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/

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