Note on the History of Immersion Theory

Abstract

In his survey of Smale’s work in Differential Topology, Moe Hirsch observed that there seemed to be no significant results in immersion theory published in the period between the work of H. Whitney in the forties and Smale’s classification of immersions of spheres (1958, 1959). This observation overlooks the C ¹-isometric immersion results of J. Nash (1954) and N. Kuiper (1955). At the time, the Nash isometric immersion theory was viewed as a separate result in Riemannian geometry; it had no relevance to Smale’s immersion theory or to the flowering of immersion-theoretic topology that took place in the sixties following Smale’s ground-breaking results. However, the Nash theory had an unexpected role to play as a precursor to M. Gromov’s remarkable refinement of immersion theory, known as Convex Integration Theory (1973). The history of immersion theory is rich and complex. It is perhaps appropriate on the occasion of the Smalefest, at which there were interesting informal discussions on the history of Differential Topology, to tie together some historical loose ends on the relation of the Nash C ¹-isometric immersion theory to these much later developments in immersion-theoretic topology due to M. Gromov.