Abstract:

Homogeneous spaces are at the heart of Riemannian geometry, providing a vast "playground" of examples, however this class of spaces is vast and a complete classification is probably out of reach. For this reason many subclasses have been studied, most famously the Riemannian symmetric spaces which were classified a century ago by Elie Cartan (subject to some additional assumptions). Weakening the symmetric space conditions slightly gives the notion of reductive homogeneous spaces, which is again too broad to classify. The goal of this talk is to mention the "in-between" class of geodesic orbit ("G.O.") spaces, and summarize what is known about them.