Question

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?

Answer 1

Bowick and Rajeev computed the Ricci curvature of $Diff(S^1)/S^1$, which is an infinite dimensional Kahler manifold, in appendix A of their article: Nucl.Phys.B293:348,1987 (which can be found here). The Ricci curvature appears in the following setting. They consider the lift of the algebra of holomorphic vector fields on $Diff(S^1)/S^1$ to a line bundle which represents the bosonic string. This lift is not an isomorphism and the lifted action closes to a central extention (which is the celebrated Virasoro algebra). The Ricci curvature is the contribution to the central extention of the determinant line bundle (which is one component of the bosonic string's line bundle). It is interesting that for almost all Kahler metrics, the trace operation is finite and there is no need in a regularization of the sum.

Answer 2

The PhD thesis of Dan Freed contains the computation of the Ricci curvature of an infinite-dimensional Kähler manifold. I don't have a scanned copy of the thesis with me, but it's probably condensed in his 1988 paper in the Journal of Differential Geometry: The Geometry of Loop Groups.

Answer 3

There is a paper of Lott and Villani that defines the concept of nonnegative Ricci curvature in an arbitrary measured length space:

http://math.berkeley.edu/~lott/LottVillani.pdf

There is a dimension parameter which can be taken to infinity in that definition.

In his proof of the Poincare and geometrisation conjectures, Perelman used a heuristic argument in which he formally applied the Bishop-Gromov inequality to an infinite-dimensional, formally Ricci flat manifold to obtain the monotonicity of what is now known as the Perelman reduced volume; see Section 6 of

http://arxiv.org/abs/math.DG/0211159

or my notes on this at

http://terrytao.wordpress.com/2008/04/27/285g-lecture-9-comparison-geometry-the-high-dimensional-limit-and-perelman-reduced-volume/

EDIT: Some formal computations of Ricci and Riemann curvature on the (infinite dimensional) space of volume-preserving diffeomorphisms also appear in a famous paper of Arnold:

http://www.ams.org/mathscinet-getitem?mr=202082