Although this is not directly about Riemannian geometry, his paper [Arc structure of Singularities](http://dx.doi.org/10.1215/S0012-7094-95-08103-4) (1995) certainly classifies as sufficiently geometrical. By all accounts, this paper was actually conceived some 30 years or more earlier. Ex post facto, the contents of that paper were way ahead of their time. 

In some twenty years, much has been written (and will be written) about the Nash problem for arc spaces of singular varieties.