature data to distinguish regional links from teleconnections. Our results confirm that teleconnection patterns are strongly influenced by the El Ni\~no type. EP events have primarily tropical teleconnections whereas CP events involve tropical-extratropical connections, particularly in the Pacific. Moreover, the central Pacific region does not have many teleconnections, even during CP events. It is mainly the eastern Pacific that mediates the remote influences for both El Ni\~no types.