Welcome to the Finsler Geometry Newsletter.
The aim of the Newsletter is to promote the interaction between researchers in convex, integral, metric, and symplectic geometry by providing them with a quick, accessible medium for communicating ideas, announcements, examples, counter-examples, and remarks. At the time I started the Newletter wikis, blogs and forums were not so common as they are now and the primitive infrastructure I designed was never up to my expectations. In order to take advantage of the new technology, the Newsletter has moved to Wikidot.
The approach to Finsler geometry that we adhere to in these pages is a mixture of metric geometry , variational calculus, and Hamiltonian dynamics. René Thom once despairingly wrote that mathematicians prefer computing to understanding for it is less painful and more convincing. All the more reason for dedicating a website to geometric insight and understanding in the theory of Finsler spaces.
About Finsler Geometry
Finsler manifolds, manifolds whose tangent spaces carry a norm that varies smoothly with the base point, were born prematurely in 1854 together with the Riemannian counterparts in Riemann's ground-breaking Habilitationsvortrag. I say prematurely because in 1854 Minkowski's work on normed spaces and convex bodies was still forty three years away, and thus not even the infinitesimal geometry on which Finsler manifolds are based was understood at the time. Apparently, Riemann did not know what to make of these 'more general class' of manifolds whose element of arc-length does not originate from a scalar product and, fatefully, put in a bad word for them:
Investigation of this more general class would actually require no essential different principles, but it would be rather time-consuming and throw relatively little new light on the study of Space, especially since the results cannot be expressed geometrically.
Given the awe in which we rightfully regard Riemann's achievements and uncanny geometrical intuition, it is tempting to take the above quotation out of historical context and to dismiss Finsler geometry altogether. But, if we think of the great advances in convex geometry, the calculus of variations, integral geometry, the theory of metric spaces, and symplectic geometry that have taken place since 1854, then we may be moved to reassess Riemann's statement and to consider applying these new tools to develop the subject in a way that Riemann could not have foreseen.
J.C. Alvarez Paiva, Some problems on Finsler geometry Chapter I in “Handbook of Differential
Geometry”, Vol. 2, F. Dillen and L. Verstraelen (eds.), North-Holland, 2006, 1–33.
