Banach Manifolds of Fiber Bundle Sections

In the past several years significant progress has been made in our understanding of infinite dimensional manifolds. Research in this area has split into two quite separate branches ; first a study of the theory of abstract Banach manifolds, and secondly a detailed study of the properties of certain classes of concrete manifolds that arise as spaces of differentiable maps or more generally as spaces of sections of fiber bundles. A survey of the remarkable progress made in the first mentioned area, (i.e. infinite dimensional differential topology) will be found in the reports to this Congress by N. Kuiper and R. Anderson. Here I would like to survey a part of the recent work in the second area, which for obvious reasons (made explicit in § 1 of 112]) has come to be called non-linear global analysis. 1 shall also attempt to indicate what in my opinion are fruitful directions of current research and hazard a few guesses for the near future. An attempt to be comprehensive would be futile since the subject shades off imperceptibly into many extremely active classical fields of mathematics for which in fact it plays the role of "foundations" (e.g. non-linear partial differential equations and continuum mechanics). I shall therefore concentrate on those few topics which have most engaged my personal interest, particularly the intrinsic structures of manifolds of sections and applications to the calculus of variations. I shall also not attempt to cover research prior to 1966 which is surveyed in the excellent and comprehensive review article [4] of James Eells Jr.