Geometry in a Fréchet Context A Projective Limit Approach London Mathematical Society Lecture Note Series

Many geometrical features of manifolds and fibre bundles modelled on Fréchet spaces either cannot be defined or are difficult to handle directly. This is due to the inherent deficiencies of Fréchet spaces; for example, the lack of a general solvability theory for differential equations, the non-existence of a reasonable Lie group structure on the general linear group of a Fréchet space, and the non-existence of an exponential map in a Fréchet?Lie group. In this book, the authors describe in detail a new approach that overcomes many of these limitations by using projective limits of geometrical objects modelled on Banach spaces. It will appeal to researchers and graduate students from a variety of backgrounds with an interest in infinite-dimensional geometry. The book concludes with an appendix outlining potential applications and motivating future research.

Preface; 1. Banach manifolds and bundles; 2. Fréchet spaces; 3. Fréchet manifolds; 4. Projective systems of principal bundles; 5. Projective systems of vector bundles; 6. Examples of projective systems of bundles; 7. Connections on plb-vector bundles; 8. Geometry of second order tangent bundles; Appendix. Further study.

C. T. J. Dodson is Emeritus Professor of Mathematics at the University of Manchester.
George Galanis is Associate Professor in the Section of Mathematics at the Hellenic Naval Academy in Piraeus, Greece.
Efstathios Vassiliou is a former Associate Professor in the Department of Mathematics at the University of Athens. Since his retirement he has been a staff member in the postgraduate program on Didactics and Methodology of Mathematics in the same department.