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
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