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.
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
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
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