Resumen: In 2006 Lott, Villani and Sturm defined the notion of synthetic Ricci curvature bound on a metric measure space. This definition is formulated in terms of the convexity of an entropy functional along geodesics in the space of probability measures and is known as the Curvature-Dimension condition (CD(K,N)). It is known that in the smooth case this condition is equivalent to having a lower bound on the Ricci curvature.
Later Gigli, Mondino and Savaré made several refinements, particularly in the structure of associated Sobolev spaces, in order to avoid pathological behaviour such as excessive branching of geodesics and Finsler geometries. Their condition is called Riemannian Curvature-Dimension condition (RCD(K,N)).
Isometric actions on Riemannian manifolds have been a useful tool to investigate the interaction between the topology and the Riemannian metric a manifold might admit. A major result in this area is the theorem of Myers-Steenrod stating that the isometry group of a Riemannian manifold is a Lie group. In this talk I will look at the isometry group of an RCD(K,N) space, prove that it is a Lie group and, if time permits, I will discuss what can be done to ensure that a compact Lie group acts by measure preserving isometries.