Foreword Ever since Riemann's 1854 address and Finsler's 1918 thesis, the subject of Riemann-Finsler geometry (Finsler geometry for short) has undergone slow but steady development. However, its metamorphosis has not kept pace with that of one of its special cases, namely Riemannian geometry. Furthermore, the successes of the latter have been so spectacular that they tend to overshadow the moderate progress of the parent subject. Happily, recent works indicate that this gap is narrowing. It is now clear that in the appropriate framework, a good number of results generalize readily from the Riemannian to the Finsler setting. In fact, there is evidence to suggest that the 'quadratic restriction' is seldom necessary. Finsler geometry also plays a role in some other areas of mathematics, as well as in biology, control theory, engineering, and physics. In such sectors, it is quite often naturally the requisite tool. Given all these considerations, it was deemed prudent to have a cross section of Finsler geometers assess the current status of the subject. It was also high time to have a first event of this nature in the United States. The AMS-IMS-SIAM Joint Summer Research Conferences in the Mathematical Sciences provided just the right platform for such an event. Ours was supported by a grant from the NSF to the AMS. We wish to express our gratitude to these organizations. Our modest budget put severe constraints on how many speakers we can afford to invite. Also, among those who had initially agreed to attend, some had to change their plans at (or close to) the last minute due to unforeseen personal matters. These two factors account for the conspicuous absence of many distinguished practitioners, and for the crudeness of our 'cross section'. This proceedings volume constitutes a report of such a gathering. It consists of six separate sections, three pure and three applied. We have arranged to have a fairly detailed preface specially written for each of the six sections. Also, each paper terminates with a short list of open problems. Hopefully these features will render the present volume useful to a wider audience than usual. Finally, we would like to acknowledge the help of Wayne Drady during the conference, Brad Lackey for debugging some source files, Sandra Shaw for designing the section title pages, Donna Harmon, Maria Marquis, Anne Newcomb, Deborah Smith and the AMS Production Staff for their patient guidance. D. Bao, S.S. Chern, and Z. Shen July 1995, Seattle vii
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