Compactifying the fibers of jet spaces, Nash blow-up, and Contact Geometry

The space $J^1 (M, R)$ of 1-jets of real-valued functions on an (n-1)- manifold M provides one of the standard examples of a contact manifold, its dimension being $2n +1$. The space $J^k = J^k (M, R)$ of k-jets of functions also has a canonical distribution (subbundle of the tangent bundle) and the k-jet of any smooth function on $M$ is an integral submanifold for this distribution. The contact group, which contains the diffeomorphism group of $M \times R$ acts on $J^k$ in such a way as to preserve this distribution. And this action is transitive. The fibers of $J^k \to M$ are not compact, rather they are affine spaces. We show how the Cartan prolongation procedure provides for a minimal compactification of these fibers in such a way that the contact group still acts on the whole space by symmetries. However the action is no longer transitive when $k > 2$, and the fibers are no longer smooth spaces when $k > 3$, and $n>2$. The central problem is to classify the orbits of this action. The key to understanding the orbits is to realize points of the compactified space as the iterated prolongations of singular hypersurface in $M \times R$. The process of prolonging hypersurfaces is identical to the algebraic geometer's `Nash blow-up'. This process leads to a complete resolution of the classification problem when $n=2$ -- the case of curves in the plane, and is mostly wide open when $n = 3$ -- the case of surfaces in space.