Intrinsic Geometric Structure of Subcartesian Spaces

Abstract

Every subset S of a Cartesian space Rd, endowed with differential structure C∞(S) generated by restrictions to S of functions in C∞(Rd), has a canonical partition M(S) by manifolds, which are orbits of the family X(S) of all derivations of C∞(S) that generate local one-parameter groups of local diffeomorphisms of S. This partition satisfies the frontier condition, Whitney’s conditions A and B. If M(S) is locally finite, then it satisfies all definitions of stratification of S. This result extends to Hausdorff locally Euclidean differential spaces. The partition M(S) of a subcartesian space S by smooth manifolds provides a measure for the applicability of differential geometric methods to the study of the geometry of S. If all manifolds in M(S) are single points, we cannot expect differential geometry to be an effective tool in the study of S. On the other extreme, if M(S) contains only one manifold M, then the subcartesian space S is a manifold, S=M, and it is a natural domain for differential geometric techniques.