Contemporary Mathematics
Volume 196, 1996
Preface for
Bao, S.S. Chern and Z. Sherr
Historical remarks
Finsler's 1918 thesis treats curves and surfaces in a general metric space, in
the spirit of classical differential geometry. The geometry of the space itself was
formulated earlier by Riemann in his great address of 1854 (see
for an English
translation). Riemann saw the difference between the general case and what is
now known as Riemannian manifolds, and made the remark: "The study of the
metric which is the fourth root of a quartic differential form is quite time-consuming
(Zeitraubend) and does not throw new light to the problem." Fortunately, in more
than a century of mathematical developments, manifold theory has acquired the
notions and tools which could give an adequate and efficient treatment of Finsler
geometry to satisfy Riemann. A quick summary of such is given in the next two
Finsler geometry is closely related to the calculus of variations. There are
two fundamental problems: the Plateau problem and the geometry of the simple
integral. In his famous address at the 1900 International Congress in Paris, Hilbert
devoted the last problem, namely Problem 23, to the subject. His main concern
was the geometry of an integral, to which he included both simple and double
integrals. He gave an extensive discussion of his invariant integral. In today's
terminology, this is a particular linear differential form which is globally defined
on the slit tangent bundle T M \ { 0}. Being constant along the rays (through the
origin) in each tangent space, this "Hilbert form" also lives on the sphere bundle
and defines a contact structure there (see also §9). The implication of such
data is enormous. For example, a connection of considerable utility was derived
by using exterior differentiation on the Hilbert form.
2. A modern setting
Let us review a setup which is appropriate for formulating local Finsler ge-
ometry. Let M be an n-dimensional manifold, and T M (resp. T* M) its tangent
Mathematics Subject Classification.
53B40, 53C60.
1996 American Mathematical Society
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