Contemporary Mathematics

Volume 196, 1996

Preface for

FINSLER GEOMETRY OVER THE REALS

D.

Bao, S.S. Chern and Z. Sherr

1.

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

[Sp]

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

sections.

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

SM

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

(see

[Chl]

or

[Ch3])

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

1991

Mathematics Subject Classification.

53B40, 53C60.

©

1996 American Mathematical Society

3

http://dx.doi.org/10.1090/conm/196/02425