Is there a good notion of "Ricci curvature" in infinite dimensions?
My intuitive understanding of Ricci curvature is that it is some kind of an "average" of the curvature tensor over "different directions." In finite dimensions, taking the sum and taking the average makes little difference...
But in infinite dimensions, naively computing the curvature and taking the trace doesn't seem like it'll give anything convergent.
Is there some analogous notion of "average" curvature for Riemannian (Banach or Hilbert) Manifolds or some smart way of taking the "trace" of the "curvature tensor" so that it could still be called Ricci curvature and it'll be meaningful geometrically?