Preface

This volume is a collection of papers on the subject of manifolds. Manifolds are an impor‐ tant and crucial structure in modern mathematics. They have been intensively investigated over the last 60 years and provide a foundation over which much of modern differential geometry has developed. The contributions represented here investigate manifolds of par‐ ticular types, such as symplectic manifolds and submanifolds. They discuss what can be learned about manifolds by defining and studying various structures on them. In particular, there is a paper covering operator actions on manifolds and their spectral properties. There is a paper on Bonnet surfaces which emphasizes the important role played by differential equations in the study of manifolds and surfaces in particular. A paper on bifurcations and manifolds and two papers on the application of manifolds to some areas of applied mathe‐ matics are presented as well. Finally, there is a paper on symplectic affine actions on mani‐ folds.

This book has been put together by an international group of invited authors, and it is a pleasure to thank them for their hard work and significant contributions to this volume. I gratefully acknowledge with great thanks the assistance and help provided by Ms. Iva Lip‐ ovic who was the publishing manager throughout this process as well as the InTech pub‐ lishing group for the opportunity to edit this volume on the subject of manifolds.

> **Paul Bracken** Department of Mathematics, University of Texas, Edinburg, TX, USA

**Section 1**

**Hopf Bifurcations**

Provisional chapter
